From 1740b58a28c0daa6fb30285493aade6c8ff9f6c4 Mon Sep 17 00:00:00 2001 From: Yevhen Hrubiian Date: Tue, 8 Jul 2025 18:15:59 +0300 Subject: [PATCH 01/25] started work on bp lecture --- lectures/1-8-commitments.tex | 19 +- lectures/2-9-bulletproofs.tex | 434 ++++++++++++++++++++++++++++++++++ 2 files changed, 451 insertions(+), 2 deletions(-) create mode 100644 lectures/2-9-bulletproofs.tex diff --git a/lectures/1-8-commitments.tex b/lectures/1-8-commitments.tex index 93bc5c1..39356a6 100644 --- a/lectures/1-8-commitments.tex +++ b/lectures/1-8-commitments.tex @@ -212,9 +212,24 @@ \subsection{Vector commitments} \subsection{Polynomial commitment} Polynomial commitment can be used to prove that the commited polynomial satisfies certain properties $P(x_1, x_2, \ldots, x_n) = y$, without revealing what the polynomial is. -The commitment is generally succint, which means that it is much smaller than the polynomial it represents. +The commitment is generally succint, which means that it is much smaller than the polynomial it represents. Typically, polynomial commiment scheme is a heart of any complex \textit{zero-knowledge} proof system. -\textbf{The KZG polynomial commitment scheme} +\subsubsection{Pedersen polynomial commitment scheme} + +Let prover $\mathcal{P}$ wants to convince verifier $\mathcal{V}$ that he knows evaluation of polynomial $p(x) = a_d x^{d} + \dots + a_1 x + a_0 \in \mathbb{F}_{q}[x]$ at some $u \in \mathbb{F}_q$: $p(u) = y$ without revealing anything about $p(x)$. Let $\mathbb{G}$ - cyclic additive group of order $q$ where \textit{discrete logarithm} problem is hard. $G, H \in \mathbb{G}$ - points with unknown discrete logarithms between them. + +\textbf{Simple polynomial commitment scheme} +\begin{enumerate} + \item \textit{Commit to polynomial}. Prover selects random $\gamma_i \xleftarrow{R} \mathbb{F}_q$ for all $i \in [d]$, computes Pedersen commitments for each coefficent: $\forall i \in [d]: C_i = [a_i]G + [\gamma_i]H$ and sends them to Verifier. + \item \textit{Challenge}. Verifier samples random $u \xleftarrow{R} \mathbb{F}_q$ and sends it to Prover + \item \textit{Proof of evaluation}. Prover evaluates blinded polynomial $h(x) = \gamma_d x^{d} + \dots + \gamma_1 x + \gamma_0$ at $u$: $\pi = h(u)$, original polynomial at $u$: $y = p(u)$ and sends pair $(y, \pi)$ to Verifier. + \item \textit{Verify the proof}. Verifier performs check $$[u^d]C_d + \dots + [u]C_1 + C_0 \stackrel{?}{=} [y]G + [\pi]H$$ +\end{enumerate} + +Due to Schwartz-Zippel lemma, the probability that Prover can cheat and convince Vertifier that he knows valid evaluation to not-commited polynomial is at most $\frac{1}{q}$, hence the scheme is sound. +However, the scheme is not succinct, as the size of the commitment grows linearly with the degree of the polynomial. + +\subsubsection{The KZG polynomial commitment scheme} The KZG (Kate-Zaverucha-Goldberg) is a polynomial commitment scheme: diff --git a/lectures/2-9-bulletproofs.tex b/lectures/2-9-bulletproofs.tex new file mode 100644 index 0000000..4ca2dfa --- /dev/null +++ b/lectures/2-9-bulletproofs.tex @@ -0,0 +1,434 @@ +\documentclass[../lecture-notes-148x210.tex]{subfiles} + +\begin{document} +\subsection{Introduction} + +\textbf{Bulletproofs} is a non-interactive zero-knowledge proof protocol with logarithmically sized proofs without a trusted setup. Originally, \textbf{bulletproofs} was developed to provide efficient inner-product proofs for range proofs in application to confidential transactions, but it applies to arbitrary arithmetic circuit (possibly encoded in R1CS). Technically, the protocol is built in an interactive fashion (like $\Sigma$-protocols) and made non-interactive with a Fiat-Shamir transform. One key feature that differs it from $\Sigma$-protocols is the number of challenges from a verifier $\mathcal{V}$: in $\Sigma$-protocols there is only one challenge, while \textbf{bulletproofs} implies a logarithmic in circuit size number of queries. The main advantages of \textbf{bulletproofs} are an absence of a trusted setup, security that relies on the \textit{discrete-logarithm} assumption without any other auxiliary structures like bilinear pairings, however, the main disadvantage of \textbf{bulletproofs} is linear in circuit size verification time though still efiicient for small circuits. + +Hence, we will describe some preliminaries. + +\subsection{Proving non-linear relations with $\Sigma$-protocols} + +\subsection{Inner-product argument} + +Firstly, we describe the protocol for the \textbf{inner-product argument} - core component of the \textbf{bulletproofs} protocol. After that we will apply it to range proofs and arithmetic circuits. We have already seen that inner-products are the main ingridients for R1CS language because any R1CS relation could be seen as a batch of inner-products though it's not the most efficient representation and we'll see how to amortize all the constraints into inner-products more efficiently. + +Let $\mathbb{G}$ - cyclic group of prime order $p$ written additively, $\mathbf{G} = (G_1, \dots, G_n), \mathbf{H} = (H_1, \dots, H_n) \in \mathbb{G}^n$ - vectors of independent generators. We denote by $\langle \mathbf{a,b} \rangle$ - inner product of vectors $\mathbf{a} = (a_1, \dots, a_n), \mathbf{b} = (b_1, \dots, b_n) \in \mathbb{F}_p^n$ and $\langle \mathbf{a,G} \rangle = \sum_{i=1}^n [a_i] G_i \in \mathbb{G}$ - inner product of vector $\mathbf{a}$ with vector of generators $\mathbf{G}$. + +The \textbf{inner-product argument} allows to prove that two vectors $\mathbf{a,b} \in \mathbb{F}_p^n$ satisfy the relation: +$$\mathcal{R}_{ip} = \{ (\mathbf{G,H}, P, c; \mathbf{a,b}) \vert P = \langle \mathbf{a,G} \rangle + \langle \mathbf{b,H} \rangle \wedge \langle \mathbf{a,b} \rangle = c \}$$ +We refer to $P \in \mathbb{G}$ as a binding Pedersen vector commitment to $\mathbf{a,b}$. + +Trivial way to prove the relation is to send $\mathbf{a,b}$ to the verifier $\mathcal{V}$, but it is not a zero-knowledge proof nor efficient due to linear in $n$ size of the proof. We want to build a zero-knowledge proof of the relation $\mathcal{R}_{ip}$ with logarithmic in $n$ size of the proof. + +Firtsly, let's combine statements $P = \langle \mathbf{a,G} \rangle + \langle \mathbf{b,H} \rangle \wedge \langle \mathbf{a,b} \rangle = c$ into a single statement by multiplying the second one by a random $r \in \mathbb{F}_p$ and some orthogonal generator $B \in \mathbb{G}$, summing up: +$$ +\mathcal{R}'_{ip} = \{ (\mathbf{G,H}, P', c; \mathbf{a,b}) \vert P' = \langle \mathbf{a,G} \rangle + \langle \mathbf{b,H} \rangle + [\langle \mathbf{a,b} \rangle]Q \} +$$ +Where $P' = P + [cr]B, Q=[r]B$. Intuitively, if prover $\mathcal{P}$ can prove $\mathcal{R}'_{ip}$ for all $r \in \mathbb{F}_p$, then it can prove $\mathcal{R}_{ip}$ for any valid witness. We use such transformation to compress each vector in half and arrive to the same form of commitment +$$P' = \langle \mathbf{a,G} \rangle + \langle \mathbf{b,H} \rangle + [\langle \mathbf{a,b} \rangle]Q$$ + +\subsubsection{Proving $\mathcal{R}'_{ip}$} + +Here we describe the \textbf{inner-product argument} protocol $\Pi_{ip}$ for the relation $\mathcal{R}'_{ip}$. +Firstly, assuming that $n = 2^d$ define by $\mathbf{G_{lo}, G_{hi}} \in \mathbb{G}^{n/2}$ -- lower and higher halves of vector $\mathbf{G}$ and $\mathbf{a_{lo}, a_{hi}} \in \mathbb{F}_{n/2}$ -- lower and higher halves of $\mathbf{a} \in \mathbb{F}_p^{n}$. + +Let $u_k \in \mathbb{F}_p$ - be some scalar, define compressed vectors: +\begin{align} +\mathbf{a}^{(k-1)} &= \mathbf{a}_{lo} \cdot u_k + u_k^{-1} \cdot \mathbf{a}_{hi} \\ +\mathbf{b}^{(k-1)} &= \mathbf{b}_{lo} \cdot u_k^{-1} + u_k \cdot \mathbf{b}_{hi} \\ +\mathbf{G}^{(k-1)} &= \mathbf{G}_{lo} \cdot u_k^{-1} + u_k \cdot \mathbf{G}_{hi} \\ +\mathbf{H}^{(k-1)} &= \mathbf{H}_{lo} \cdot u_k + u_k^{-1} \cdot \mathbf{H}_{hi} +\end{align} + +Define $P_k \gets P'$ and define $P_{k-1}$ using compressed vectors to have the same form as $P_k$: +$$P_{k-1} = \langle \mathbf{a}^{(k-1)}, \mathbf{G}^{(k-1)} \rangle + \langle \mathbf{b}^{(k-1)}, \mathbf{H}^{(k-1)} \rangle + [\langle \mathbf{a}^{(k-1)}, \mathbf{b}^{(k-1)} \rangle]Q $$ + +\subsection{STARK-friendly fields} +In general, STARK protocol can work over any field $\mathbb{F}$ with high two-adicity. The primary +reason for that is that STARKs can work only with NTT-friendly fields, and the +NTT-friendly fields are the fields where we can select the multiplicative +subgroup of order $2^k$ for sufficiently many values of $k$. + +\begin{definition} + We call \textbf{two-adicity fields}, the fields where we can select the + multiplicative subgroup of order $2^k$ for sufficiently many values of $k$. + In this case, the field order $p$ is typically of form $p = 2^m \cdot p' + + 1$ where $p'$ is a small integer. +\end{definition} + +To be honest, all protocol steps are followed with powers of two. It will be shown, why the groups we are working over must be of size $2^k$ and why the input data also follows this rule. As the result, the maximum size of the statement that we can prove using the STARK protocol is strictly depends on the size of two-adicity subgroup (that is why we label some fields to have \textit{high two-adicity} or \textit{low two-adicity}). + +\begin{remark} +In our initial discussion we consider using field over prime modulus $p = 3\cdot +2^{30} + 1$ and subgroups of size $2^{13}$ and $2^{10}$. +\end{remark} + +As we will work in the new subgroup we may want to specify the subgroup +generator to be used in future equations. So, for the multiplicative group +generator $w \in \mathbb{F}_p^{\times}$, the generator of the subgroup of order +$2^k$ is $\omega_k = w^\frac{p - 1}{2^k}$, as was shown in the NTT section. + +\begin{example} +For the prime field $\mathbb{F}_p$ where $p = 3\cdot 2^{30} + 1$, the order of $\mathbb{F}^{\times}_p$ is $p-1 = 3\cdot 2^{30}$. If we take $w = 5$ as the primitive element, the multiplicative subgroup of $2^{13}$ elements generator will be $\omega = 5^{3\cdot 2^{17}}$ +\end{example} + +This kind of subgroups comes with very useful property: for each element in +two-adicity subgroup $\mathbb{H}$, the additive inverse element +can be calculated by a simple equation over the element power. + +\begin{proposition} + Suppose $\mathbb{H} \leq \mathbb{F}_p$ is a subgroup of order $r$ with + generator $h = w^{(p-1)/r}$. Then, the additive inverse for $x = h^i \in + \mathbb{H}$ is $h^j$ where $j = i + \frac{r}{2} \pmod{r}$. +\end{proposition} + +\textbf{Proof.} The sum of $x$ and $-x$ must equal to zero modulo $p$, so: +\begin{equation*} + \begin{aligned} + x + (-x) &= w^{(p-1)i/r} + w^{(p-1)j/r} = w^{(p-1)i/r}(1 + w^{(p-1)(j-i)/r}) \\ &= w^{(p-1)i/r}(1 + w^{(p-1)/2}). + \end{aligned} +\end{equation*} + +Now note that $w^{(p-1)/2} = -1$ which completes the proof. $\blacksquare$ + +\begin{remark} +The equation $w^{p - 1} = 1$ is obtained from the order property of the +primitive element $w$ in the multiplicative group $\mathbb{F}^{\times}_p$. +\end{remark} +\begin{remark} +This provides us with an additional important property beyond element's power +computation: when working with a negative element, its power shift equals half +the size of the subgroup so, squaring the elements within this subgroup results +in a smaller subgroup, reduced by a factor of two. Consequently, to compute the +square of the subgroup, it suffices to square only the first half of its +elements (powers $0, 1, 2, 3, \dots, \frac{r}{2}$). +\end{remark} + +\subsection{Protocol definition} + +\subsubsection{Trace, evaluation domain and commitment} +Now, we are going to prove that some statement holds on the given sequence of elements. + +\begin{definition} +We call \textbf{trace} a sequence of elements from $\mathbb{F}$ that represents our witness. This sequence contains private and public values together and follows certain constraints. +\end{definition} + +\begin{example} +The \textbf{Fibonacci square sequence} is a sequence of elements defined over +$\mathbb{F}$ as follows: +\begin{equation*} +a_{j+2} = a_{j+1}^2 + a_{j}^2 +\end{equation*} + +Then we can, for example, prove the following statement: \textcolor{blue!60!black}{\textit{I know a field +element $w \in \mathbb{F}$ such that the $k^{\text{th}}$ element of the Fibonacci +square sequence ($a_k$) starting with $x$ and $w$ is $y$.}} Formally, +this can be written as: +\begin{equation*} + \mathcal{R}_{\text{Fib}} = \left\{ \begin{matrix} + \textbf{Public Statement:} \; (x, y, k) \\ + \textbf{Witness:} \; w + \end{matrix} \;\Big|\; \begin{matrix} + a_0 = x, a_1 = w, a_k = y \; \text{with} \\ a_{j+2} = a_{j+1}^2 + a_j^2 \; \text{for all} \; j \in [k] + \end{matrix} + \right\} +\end{equation*} + +For concreteness, let us take $k=1023$, $x = 1$, and $y=2338775057$. +\end{example} + +Following the Unisolvence Theorem, the trace $\{a_j\}_j$ is implied to be an evaluation +of some unknown \textbf{trace polynomial} of degree equal to the length of the +sequence $\{a_j\}_j$. Also, to be +evaluable on the two-adicity subgroup, the size of the trace has to be a power +of two. + +\begin{definition} +We call \textbf{domain} a two-adicity subgroup $\mathbb{G} \leq +\mathbb{F}^{\times}$ where we evaluate our polynomials. +\end{definition} + +\begin{example} +In our example, we put trace a sequence $\{a_j\}_j$ of first $1023$ elements of +the Fibonacci square sequence over $\mathbb{F}_p$, where $p=3\cdot 2^{30} + 1$. +\begin{equation*} +1, 1, 2, 5, 29, \ldots +\end{equation*} +To interpolate our trace polynomial we select as a domain a two-adicity subgroup +of $2^{10}$ elements from $\mathbb{F}^\times_p$ with a generator $g = +5^{\frac{3\cdot 2^{30}}{2^{10}}} = 5^{3 \cdot 2^{20}}$ (here $5$ is the +primitive element in the multiplicative group $\mathbb{F}^\times_p$). That being +said, $\mathbb{G} = \{g^i\}_{i \in [1024]}$. +\end{example} + +Next, using the Lagrange interpolation over $(g^j, a_j)_{j \in [k]}$ points +we compute a trace polynomial $f \in \mathbb{F}[x]$. Note that the interpolation +can be done in $O(k\log k)$, as shown in NTT section. + +\begin{definition} +We call \textbf{evaluation domain} a two-adicity coset $\mathbb{E} = w\mathbb{H} +\leq \mathbb{F}_p^{\times}$, where $\mathbb{H} \leq \mathbb{F}_p^{\times}$ is a +two-adicity subgroup, that is larger $\rho \in \mathbb{N}$ times (typically a +relatively small constant) than the domain. In other words, +$\text{ord}(\mathbb{H}) = \rho \cdot \text{ord}(\mathbb{G})$. +\end{definition} + +\begin{example} +In our case we select a two-adicity subgroup $\mathbb{H}$ of $2^{13}$ elements +from $\mathbb{F}_p^\times$ with $\rho = 8$ as $\mathbb{H} = \{h^i\}_{i \in +[8192]}$ where $h = 5^{3 \cdot 2^{17}}$. Then, we define the \emph{evaluation +domain} as $\mathbb{E}=5\mathbb{H} = \{5h^i\}_{i \in [8192]}$. +\end{example} + +We build a Merkle tree over the values $\{f(e)\}_{e \in \mathbb{E}}$ and label +its root as a \textbf{trace polynomial commitment}. This approach will also be +used to commit other polynomials during the protocol walkthrough. + +The \textbf{constraints} in STARK protocol are expressed as polynomials +evaluated over the trace cells, which are satisfied if and only if the +computations are correct. + +\begin{example} +Obviously, our initial statement consists of the following three requirements: +\begin{enumerate} + \item The element $a_0$ is equal to $1$; + \item The element $a_{1022}$ is equal to $2338775057$; + \item Each element $a_{i+2}$ is equal to $a_{i+1}^2 + a_{i}^2$. +\end{enumerate} +\end{example} +To verify that our committed trace polynomial satisfies all constraints, we can +check that it has corresponding roots. In particular, according to the selected +interpolation points $\{(g^i, a_i)\}_{i \in [k]}$, the relation $r(a_i, a_j) = +0$ can be rewritten as $r(f(g^i), f(g^j)) = 0$. +\begin{example} +For our Fibonacci trace we have the following constraints to be checked over the interpolated polynomial: +\begin{enumerate} + \item \textit{The element $a_0$ is equal to $1$} translated to: $f(x)-1$ has root at $x = g^0 = 1$; + \item \textit{The element $a_{1022}$ is equal to $2338775057$} translated to: $f(x) - 2338775057$ has root at $x = g^{1022}$; + \item \textit{Each element $a_{i+2}$ is equal to $a_{i+1}^2 + a_{i}^2$} translated to: $f(g^2x) - f(gx)^2 - f(x)^2$ has roots in $\mathbb{G} \setminus \{g^{1021}, g^{1022}, g^{1023}\}$ +\end{enumerate} +\end{example} + +To ensure that the specified polynomials have roots in given values, we can use the following property: if polynomial $f(x) \in \mathbb{F}[x]$ has root in $x_0$ then the $\frac{f(x)}{x - x_0}$ is also a polynomial in $\mathbb{F}[x]$. + +\begin{example} +Finally, we define the following STARK constraints: +\vspace{-1mm} +\begin{gather*} + p_0(x) = \frac{f(x)-1}{x - 1} \\ + p_1(x) = \frac{f(x) - 2338775057}{x - g^{1022}} \\ + p_2(x) = \frac{f(g^2x) - f(gx)^2 - f(x)^2}{\prod_{i=0}^{1020} (x - g^i)} +\end{gather*} +\vspace{-1mm} +Unfortunately, the $p_2$ polynomial still looks inconvenient to work with, so we +may want to simplify it (this is not a part of the protocol in general, but you +always may want to simplify your equations to achieve better proving time). Note +that $p_2$ is \textit{almost} a vanishing polynomial of $\mathbb{G}$, which has +a form $x^{\text{ord}(\mathbb{G})} - 1$, except for points $g^{1021}, +g^{1022}, g^{1023}$. In other words, we can simplify the denominator as: +\begin{xequation*} + \prod_{i=0}^{1020} (x - g^i) = \frac{x^{1024} - 1}{(x-g^{1021})(x-g^{1022})(x-g^{1023})} +\end{xequation*} + +Note, that while evaluating our polynomial on a larger domain +then $\mathbb{G}$ we should only ensure that the resulting polynomial still +holds the relation $f(g^i) = a_i$, so it is acceptable to use properties that +only work over $\mathbb{G}$. So, finally we have: + \begin{xequation} + p_2(x) = \frac{(f(g^2x) - f(gx)^2 - f(x)^2)(x - g^{2021})(x - g^{2022})(x - g^{2024})}{x^{1024} - 1} + \end{xequation} +\end{example} + +In addition, there is one obvious requirement for the STARK constraints: the +verifier should be able to compute the constraints polynomials $p_i(x)$ using +only the given trace polynomial evaluations for the certain $x$. + +\begin{remark} +In our Fibonacci example, verifier can check the constraint polynomials +evaluation by requesting only $f(x)$, $f(gx)$ and $f(g^2x)$ --- the values +committed in the trace polynomial commitment. +\end{remark} + +To combine all our constraints into a single polynomial, we can follow a +commonly used principle by taking a linear combination with the challenges from +the verifier. In particular, after receiving trace polynomial commitment from +the prover, the verifier selects scalars $\alpha_1,\dots,\alpha_m$ and sends it +to the prover. Then, the prover puts the \textbf{composition polynomial} as: +\begin{xequation*} + \text{CP}(x) := \sum_{j = 1}^m \alpha_j\cdot p_j(x) +\end{xequation*} +Additionally, prover also commits this polynomial by evaluating on the evaluation domain and building a Merkle tree. + +\begin{example} +The Fibonacci composition polynomial looks like as follows: + \begin{gather*} + \text{CP}(x) = \alpha_0 p_0(x) + \alpha_1 p_1(x) + \alpha_2 p_2(x) =\\ + \alpha_0 \frac{f(x)-1}{x - 1} + \alpha_1 \frac{f(x) - 2338775057}{x - g^{1022}} + \\ + \alpha_2 \frac{(f(g^2x) - f(gx)^2 - f(x)^2)(x - g^{2021})(x - g^{2022})(x - g^{2024})}{x^{1024} - 1} + \end{gather*} +\end{example} + +\subsubsection{FRI protocol} +In general, our goal is to verify that the committed polynomial $\text{CP}(x)$ +satisfies all our constraints, by checking it's evaluation at a random point +from the evaluation domain that the verifier selects. Anyway, we can face the +problem when the malicious prover constructs a larger polynomial that accepts +lots of possible roots from our field (even $2^{64}$ field is still insecure for +just checking the evaluation at one point). That is why we have to make sure +that the committed polynomial degree lies in the acceptable range (the upper +bound depends on the trace size). + +The final stage of the STARK protocol is a \textbf{Fast Reed-Solomon IOP of Proximity (FRI)}. FRI is a protocol between a prover and a verifier, which establishes that a given evaluation belongs to a polynomial of low-degree. In this context \textit{low} means no more than $\rho$ times bigger than the trace. + +The key idea of FRI protocol is to move from a polynomial of degree $n$ to a +polynomial of degree $n/2$ until we get a constant value. Let's consider the +polynomial $z_0(x) = \sum_i a_i\cdot x^i$ of degree $n=2^t$ and the evaluation +domain $\mathbb{E}_0 = \mathbb{E}$. We suppose to group the \textit{odd} and the +\textit{even} coefficients of the $z_0$ together into the two separate +polynomials($z_0^O$ and $z_0^E$ respectively): +\begin{xequation*} + \begin{aligned} + z_0^O(x^2) = \sum_{i=0}^{n/2} (a_{2i+1}\cdot x^{2i}), \quad z_0^E(x^2) = \sum_{i=0}^{n/2} (a_{2i}\cdot x^{2i}) +\end{aligned} +\end{xequation*} + +Or, in a more comfortable form (we have already examined why searching of $-x$ +can be done easily in our two-adicity subgroup): +\begin{xequation*} + \begin{aligned} + z_0^E(x^2) = \frac{z_0(x) + z_0(-x)}{2}, \quad z_0^O(x^2) = \frac{z_0(x) - z_0(-x)}{2x} + \end{aligned} +\end{xequation*} + +Then, we define a next-layer of the FRI polynomial as $z_1(x^2) = z_0^E(x^2) + +\beta z_0^O(x^2)$, where $\beta$ is a challenge received from verifier. The +next-layer evaluation domain is also simple to compute: $\mathbb{E}_1 = +\{(w\cdot h_i)^2\}_{i \in [\text{ord}(\mathbb{E}_0)/2]}$ as squaring the +other elements in $\mathbb{E}_0$ will result in the same values. + +Next, we commit to the $z_1(x^2)$ using a next-layer evaluation domain +$\mathbb{E}_1$ (is also reduced by a factor two) and continue to repeat the +described operations until $z_j(x^{2^j})$ becomes constant. + +\vspace{-2mm} + +\begin{tcolorbox}[title=Interactive ZK-STARK protocol, + colback=blue!5!white, + colframe=blue!75!black, + colbacktitle=blue!25!white, + coltitle=blue!20!black, + fonttitle=\bfseries, + boxrule=1.25pt, + subtitle style={boxrule=0pt, + colback=blue!20!white, + colupper=blue!75!gray} ] + \small + + The prover and the verifier run the interactive version of the ZK-STARK + protocol. Both know the statement to be proved, that is defined by the + constraint polynomials and the field $\mathbb{F}_p$ to work over. Prover also + knows the witness to be able to generate the trace. + + \tcbsubtitle{Preparation} + \begin{itemize}[label=\ding{51}] + \item The prover interpolates trace polynomial $f(x)$ and submits its + commitment to the verifier. + \item The verifier selects challenges random $\alpha_i \in \mathbb{F}_p$ + and sends to the prover. + \item The prover builds the composition polynomial $\text{CP}(x)$ and + submits its commitment to the verifier. + \end{itemize} + + \tcbsubtitle{FRI} + \begin{itemize}[label=\ding{51}] + \item The verifier selects random $j \in [\text{ord}(\mathbb{E})]$, sets + $c \gets w\cdot h^j$ and sends it to the prover. + \item The prover responds with the $\text{CP}(c), \text{CP}(-c)$ and all + $f(x)$ required to check $\text{CP}$ evaluation with corresponding Merkle + proofs to them. + \item The verifier checks Merkle proofs and the evaluation of + $\text{CP}(c)$ by evaluating the constraints polynomials $p_j(c)$. + \item The prover and the verifier go through the FRI protocol for + $z_0(x) = CP(x)$ where the prover commits to the layer-$j$ polynomial + $z_j(x)$, the verifier selects a challenge $\beta$ and queries from the + prover $z_j(c), z_j(-c)$ to compute $z_{j+1}(c)$ until $z_k(x), j \leq + \log_2(\deg \text{CP})$ becomes constant. + \end{itemize} + +\end{tcolorbox} + +The non-interactive version of the presented protocol can be easily built +obtaining the Fiat-Shamir heuristics. + +The soundness of the presented STARK protocol follows from the impossibility to +commit any possible evaluation of the forgery $\text{CP}(x)$ over evaluation +domain $\mathbb{E}$ and simultaneously prove that $\text{CP}(x)$ is a low-degree +polynomial by the FRI protocol. Since the size of $\mathbb{E}$ is $\rho$ times bigger +then the maximum allowed polynomial degree (that directly depends on the size of +the trace), the attacker either can't construct such a polynomial or can't +construct a low-degree polynomial, so a valid low-degree composition polynomial +can only be obtained using a valid trace. + +\vspace{-2mm} + +\begin{example} + Finally, let's overview the first steps of the ZK-STARK protocol applied to our Fibonacci example: + + \begin{enumerate} + \item The protocol defines the public constraints such as 2023-th + element of sequence, field $\mathbb{F}_p$, etc. + \item The prover generates the trace $a$ where $a_0 = 1, a_1 = 3141592, + a_i = a_{i-1}^2 + a_{i-2}^2$, evaluates the trace polynomial $f(x)$ over + the evaluation domain and sends it's commitments to the verifier. + \item The verifier selects challenges $\alpha_0, \alpha_1, \alpha_2 \in + \mathbb{F}$ and shares them with the prover. + \item The prover evaluates the composition polynomial $\text{CP}(x)$ + over evaluation domain and sends it's commitments to the verifier. + \item The verifier selects random $i \in [8192-16]$, puts $c = 5\cdot + h^i$ and sends it to the prover. + \item The prover responds with the $f(c), f(gc), f(g^2c), \text{CP}(c), + \text{CP}(-c)$ and corresponding Merkle proofs to them. + \item The verifier checks Merkle proofs and the evaluation of + $\text{CP}(c)$ by evaluating the constraint polynomials $p_0(c), p_1(c), + p_2(c)$. + \item The prover and the verifier go through the FRI protocol for + $z_0(x) = \text{CP}(x)$ until $z_i(x), i \in [12]$ becomes constant. + \end{enumerate} + +\end{example} + +\subsection{Protocol security} +Most of the existing versions of the STARK protocol leverage on several +optimizations to achieve better proving and verification time. The key point +here is that each FRI query check adds $\log_2(\rho)$ bits of security, so we can +skip some of these checks if the security level is already satisfied. One more +optimization is to include a proof-of-work computation into the protocol that +should be done before FRI with dependency on the committed values. It can be +useful because the verification of the proof-of-work is less expensive then the +verification of the FRI step while still increases the computation cost for the +malicious prover. + +More precisely, let's assume that the desired security level of the protocol is +$\lambda$. First of all, we obviously have to use a proper collision-resistant +hash function with $2\lambda$ bits output. Then, according to the StarkWare's +definition of the STARK protocol, the resulting security is defined as follows: +\begin{xequation*} + \lambda \geq \min\{ \delta + \log_2(\rho) \cdot s, \log_2(|\mathbb{F}|) \} - 1 +\end{xequation*} +where $\delta$ -- number of the proof-of-work bits, $s$ -- number of the FRI queries. + +\begin{example} +If the protocol is deployed over $256$-bit field and the domain ratio is $\rho = +8$, to achieve the $128$ bit security we can for example execute $33$ FRI query +and evaluate $29$ proof-of-work bits: $\min\{29+3\cdot 33, 256\} = 128$. +\end{example} + +\subsection*{Acknowledgements} + +This work was inspired by \href{https://starkware.co/stark-101/}{``STARK-101''} +course by StarkWare and +\href{https://vitalik.eth.limo/general/2017/11/09/starks_part_1.html}{``STARKs''} +series by Vitalik Buterin. + +\end{document} From 00aa223417e11a13931361e8156cd1aefcac50c1 Mon Sep 17 00:00:00 2001 From: Yevhen Hrubiian Date: Wed, 9 Jul 2025 19:52:37 +0300 Subject: [PATCH 02/25] added inner-product proto draft --- lectures/2-9-bulletproofs.tex | 84 +++++++++++++++++++++++++++++++---- 1 file changed, 75 insertions(+), 9 deletions(-) diff --git a/lectures/2-9-bulletproofs.tex b/lectures/2-9-bulletproofs.tex index 4ca2dfa..d0934e9 100644 --- a/lectures/2-9-bulletproofs.tex +++ b/lectures/2-9-bulletproofs.tex @@ -23,27 +23,93 @@ \subsection{Inner-product argument} Firtsly, let's combine statements $P = \langle \mathbf{a,G} \rangle + \langle \mathbf{b,H} \rangle \wedge \langle \mathbf{a,b} \rangle = c$ into a single statement by multiplying the second one by a random $r \in \mathbb{F}_p$ and some orthogonal generator $B \in \mathbb{G}$, summing up: $$ -\mathcal{R}'_{ip} = \{ (\mathbf{G,H}, P', c; \mathbf{a,b}) \vert P' = \langle \mathbf{a,G} \rangle + \langle \mathbf{b,H} \rangle + [\langle \mathbf{a,b} \rangle]Q \} +\mathcal{R}'_{ip} = \{ (\mathbf{G,H}, Q, P'; \mathbf{a,b}) \vert P' = \langle \mathbf{a,G} \rangle + \langle \mathbf{b,H} \rangle + [\langle \mathbf{a,b} \rangle]Q \} $$ Where $P' = P + [cr]B, Q=[r]B$. Intuitively, if prover $\mathcal{P}$ can prove $\mathcal{R}'_{ip}$ for all $r \in \mathbb{F}_p$, then it can prove $\mathcal{R}_{ip}$ for any valid witness. We use such transformation to compress each vector in half and arrive to the same form of commitment $$P' = \langle \mathbf{a,G} \rangle + \langle \mathbf{b,H} \rangle + [\langle \mathbf{a,b} \rangle]Q$$ -\subsubsection{Proving $\mathcal{R}'_{ip}$} +\subsubsection{Inner-product compression} -Here we describe the \textbf{inner-product argument} protocol $\Pi_{ip}$ for the relation $\mathcal{R}'_{ip}$. -Firstly, assuming that $n = 2^d$ define by $\mathbf{G_{lo}, G_{hi}} \in \mathbb{G}^{n/2}$ -- lower and higher halves of vector $\mathbf{G}$ and $\mathbf{a_{lo}, a_{hi}} \in \mathbb{F}_{n/2}$ -- lower and higher halves of $\mathbf{a} \in \mathbb{F}_p^{n}$. +Here we describe \textbf{inner-product compression} algorithm -- main building block of the interactive \textbf{inner-product} procotol. +Firstly, assuming that $n = 2^d$ define by $\mathbf{G_{lo}} = (G_1, \dots, G_{n/2}), \mathbf{G_{hi}} = (G_{n/2+1},\dots, G_n) \in \mathbb{G}^{n/2}$ -- lower and higher halves of vector $\mathbf{G}$ and $\mathbf{a_{lo}} = (a_1, \dots, a_{n/2}), \mathbf{a_{hi}} = (a_{n/2+1},\dots,a_n) \in \mathbb{F}_{n/2}$ -- lower and higher halves of $\mathbf{a} \in \mathbb{F}_p^{n}$. Let $u_k \in \mathbb{F}_p$ - be some scalar, define compressed vectors: \begin{align} -\mathbf{a}^{(k-1)} &= \mathbf{a}_{lo} \cdot u_k + u_k^{-1} \cdot \mathbf{a}_{hi} \\ -\mathbf{b}^{(k-1)} &= \mathbf{b}_{lo} \cdot u_k^{-1} + u_k \cdot \mathbf{b}_{hi} \\ -\mathbf{G}^{(k-1)} &= \mathbf{G}_{lo} \cdot u_k^{-1} + u_k \cdot \mathbf{G}_{hi} \\ -\mathbf{H}^{(k-1)} &= \mathbf{H}_{lo} \cdot u_k + u_k^{-1} \cdot \mathbf{H}_{hi} + \mathbf{a}^{(k-1)} &= \mathbf{a_{lo}} \cdot u_k + u_k^{-1} \cdot \mathbf{a_{hi}} \\ + \mathbf{b}^{(k-1)} &= \mathbf{b_{lo}} \cdot u_k^{-1} + u_k \cdot \mathbf{b_{hi}} \\ + \mathbf{G}^{(k-1)} &= \mathbf{G_{lo}} \cdot u_k^{-1} + u_k \cdot \mathbf{G_{hi}} \\ + \mathbf{H}^{(k-1)} &= \mathbf{H_{lo}} \cdot u_k + u_k^{-1} \cdot \mathbf{H_{hi}} \end{align} -Define $P_k \gets P'$ and define $P_{k-1}$ using compressed vectors to have the same form as $P_k$: +Define $P_k \gets P' = \langle \mathbf{a,G} \rangle + \langle \mathbf{b,H} \rangle + [\langle \mathbf{a,b} \rangle]Q$ and define $P_{k-1}$ using compressed vectors to have the same form as $P_k$: $$P_{k-1} = \langle \mathbf{a}^{(k-1)}, \mathbf{G}^{(k-1)} \rangle + \langle \mathbf{b}^{(k-1)}, \mathbf{H}^{(k-1)} \rangle + [\langle \mathbf{a}^{(k-1)}, \mathbf{b}^{(k-1)} \rangle]Q $$ +Subsituting compressed vectors and applying bilinearity property of inner product we get: +\begin{align} + P_{k-1} = & \langle \mathbf{a_{lo}}, \mathbf{G_{lo}}\rangle + \langle \mathbf{a_{hi}}, \mathbf{G_{hi}}\rangle &+ u_k^2\langle \mathbf{a_{lo}}, \mathbf{G_{hi}}\rangle + u_k^{-2}\langle \mathbf{a_{hi}}, \mathbf{G_{lo}}\rangle + \\ + & \langle \mathbf{b_{lo}}, \mathbf{H_{lo}}\rangle + \langle \mathbf{b_{hi}}, \mathbf{H_{hi}}\rangle &+ u_k^2\langle \mathbf{b_{hi}}, \mathbf{H_{lo}}\rangle + u_k^{-2}\langle \mathbf{b_{lo}}, \mathbf{H_{hi}}\rangle + \\ + & [\langle \mathbf{a_{lo}}, \mathbf{b_{lo}}\rangle + \langle \mathbf{a_{hi}}, \mathbf{b_{hi}}\rangle]Q &+ [u_k^2\langle \mathbf{a_{lo}}, \mathbf{b_{hi}}\rangle + u_k^{-2}\langle \mathbf{a_{hi}}, \mathbf{b_{lo}}\rangle]Q +\end{align} + +Note that $\langle \mathbf{a_{lo}}, \mathbf{G_{lo}}\rangle + \langle \mathbf{a_{hi}}, \mathbf{G_{hi}}\rangle = \langle \mathbf{a,G}\rangle$ so that the first two columns of $P_{k-1}$ definition precisecly contains $P_{k} = P'$: +$$P_{k} = \langle \mathbf{a_{lo}}, \mathbf{G_{lo}}\rangle + \langle \mathbf{a_{hi}}, \mathbf{G_{hi}}\rangle + \langle \mathbf{b_{lo}}, \mathbf{H_{lo}}\rangle + \langle \mathbf{b_{hi}}, \mathbf{H_{hi}}\rangle + [\langle \mathbf{a_{lo}}, \mathbf{b_{lo}}\rangle + \langle \mathbf{a_{hi}}, \mathbf{b_{hi}}\rangle]Q$$ + +Define cross-terms $L_k, R_k$ of $P_{k-1}$ such that: +\begin{align} + P_{k-1} &= P_k + [u_k^2] L_k + [u_k^{-2}] R_k \\ + L_{k} &= \langle \mathbf{a_{lo}}, \mathbf{G_{hi}}\rangle + \langle \mathbf{b_{hi}}, \mathbf{H_{lo}}\rangle + [\langle \mathbf{a_{lo}}, \mathbf{b_{hi}}\rangle]Q \\ + R_{k} &= \langle \mathbf{a_{hi}}, \mathbf{G_{lo}}\rangle + \langle \mathbf{b_{lo}}, \mathbf{H_{hi}}\rangle + [\langle \mathbf{a_{hi}}, \mathbf{b_{lo}}\rangle]Q +\end{align} + +Here we come up with some kind of statement compression algorithm reducing size of all vectors in half per compression step. Repeating comression algorithm $k$ times we end up with vectors $\mathbf{a}^{(0)}, \mathbf{b}^{(0)}, \mathbf{G}^{(0)}, \mathbf{H}^{(0)}$ each of length one and $P_0$ containing all accumulated cross-terms: +\begin{align} + P_0 &= [a_1^{(0)}]G_1^{(0)} + [b_1^{(0)}]H_1^{(0)} + [a_1^{(0)}b_1^{(0)}]Q \\ + P_0 &= P_k + \sum_{i=1}^d ([u_i^2]L_i + [u_i^{-2}]R_i) +\end{align} + +Recalling that $P_k = P'$, the final compressed statement asserting inner-product value will have the following form: +$$P' + \sum_{i=1}^d ([u_i^2]L_i + [u_i^{-2}]R_i) = [a_1^{(0)}]G_1^{(0)} + [b_1^{(0)}]H_1^{(0)} + [a_1^{(0)}b_1^{(0)}]Q$$ + +\subsubsection{Proving $\mathcal{R}'_{ip}$} + +Let's describe the \textbf{inner-product} protocol $\Pi_{ip}$ for relation $\mathcal{R}'_{ip}$. +\begin{definition} + The \textbf{inner-product} protocol $\Pi_{ip} = (\mathcal{P}, \mathcal{V})$ for relation $\mathcal{R}'_{ip} = \{ (\mathbf{G,H}, Q, P'; \mathbf{a,b}) \vert P' = \langle \mathbf{a,G} \rangle + \langle \mathbf{b,H} \rangle + [\langle \mathbf{a,b} \rangle]Q \}$, where all vectors have length $n=2^d$ with prover $\mathcal{P}$, verifier $\mathcal{V}$ is defined as follows: + \begin{itemize} + \item Prover $\mathcal{P}$ sets $$(k, \mathbf{a}^{(k)}, \mathbf{b}^{(k)}, \mathbf{G}^{(k)}, \mathbf{H}^{(k)}, P_k) \gets (n, \mathbf{a,b,G,H},P')$$ + \item Verifier $\mathcal{V}$ sets $$(k, \mathbf{G}^{(k)}, \mathbf{H}^{(k)}, P_k) \gets (n, \mathbf{G,H},P')$$ + \item While $k > 0$ then: + \begin{itemize} + \item Prover $\mathcal{P}$ computes + \begin{align*} + L_{k} &= \langle \mathbf{a_{lo}}^{(k)}, \mathbf{G_{hi}}^{(k)}\rangle + \langle \mathbf{b_{hi}}^{(k)}, \mathbf{H_{lo}}^{(k)}\rangle + [\langle \mathbf{a_{lo}}^{(k)}, \mathbf{b_{hi}}^{(k)}\rangle]Q \\ + R_{k} &= \langle \mathbf{a_{hi}}^{(k)}, \mathbf{G_{lo}}^{(k)}\rangle + \langle \mathbf{b_{lo}}^{(k)}, \mathbf{H_{hi}}^{(k)}\rangle + [\langle \mathbf{a_{hi}}^{(k)}, \mathbf{b_{lo}}^{(k)}\rangle]Q + \end{align*} + \item $\mathcal{P}$ sends $(L_k, R_k)$ to $\mathcal{V}$ + \item $\mathcal{V}$ draws challenge $u_k \xleftarrow{R} \mathbb{F}_p$ and sends it to $\mathcal{P}$ + \item Both $\mathcal{P}$ and $\mathcal{V}$ compute: + \begin{align*} + P_{k-1} &= P_k + [u_k^2] L_k + [u_k^{-2}] R_k \\ + \mathbf{G}^{(k-1)} &= \mathbf{G_{lo}}^{(k)} \cdot u_k^{-1} + u_k \cdot \mathbf{G_{hi}}^{(k)} \\ + \mathbf{H}^{(k-1)} &= \mathbf{H_{lo}}^{(k)} \cdot u_k + u_k^{-1} \cdot \mathbf{H_{hi}}^{(k)} + \end{align*} + \item $\mathcal{P}$ computes: + \begin{align*} + \mathbf{a}^{(k-1)} &= \mathbf{a_{lo}}^{(k)} \cdot u_k + u_k^{-1} \cdot \mathbf{a_{hi}}^{(k)} \\ + \mathbf{b}^{(k-1)} &= \mathbf{b_{lo}}^{(k)} \cdot u_k^{-1} + u_k \cdot \mathbf{b_{hi}}^{(k)} \\ + \end{align*} + \end{itemize} + \item Prover $\mathcal{P}$ sends $(a,b) \gets (\mathbf{a}_1^{(0)}, \mathbf{b}_1^{(0)})$ to verifier $\mathcal{V}$ + \item Verifier performs final check: + $$P' + \sum_{i=1}^d ([u_i^2]L_i + [u_i^{-2}]R_i) = [a]G_1^{(0)} + [b]H_1^{(0)} + [ab]Q$$ + outputs \textbf{accept} if equality holds and \textbf{reject} otherwise. + \end{itemize} +\end{definition} + +\subsection{Range proofs} + +\subsection{Arithmetic circuits proofs} + \subsection{STARK-friendly fields} In general, STARK protocol can work over any field $\mathbb{F}$ with high two-adicity. The primary reason for that is that STARKs can work only with NTT-friendly fields, and the From c2752c1c18d12a3d09b1c213d0430ab91250f99f Mon Sep 17 00:00:00 2001 From: Yevhen Hrubiian Date: Thu, 10 Jul 2025 20:17:09 +0300 Subject: [PATCH 03/25] add figure --- lectures/2-9-bulletproofs.tex | 476 +++++++--------------------------- 1 file changed, 92 insertions(+), 384 deletions(-) diff --git a/lectures/2-9-bulletproofs.tex b/lectures/2-9-bulletproofs.tex index d0934e9..9253672 100644 --- a/lectures/2-9-bulletproofs.tex +++ b/lectures/2-9-bulletproofs.tex @@ -28,6 +28,16 @@ \subsection{Inner-product argument} Where $P' = P + [cr]B, Q=[r]B$. Intuitively, if prover $\mathcal{P}$ can prove $\mathcal{R}'_{ip}$ for all $r \in \mathbb{F}_p$, then it can prove $\mathcal{R}_{ip}$ for any valid witness. We use such transformation to compress each vector in half and arrive to the same form of commitment $$P' = \langle \mathbf{a,G} \rangle + \langle \mathbf{b,H} \rangle + [\langle \mathbf{a,b} \rangle]Q$$ +\begin{definition} + The \textbf{inner-product} protocol $\Pi_{ip} = (\mathcal{P}, \mathcal{V})$ for relation $\mathcal{R}_{ip} = \{ (\mathbf{G,H}, P, c; \mathbf{a,b}) \vert P = \langle \mathbf{a,G} \rangle + \langle \mathbf{b,H} \rangle \wedge \langle \mathbf{a,b} \rangle = c \}$ with prover $\mathcal{P}$, verifier $\mathcal{V}$ is defined as follows: + \begin{itemize} + \item Parties $\mathcal{P, V}$ agree on some group element $B \in \mathbb{G}$ with unknown discrete log + \item Verifier $\mathcal{V}$ samples random $r \xleftarrow{R} \mathbb{F}_p$ and sends it to prover $\mathcal{P}$ + \item Parties $\mathcal{P, V}$ compute $Q \gets [r]B$ and $P' = P + [c]Q$ + \item Parties run protocol $\Pi'_{ip}$ for relation $\mathcal{R}'_{ip}$ on input $(\mathbf{G,H}, Q, P'; \mathbf{a,b})$ + \end{itemize} +\end{definition} + \subsubsection{Inner-product compression} Here we describe \textbf{inner-product compression} algorithm -- main building block of the interactive \textbf{inner-product} procotol. @@ -72,9 +82,9 @@ \subsubsection{Inner-product compression} \subsubsection{Proving $\mathcal{R}'_{ip}$} -Let's describe the \textbf{inner-product} protocol $\Pi_{ip}$ for relation $\mathcal{R}'_{ip}$. +Let's describe the \textbf{inner-product} protocol $\Pi'_{ip}$ for relation $\mathcal{R}'_{ip}$. \begin{definition} - The \textbf{inner-product} protocol $\Pi_{ip} = (\mathcal{P}, \mathcal{V})$ for relation $\mathcal{R}'_{ip} = \{ (\mathbf{G,H}, Q, P'; \mathbf{a,b}) \vert P' = \langle \mathbf{a,G} \rangle + \langle \mathbf{b,H} \rangle + [\langle \mathbf{a,b} \rangle]Q \}$, where all vectors have length $n=2^d$ with prover $\mathcal{P}$, verifier $\mathcal{V}$ is defined as follows: + The \textbf{inner-product} protocol $\Pi'_{ip} = (\mathcal{P}, \mathcal{V})$ for relation $\mathcal{R}'_{ip} = \{ (\mathbf{G,H}, Q, P'; \mathbf{a,b}) \vert P' = \langle \mathbf{a,G} \rangle + \langle \mathbf{b,H} \rangle + [\langle \mathbf{a,b} \rangle]Q \}$, where all vectors have length $n=2^d$ with prover $\mathcal{P}$, verifier $\mathcal{V}$ is defined as follows: \begin{itemize} \item Prover $\mathcal{P}$ sets $$(k, \mathbf{a}^{(k)}, \mathbf{b}^{(k)}, \mathbf{G}^{(k)}, \mathbf{H}^{(k)}, P_k) \gets (n, \mathbf{a,b,G,H},P')$$ \item Verifier $\mathcal{V}$ sets $$(k, \mathbf{G}^{(k)}, \mathbf{H}^{(k)}, P_k) \gets (n, \mathbf{G,H},P')$$ @@ -89,7 +99,6 @@ \subsubsection{Proving $\mathcal{R}'_{ip}$} \item $\mathcal{V}$ draws challenge $u_k \xleftarrow{R} \mathbb{F}_p$ and sends it to $\mathcal{P}$ \item Both $\mathcal{P}$ and $\mathcal{V}$ compute: \begin{align*} - P_{k-1} &= P_k + [u_k^2] L_k + [u_k^{-2}] R_k \\ \mathbf{G}^{(k-1)} &= \mathbf{G_{lo}}^{(k)} \cdot u_k^{-1} + u_k \cdot \mathbf{G_{hi}}^{(k)} \\ \mathbf{H}^{(k-1)} &= \mathbf{H_{lo}}^{(k)} \cdot u_k + u_k^{-1} \cdot \mathbf{H_{hi}}^{(k)} \end{align*} @@ -104,397 +113,96 @@ \subsubsection{Proving $\mathcal{R}'_{ip}$} $$P' + \sum_{i=1}^d ([u_i^2]L_i + [u_i^{-2}]R_i) = [a]G_1^{(0)} + [b]H_1^{(0)} + [ab]Q$$ outputs \textbf{accept} if equality holds and \textbf{reject} otherwise. \end{itemize} -\end{definition} - -\subsection{Range proofs} - -\subsection{Arithmetic circuits proofs} - -\subsection{STARK-friendly fields} -In general, STARK protocol can work over any field $\mathbb{F}$ with high two-adicity. The primary -reason for that is that STARKs can work only with NTT-friendly fields, and the -NTT-friendly fields are the fields where we can select the multiplicative -subgroup of order $2^k$ for sufficiently many values of $k$. - -\begin{definition} - We call \textbf{two-adicity fields}, the fields where we can select the - multiplicative subgroup of order $2^k$ for sufficiently many values of $k$. - In this case, the field order $p$ is typically of form $p = 2^m \cdot p' + - 1$ where $p'$ is a small integer. -\end{definition} - -To be honest, all protocol steps are followed with powers of two. It will be shown, why the groups we are working over must be of size $2^k$ and why the input data also follows this rule. As the result, the maximum size of the statement that we can prove using the STARK protocol is strictly depends on the size of two-adicity subgroup (that is why we label some fields to have \textit{high two-adicity} or \textit{low two-adicity}). - -\begin{remark} -In our initial discussion we consider using field over prime modulus $p = 3\cdot -2^{30} + 1$ and subgroups of size $2^{13}$ and $2^{10}$. -\end{remark} - -As we will work in the new subgroup we may want to specify the subgroup -generator to be used in future equations. So, for the multiplicative group -generator $w \in \mathbb{F}_p^{\times}$, the generator of the subgroup of order -$2^k$ is $\omega_k = w^\frac{p - 1}{2^k}$, as was shown in the NTT section. - -\begin{example} -For the prime field $\mathbb{F}_p$ where $p = 3\cdot 2^{30} + 1$, the order of $\mathbb{F}^{\times}_p$ is $p-1 = 3\cdot 2^{30}$. If we take $w = 5$ as the primitive element, the multiplicative subgroup of $2^{13}$ elements generator will be $\omega = 5^{3\cdot 2^{17}}$ -\end{example} - -This kind of subgroups comes with very useful property: for each element in -two-adicity subgroup $\mathbb{H}$, the additive inverse element -can be calculated by a simple equation over the element power. - -\begin{proposition} - Suppose $\mathbb{H} \leq \mathbb{F}_p$ is a subgroup of order $r$ with - generator $h = w^{(p-1)/r}$. Then, the additive inverse for $x = h^i \in - \mathbb{H}$ is $h^j$ where $j = i + \frac{r}{2} \pmod{r}$. -\end{proposition} - -\textbf{Proof.} The sum of $x$ and $-x$ must equal to zero modulo $p$, so: -\begin{equation*} - \begin{aligned} - x + (-x) &= w^{(p-1)i/r} + w^{(p-1)j/r} = w^{(p-1)i/r}(1 + w^{(p-1)(j-i)/r}) \\ &= w^{(p-1)i/r}(1 + w^{(p-1)/2}). - \end{aligned} -\end{equation*} - -Now note that $w^{(p-1)/2} = -1$ which completes the proof. $\blacksquare$ - -\begin{remark} -The equation $w^{p - 1} = 1$ is obtained from the order property of the -primitive element $w$ in the multiplicative group $\mathbb{F}^{\times}_p$. -\end{remark} -\begin{remark} -This provides us with an additional important property beyond element's power -computation: when working with a negative element, its power shift equals half -the size of the subgroup so, squaring the elements within this subgroup results -in a smaller subgroup, reduced by a factor of two. Consequently, to compute the -square of the subgroup, it suffices to square only the first half of its -elements (powers $0, 1, 2, 3, \dots, \frac{r}{2}$). -\end{remark} - -\subsection{Protocol definition} - -\subsubsection{Trace, evaluation domain and commitment} -Now, we are going to prove that some statement holds on the given sequence of elements. -\begin{definition} -We call \textbf{trace} a sequence of elements from $\mathbb{F}$ that represents our witness. This sequence contains private and public values together and follows certain constraints. + This protocol is illustrated in \Cref{fig:interactive_ip}. \end{definition} -\begin{example} -The \textbf{Fibonacci square sequence} is a sequence of elements defined over -$\mathbb{F}$ as follows: -\begin{equation*} -a_{j+2} = a_{j+1}^2 + a_{j}^2 -\end{equation*} - -Then we can, for example, prove the following statement: \textcolor{blue!60!black}{\textit{I know a field -element $w \in \mathbb{F}$ such that the $k^{\text{th}}$ element of the Fibonacci -square sequence ($a_k$) starting with $x$ and $w$ is $y$.}} Formally, -this can be written as: -\begin{equation*} - \mathcal{R}_{\text{Fib}} = \left\{ \begin{matrix} - \textbf{Public Statement:} \; (x, y, k) \\ - \textbf{Witness:} \; w - \end{matrix} \;\Big|\; \begin{matrix} - a_0 = x, a_1 = w, a_k = y \; \text{with} \\ a_{j+2} = a_{j+1}^2 + a_j^2 \; \text{for all} \; j \in [k] - \end{matrix} - \right\} -\end{equation*} - -For concreteness, let us take $k=1023$, $x = 1$, and $y=2338775057$. -\end{example} - -Following the Unisolvence Theorem, the trace $\{a_j\}_j$ is implied to be an evaluation -of some unknown \textbf{trace polynomial} of degree equal to the length of the -sequence $\{a_j\}_j$. Also, to be -evaluable on the two-adicity subgroup, the size of the trace has to be a power -of two. +\begin{figure}[h!] + \centering + % TikZ inner-product protocol diagram with highlighted loop + \begin{tikzpicture}[>=Stealth,thick,node distance=5cm,auto] + \usetikzlibrary{fit,backgrounds,positioning} + % Prover and Verifier nodes + \node[inner sep=0pt] (P) {\begin{tabular}{c}\includegraphics[width=1.2cm]{lectures/images/common/prover.png}\\Prover $\mathcal{P}$\end{tabular}}; + \node[inner sep=0pt,right=of P] (V) {\begin{tabular}{c}\includegraphics[width=1.2cm]{lectures/images/common/verifier.png}\\Verifier $\mathcal{V}$\end{tabular}}; + % Dashed lifelines extended + \draw[dashed,line width=0.3mm] ([yshift=-0cm]P.south) -- ++(0,-12cm); + \draw[dashed,line width=0.3mm] ([yshift=-0cm]V.south) -- ++(0,-12cm); + + % Step 1: Begin loop + \node[coordinate] (Ploopstart) at ([yshift=-1cm]P.south) {}; + \node[coordinate] (Vloopstart) at ([yshift=-1cm]V.south) {}; + + \node[draw,rounded corners,fill=white,below=1cm of P,align=left] (LR) {% + \small $L_{k} \gets \langle \mathbf{a_{lo}}^{(k)}, \mathbf{G_{hi}}^{(k)}\rangle + $\\ + \small $\langle \mathbf{b_{hi}}^{(k)}, \mathbf{H_{lo}}^{(k)}\rangle + [\langle \mathbf{a_{lo}}^{(k)}, \mathbf{b_{hi}}^{(k)}\rangle]Q $ \\ + \small $R_{k} \gets \langle \mathbf{a_{hi}}^{(k)}, \mathbf{G_{lo}}^{(k)}\rangle + $\\ + \small $\langle \mathbf{b_{lo}}^{(k)}, \mathbf{H_{hi}}^{(k)}\rangle + [\langle \mathbf{a_{hi}}^{(k)}, \mathbf{b_{lo}}^{(k)}\rangle]Q$ + }; + + % Step 2: Prover sends L_k, R_k + \node[coordinate] (Vloop1) at ([yshift=-3.5cm]V.south) {}; + \node[coordinate] (Ploop1) at ([yshift=-3.5cm]P.south) {}; + \draw[->] (Ploop1) -- node[above]{Send $(L_k,R_k)$} (Vloop1); + + % Step 3: Challenge + \node[draw,rounded corners,fill=white,below=4cm of V,align=left] (pchelV) {% + \small $u_k \xleftarrow{R} \mathbb{F}_p$ + }; + + + % Step 2: Verifier sends u_k + \node[coordinate] (Vloop2) at ([yshift=-5cm]V.south) {}; + \node[coordinate] (Ploop2) at ([yshift=-5cm]P.south) {}; + \draw[->] (Vloop2) -- node[above]{Send $u_k$} (Ploop2); + + % Step 3: Updates + \node[draw,rounded corners,fill=white,below=5.5cm of P,align=left] (updatesP) {% + \small $\mathbf{G}^{(k-1)}\gets \mathbf{G}_{lo}^{(k)}u_k^{-1}+u_k\mathbf{G}_{hi}^{(k)}$\\ + \small $\mathbf{H}^{(k-1)}\gets \mathbf{H}_{lo}^{(k)}u_k+u_k^{-1}\mathbf{H}_{hi}^{(k)}$\\ + \small $\mathbf{a}^{(k-1)}\gets \mathbf{a}_{lo}^{(k)}u_k+u_k^{-1}\mathbf{a}_{hi}^{(k)}$\\ + \small $\mathbf{b}^{(k-1)}\gets \mathbf{b}_{lo}^{(k)}u_k^{-1}+u_k\mathbf{b}_{hi}^{(k)}$ + }; + + \node[draw,rounded corners,fill=white,below=5.5cm of V,align=left] (updatesV) {% + \small $\mathbf{G}^{(k-1)}\gets \mathbf{G}_{lo}^{(k)}u_k^{-1}+u_k\mathbf{G}_{hi}^{(k)}$\\ + \small $\mathbf{H}^{(k-1)}\gets \mathbf{H}_{lo}^{(k)}u_k+u_k^{-1}\mathbf{H}_{hi}^{(k)}$ + }; + + % Loop region (highlighted) + \begin{scope}[on background layer] + \node[draw,rounded corners,fill=blue!10,inner sep=6pt,fit=(Ploopstart)(Vloopstart)(updatesP)(updatesV)(LR)] (loopBlock) {}; + \end{scope} + \node[above=0cm of loopBlock.north] {\small \textbf{Repeat for }$k \gets \overline{d..1}$}; + + % Final step: Prover sends (a,b) + \node[coordinate] (Pfinal) at ([yshift=-9cm]P.south) {}; + \node[coordinate] (Vfinal) at ([yshift=-9cm]V.south) {}; + \draw[->] (Pfinal) -- node[above]{Send $(a,b) = (\mathbf{a}^{(0)},\mathbf{b}^{(0)})$} (Vfinal); + + % Verifier final check + \node[draw,rounded corners,fill=white,below=0.5cm of Vfinal,align=left] (checkV) {% + \small Verify:\\ + $P'+\sum_{i=1}^d([u_i^2]L_i+[u_i^{-2}]R_i)=$\\ + $[a]G_1^{(0)}+[b]H_1^{(0)}+[ab]Q$ + }; + \end{tikzpicture} + \caption{Interactive inner-product protocol $\Pi'_{ip}$ between prover $\mathcal{P}$ and verifier $\mathcal{V}$ for relation $\mathcal{R}'_{ip}$} + \label{fig:interactive_ip} +\end{figure} -\begin{definition} -We call \textbf{domain} a two-adicity subgroup $\mathbb{G} \leq -\mathbb{F}^{\times}$ where we evaluate our polynomials. -\end{definition} -\begin{example} -In our example, we put trace a sequence $\{a_j\}_j$ of first $1023$ elements of -the Fibonacci square sequence over $\mathbb{F}_p$, where $p=3\cdot 2^{30} + 1$. -\begin{equation*} -1, 1, 2, 5, 29, \ldots -\end{equation*} -To interpolate our trace polynomial we select as a domain a two-adicity subgroup -of $2^{10}$ elements from $\mathbb{F}^\times_p$ with a generator $g = -5^{\frac{3\cdot 2^{30}}{2^{10}}} = 5^{3 \cdot 2^{20}}$ (here $5$ is the -primitive element in the multiplicative group $\mathbb{F}^\times_p$). That being -said, $\mathbb{G} = \{g^i\}_{i \in [1024]}$. -\end{example} - -Next, using the Lagrange interpolation over $(g^j, a_j)_{j \in [k]}$ points -we compute a trace polynomial $f \in \mathbb{F}[x]$. Note that the interpolation -can be done in $O(k\log k)$, as shown in NTT section. +\subsection{Range proofs} -\begin{definition} -We call \textbf{evaluation domain} a two-adicity coset $\mathbb{E} = w\mathbb{H} -\leq \mathbb{F}_p^{\times}$, where $\mathbb{H} \leq \mathbb{F}_p^{\times}$ is a -two-adicity subgroup, that is larger $\rho \in \mathbb{N}$ times (typically a -relatively small constant) than the domain. In other words, -$\text{ord}(\mathbb{H}) = \rho \cdot \text{ord}(\mathbb{G})$. -\end{definition} +\subsection{Arithmetic circuits proofs} -\begin{example} -In our case we select a two-adicity subgroup $\mathbb{H}$ of $2^{13}$ elements -from $\mathbb{F}_p^\times$ with $\rho = 8$ as $\mathbb{H} = \{h^i\}_{i \in -[8192]}$ where $h = 5^{3 \cdot 2^{17}}$. Then, we define the \emph{evaluation -domain} as $\mathbb{E}=5\mathbb{H} = \{5h^i\}_{i \in [8192]}$. -\end{example} - -We build a Merkle tree over the values $\{f(e)\}_{e \in \mathbb{E}}$ and label -its root as a \textbf{trace polynomial commitment}. This approach will also be -used to commit other polynomials during the protocol walkthrough. - -The \textbf{constraints} in STARK protocol are expressed as polynomials -evaluated over the trace cells, which are satisfied if and only if the -computations are correct. - -\begin{example} -Obviously, our initial statement consists of the following three requirements: -\begin{enumerate} - \item The element $a_0$ is equal to $1$; - \item The element $a_{1022}$ is equal to $2338775057$; - \item Each element $a_{i+2}$ is equal to $a_{i+1}^2 + a_{i}^2$. -\end{enumerate} -\end{example} -To verify that our committed trace polynomial satisfies all constraints, we can -check that it has corresponding roots. In particular, according to the selected -interpolation points $\{(g^i, a_i)\}_{i \in [k]}$, the relation $r(a_i, a_j) = -0$ can be rewritten as $r(f(g^i), f(g^j)) = 0$. -\begin{example} -For our Fibonacci trace we have the following constraints to be checked over the interpolated polynomial: -\begin{enumerate} - \item \textit{The element $a_0$ is equal to $1$} translated to: $f(x)-1$ has root at $x = g^0 = 1$; - \item \textit{The element $a_{1022}$ is equal to $2338775057$} translated to: $f(x) - 2338775057$ has root at $x = g^{1022}$; - \item \textit{Each element $a_{i+2}$ is equal to $a_{i+1}^2 + a_{i}^2$} translated to: $f(g^2x) - f(gx)^2 - f(x)^2$ has roots in $\mathbb{G} \setminus \{g^{1021}, g^{1022}, g^{1023}\}$ -\end{enumerate} -\end{example} - -To ensure that the specified polynomials have roots in given values, we can use the following property: if polynomial $f(x) \in \mathbb{F}[x]$ has root in $x_0$ then the $\frac{f(x)}{x - x_0}$ is also a polynomial in $\mathbb{F}[x]$. - -\begin{example} -Finally, we define the following STARK constraints: -\vspace{-1mm} -\begin{gather*} - p_0(x) = \frac{f(x)-1}{x - 1} \\ - p_1(x) = \frac{f(x) - 2338775057}{x - g^{1022}} \\ - p_2(x) = \frac{f(g^2x) - f(gx)^2 - f(x)^2}{\prod_{i=0}^{1020} (x - g^i)} -\end{gather*} -\vspace{-1mm} -Unfortunately, the $p_2$ polynomial still looks inconvenient to work with, so we -may want to simplify it (this is not a part of the protocol in general, but you -always may want to simplify your equations to achieve better proving time). Note -that $p_2$ is \textit{almost} a vanishing polynomial of $\mathbb{G}$, which has -a form $x^{\text{ord}(\mathbb{G})} - 1$, except for points $g^{1021}, -g^{1022}, g^{1023}$. In other words, we can simplify the denominator as: -\begin{xequation*} - \prod_{i=0}^{1020} (x - g^i) = \frac{x^{1024} - 1}{(x-g^{1021})(x-g^{1022})(x-g^{1023})} -\end{xequation*} - -Note, that while evaluating our polynomial on a larger domain -then $\mathbb{G}$ we should only ensure that the resulting polynomial still -holds the relation $f(g^i) = a_i$, so it is acceptable to use properties that -only work over $\mathbb{G}$. So, finally we have: - \begin{xequation} - p_2(x) = \frac{(f(g^2x) - f(gx)^2 - f(x)^2)(x - g^{2021})(x - g^{2022})(x - g^{2024})}{x^{1024} - 1} - \end{xequation} -\end{example} - -In addition, there is one obvious requirement for the STARK constraints: the -verifier should be able to compute the constraints polynomials $p_i(x)$ using -only the given trace polynomial evaluations for the certain $x$. - -\begin{remark} -In our Fibonacci example, verifier can check the constraint polynomials -evaluation by requesting only $f(x)$, $f(gx)$ and $f(g^2x)$ --- the values -committed in the trace polynomial commitment. -\end{remark} - -To combine all our constraints into a single polynomial, we can follow a -commonly used principle by taking a linear combination with the challenges from -the verifier. In particular, after receiving trace polynomial commitment from -the prover, the verifier selects scalars $\alpha_1,\dots,\alpha_m$ and sends it -to the prover. Then, the prover puts the \textbf{composition polynomial} as: -\begin{xequation*} - \text{CP}(x) := \sum_{j = 1}^m \alpha_j\cdot p_j(x) -\end{xequation*} -Additionally, prover also commits this polynomial by evaluating on the evaluation domain and building a Merkle tree. - -\begin{example} -The Fibonacci composition polynomial looks like as follows: - \begin{gather*} - \text{CP}(x) = \alpha_0 p_0(x) + \alpha_1 p_1(x) + \alpha_2 p_2(x) =\\ - \alpha_0 \frac{f(x)-1}{x - 1} + \alpha_1 \frac{f(x) - 2338775057}{x - g^{1022}} + \\ - \alpha_2 \frac{(f(g^2x) - f(gx)^2 - f(x)^2)(x - g^{2021})(x - g^{2022})(x - g^{2024})}{x^{1024} - 1} - \end{gather*} -\end{example} - -\subsubsection{FRI protocol} -In general, our goal is to verify that the committed polynomial $\text{CP}(x)$ -satisfies all our constraints, by checking it's evaluation at a random point -from the evaluation domain that the verifier selects. Anyway, we can face the -problem when the malicious prover constructs a larger polynomial that accepts -lots of possible roots from our field (even $2^{64}$ field is still insecure for -just checking the evaluation at one point). That is why we have to make sure -that the committed polynomial degree lies in the acceptable range (the upper -bound depends on the trace size). - -The final stage of the STARK protocol is a \textbf{Fast Reed-Solomon IOP of Proximity (FRI)}. FRI is a protocol between a prover and a verifier, which establishes that a given evaluation belongs to a polynomial of low-degree. In this context \textit{low} means no more than $\rho$ times bigger than the trace. - -The key idea of FRI protocol is to move from a polynomial of degree $n$ to a -polynomial of degree $n/2$ until we get a constant value. Let's consider the -polynomial $z_0(x) = \sum_i a_i\cdot x^i$ of degree $n=2^t$ and the evaluation -domain $\mathbb{E}_0 = \mathbb{E}$. We suppose to group the \textit{odd} and the -\textit{even} coefficients of the $z_0$ together into the two separate -polynomials($z_0^O$ and $z_0^E$ respectively): -\begin{xequation*} - \begin{aligned} - z_0^O(x^2) = \sum_{i=0}^{n/2} (a_{2i+1}\cdot x^{2i}), \quad z_0^E(x^2) = \sum_{i=0}^{n/2} (a_{2i}\cdot x^{2i}) -\end{aligned} -\end{xequation*} - -Or, in a more comfortable form (we have already examined why searching of $-x$ -can be done easily in our two-adicity subgroup): -\begin{xequation*} - \begin{aligned} - z_0^E(x^2) = \frac{z_0(x) + z_0(-x)}{2}, \quad z_0^O(x^2) = \frac{z_0(x) - z_0(-x)}{2x} - \end{aligned} -\end{xequation*} - -Then, we define a next-layer of the FRI polynomial as $z_1(x^2) = z_0^E(x^2) + -\beta z_0^O(x^2)$, where $\beta$ is a challenge received from verifier. The -next-layer evaluation domain is also simple to compute: $\mathbb{E}_1 = -\{(w\cdot h_i)^2\}_{i \in [\text{ord}(\mathbb{E}_0)/2]}$ as squaring the -other elements in $\mathbb{E}_0$ will result in the same values. - -Next, we commit to the $z_1(x^2)$ using a next-layer evaluation domain -$\mathbb{E}_1$ (is also reduced by a factor two) and continue to repeat the -described operations until $z_j(x^{2^j})$ becomes constant. - -\vspace{-2mm} - -\begin{tcolorbox}[title=Interactive ZK-STARK protocol, - colback=blue!5!white, - colframe=blue!75!black, - colbacktitle=blue!25!white, - coltitle=blue!20!black, - fonttitle=\bfseries, - boxrule=1.25pt, - subtitle style={boxrule=0pt, - colback=blue!20!white, - colupper=blue!75!gray} ] - \small - - The prover and the verifier run the interactive version of the ZK-STARK - protocol. Both know the statement to be proved, that is defined by the - constraint polynomials and the field $\mathbb{F}_p$ to work over. Prover also - knows the witness to be able to generate the trace. - - \tcbsubtitle{Preparation} - \begin{itemize}[label=\ding{51}] - \item The prover interpolates trace polynomial $f(x)$ and submits its - commitment to the verifier. - \item The verifier selects challenges random $\alpha_i \in \mathbb{F}_p$ - and sends to the prover. - \item The prover builds the composition polynomial $\text{CP}(x)$ and - submits its commitment to the verifier. - \end{itemize} - \tcbsubtitle{FRI} - \begin{itemize}[label=\ding{51}] - \item The verifier selects random $j \in [\text{ord}(\mathbb{E})]$, sets - $c \gets w\cdot h^j$ and sends it to the prover. - \item The prover responds with the $\text{CP}(c), \text{CP}(-c)$ and all - $f(x)$ required to check $\text{CP}$ evaluation with corresponding Merkle - proofs to them. - \item The verifier checks Merkle proofs and the evaluation of - $\text{CP}(c)$ by evaluating the constraints polynomials $p_j(c)$. - \item The prover and the verifier go through the FRI protocol for - $z_0(x) = CP(x)$ where the prover commits to the layer-$j$ polynomial - $z_j(x)$, the verifier selects a challenge $\beta$ and queries from the - prover $z_j(c), z_j(-c)$ to compute $z_{j+1}(c)$ until $z_k(x), j \leq - \log_2(\deg \text{CP})$ becomes constant. - \end{itemize} - -\end{tcolorbox} - -The non-interactive version of the presented protocol can be easily built -obtaining the Fiat-Shamir heuristics. - -The soundness of the presented STARK protocol follows from the impossibility to -commit any possible evaluation of the forgery $\text{CP}(x)$ over evaluation -domain $\mathbb{E}$ and simultaneously prove that $\text{CP}(x)$ is a low-degree -polynomial by the FRI protocol. Since the size of $\mathbb{E}$ is $\rho$ times bigger -then the maximum allowed polynomial degree (that directly depends on the size of -the trace), the attacker either can't construct such a polynomial or can't -construct a low-degree polynomial, so a valid low-degree composition polynomial -can only be obtained using a valid trace. - -\vspace{-2mm} - -\begin{example} - Finally, let's overview the first steps of the ZK-STARK protocol applied to our Fibonacci example: - - \begin{enumerate} - \item The protocol defines the public constraints such as 2023-th - element of sequence, field $\mathbb{F}_p$, etc. - \item The prover generates the trace $a$ where $a_0 = 1, a_1 = 3141592, - a_i = a_{i-1}^2 + a_{i-2}^2$, evaluates the trace polynomial $f(x)$ over - the evaluation domain and sends it's commitments to the verifier. - \item The verifier selects challenges $\alpha_0, \alpha_1, \alpha_2 \in - \mathbb{F}$ and shares them with the prover. - \item The prover evaluates the composition polynomial $\text{CP}(x)$ - over evaluation domain and sends it's commitments to the verifier. - \item The verifier selects random $i \in [8192-16]$, puts $c = 5\cdot - h^i$ and sends it to the prover. - \item The prover responds with the $f(c), f(gc), f(g^2c), \text{CP}(c), - \text{CP}(-c)$ and corresponding Merkle proofs to them. - \item The verifier checks Merkle proofs and the evaluation of - $\text{CP}(c)$ by evaluating the constraint polynomials $p_0(c), p_1(c), - p_2(c)$. - \item The prover and the verifier go through the FRI protocol for - $z_0(x) = \text{CP}(x)$ until $z_i(x), i \in [12]$ becomes constant. - \end{enumerate} - -\end{example} - -\subsection{Protocol security} -Most of the existing versions of the STARK protocol leverage on several -optimizations to achieve better proving and verification time. The key point -here is that each FRI query check adds $\log_2(\rho)$ bits of security, so we can -skip some of these checks if the security level is already satisfied. One more -optimization is to include a proof-of-work computation into the protocol that -should be done before FRI with dependency on the committed values. It can be -useful because the verification of the proof-of-work is less expensive then the -verification of the FRI step while still increases the computation cost for the -malicious prover. - -More precisely, let's assume that the desired security level of the protocol is -$\lambda$. First of all, we obviously have to use a proper collision-resistant -hash function with $2\lambda$ bits output. Then, according to the StarkWare's -definition of the STARK protocol, the resulting security is defined as follows: -\begin{xequation*} - \lambda \geq \min\{ \delta + \log_2(\rho) \cdot s, \log_2(|\mathbb{F}|) \} - 1 -\end{xequation*} -where $\delta$ -- number of the proof-of-work bits, $s$ -- number of the FRI queries. - -\begin{example} -If the protocol is deployed over $256$-bit field and the domain ratio is $\rho = -8$, to achieve the $128$ bit security we can for example execute $33$ FRI query -and evaluate $29$ proof-of-work bits: $\min\{29+3\cdot 33, 256\} = 128$. -\end{example} \subsection*{Acknowledgements} -This work was inspired by \href{https://starkware.co/stark-101/}{``STARK-101''} -course by StarkWare and -\href{https://vitalik.eth.limo/general/2017/11/09/starks_part_1.html}{``STARKs''} -series by Vitalik Buterin. +\begin{itemize} + \item \href{https://doc-internal.dalek.rs/bulletproofs/index.html}{dalek's crate bulletproofs} + \item \href{https://eprint.iacr.org/2017/1066.pdf}{bulletproofs whitepaper} +\end{itemize} \end{document} From edbc72c81f691af6718c5161906d5cb0ac0229dc Mon Sep 17 00:00:00 2001 From: Yevhen Hrubiian Date: Fri, 11 Jul 2025 19:25:04 +0300 Subject: [PATCH 04/25] improv. --- lectures/2-9-bulletproofs.tex | 37 ++++++++++++++++++++++++++--------- 1 file changed, 28 insertions(+), 9 deletions(-) diff --git a/lectures/2-9-bulletproofs.tex b/lectures/2-9-bulletproofs.tex index 9253672..b13089e 100644 --- a/lectures/2-9-bulletproofs.tex +++ b/lectures/2-9-bulletproofs.tex @@ -44,41 +44,44 @@ \subsubsection{Inner-product compression} Firstly, assuming that $n = 2^d$ define by $\mathbf{G_{lo}} = (G_1, \dots, G_{n/2}), \mathbf{G_{hi}} = (G_{n/2+1},\dots, G_n) \in \mathbb{G}^{n/2}$ -- lower and higher halves of vector $\mathbf{G}$ and $\mathbf{a_{lo}} = (a_1, \dots, a_{n/2}), \mathbf{a_{hi}} = (a_{n/2+1},\dots,a_n) \in \mathbb{F}_{n/2}$ -- lower and higher halves of $\mathbf{a} \in \mathbb{F}_p^{n}$. Let $u_k \in \mathbb{F}_p$ - be some scalar, define compressed vectors: -\begin{align} +\begin{align*} \mathbf{a}^{(k-1)} &= \mathbf{a_{lo}} \cdot u_k + u_k^{-1} \cdot \mathbf{a_{hi}} \\ \mathbf{b}^{(k-1)} &= \mathbf{b_{lo}} \cdot u_k^{-1} + u_k \cdot \mathbf{b_{hi}} \\ \mathbf{G}^{(k-1)} &= \mathbf{G_{lo}} \cdot u_k^{-1} + u_k \cdot \mathbf{G_{hi}} \\ \mathbf{H}^{(k-1)} &= \mathbf{H_{lo}} \cdot u_k + u_k^{-1} \cdot \mathbf{H_{hi}} -\end{align} +\end{align*} Define $P_k \gets P' = \langle \mathbf{a,G} \rangle + \langle \mathbf{b,H} \rangle + [\langle \mathbf{a,b} \rangle]Q$ and define $P_{k-1}$ using compressed vectors to have the same form as $P_k$: $$P_{k-1} = \langle \mathbf{a}^{(k-1)}, \mathbf{G}^{(k-1)} \rangle + \langle \mathbf{b}^{(k-1)}, \mathbf{H}^{(k-1)} \rangle + [\langle \mathbf{a}^{(k-1)}, \mathbf{b}^{(k-1)} \rangle]Q $$ Subsituting compressed vectors and applying bilinearity property of inner product we get: -\begin{align} +\begin{align*} P_{k-1} = & \langle \mathbf{a_{lo}}, \mathbf{G_{lo}}\rangle + \langle \mathbf{a_{hi}}, \mathbf{G_{hi}}\rangle &+ u_k^2\langle \mathbf{a_{lo}}, \mathbf{G_{hi}}\rangle + u_k^{-2}\langle \mathbf{a_{hi}}, \mathbf{G_{lo}}\rangle + \\ & \langle \mathbf{b_{lo}}, \mathbf{H_{lo}}\rangle + \langle \mathbf{b_{hi}}, \mathbf{H_{hi}}\rangle &+ u_k^2\langle \mathbf{b_{hi}}, \mathbf{H_{lo}}\rangle + u_k^{-2}\langle \mathbf{b_{lo}}, \mathbf{H_{hi}}\rangle + \\ & [\langle \mathbf{a_{lo}}, \mathbf{b_{lo}}\rangle + \langle \mathbf{a_{hi}}, \mathbf{b_{hi}}\rangle]Q &+ [u_k^2\langle \mathbf{a_{lo}}, \mathbf{b_{hi}}\rangle + u_k^{-2}\langle \mathbf{a_{hi}}, \mathbf{b_{lo}}\rangle]Q -\end{align} +\end{align*} Note that $\langle \mathbf{a_{lo}}, \mathbf{G_{lo}}\rangle + \langle \mathbf{a_{hi}}, \mathbf{G_{hi}}\rangle = \langle \mathbf{a,G}\rangle$ so that the first two columns of $P_{k-1}$ definition precisecly contains $P_{k} = P'$: $$P_{k} = \langle \mathbf{a_{lo}}, \mathbf{G_{lo}}\rangle + \langle \mathbf{a_{hi}}, \mathbf{G_{hi}}\rangle + \langle \mathbf{b_{lo}}, \mathbf{H_{lo}}\rangle + \langle \mathbf{b_{hi}}, \mathbf{H_{hi}}\rangle + [\langle \mathbf{a_{lo}}, \mathbf{b_{lo}}\rangle + \langle \mathbf{a_{hi}}, \mathbf{b_{hi}}\rangle]Q$$ Define cross-terms $L_k, R_k$ of $P_{k-1}$ such that: -\begin{align} +\begin{align*} P_{k-1} &= P_k + [u_k^2] L_k + [u_k^{-2}] R_k \\ L_{k} &= \langle \mathbf{a_{lo}}, \mathbf{G_{hi}}\rangle + \langle \mathbf{b_{hi}}, \mathbf{H_{lo}}\rangle + [\langle \mathbf{a_{lo}}, \mathbf{b_{hi}}\rangle]Q \\ R_{k} &= \langle \mathbf{a_{hi}}, \mathbf{G_{lo}}\rangle + \langle \mathbf{b_{lo}}, \mathbf{H_{hi}}\rangle + [\langle \mathbf{a_{hi}}, \mathbf{b_{lo}}\rangle]Q -\end{align} +\end{align*} Here we come up with some kind of statement compression algorithm reducing size of all vectors in half per compression step. Repeating comression algorithm $k$ times we end up with vectors $\mathbf{a}^{(0)}, \mathbf{b}^{(0)}, \mathbf{G}^{(0)}, \mathbf{H}^{(0)}$ each of length one and $P_0$ containing all accumulated cross-terms: -\begin{align} +\begin{align*} P_0 &= [a_1^{(0)}]G_1^{(0)} + [b_1^{(0)}]H_1^{(0)} + [a_1^{(0)}b_1^{(0)}]Q \\ P_0 &= P_k + \sum_{i=1}^d ([u_i^2]L_i + [u_i^{-2}]R_i) -\end{align} +\end{align*} Recalling that $P_k = P'$, the final compressed statement asserting inner-product value will have the following form: -$$P' + \sum_{i=1}^d ([u_i^2]L_i + [u_i^{-2}]R_i) = [a_1^{(0)}]G_1^{(0)} + [b_1^{(0)}]H_1^{(0)} + [a_1^{(0)}b_1^{(0)}]Q$$ +\begin{equation} + P' + \sum_{i=1}^d ([u_i^2]L_i + [u_i^{-2}]R_i) = [a_1^{(0)}]G_1^{(0)} + [b_1^{(0)}]H_1^{(0)} + [a_1^{(0)}b_1^{(0)}]Q + \label{eq:ip-final-compressed} +\end{equation} \subsubsection{Proving $\mathcal{R}'_{ip}$} @@ -191,7 +194,23 @@ \subsubsection{Proving $\mathcal{R}'_{ip}$} \label{fig:interactive_ip} \end{figure} +As we can see, overall communication complexity of $\Pi_{ip}$ is $2\log_2 n \mathbb{G} + 2 \mathbb{F}_p$ so we come up with logarithmic proof size for our inner-product relation $\mathcal{R}_{ip}$. Now one might ask whether $\Pi_{ip}$ has any desired properties such as completeness, soundness and zero-knowledge. It appears that the first two holds under some generalizations needed for security proofs but not zero-knowledge (indeed, if $n=1$ then $\mathcal{P}$ sends witness pair $a,b$ directly). +\begin{theorem}[Inner-Product Argument] + The argument system $\Pi_{ip}$ for relation $\mathcal{R}_{ip}$ has \textit{perfect completeness and statistical witness-extended emulation} for either extracting a non-trivial discrete logarithm relation between $\mathbf{G,H}, Q$ or extracting valid witness $\mathbf{a,b}$. +\end{theorem} + +\textbf{Proof idea}. \textit{Perfect completeness} of $Pi_{ip}$ follows because $Pi_{ip}$ converts instance of $\mathcal{R}_{ip}$ to instance of $\mathcal{P}'_{ip}$ and $\Pi'_{ip}$ is trivially complete by construction due to \Cref{eq:ip-final-compressed}. Notation \textit{statistical witness-extended emulation} generalizes special soundness in the way applicable for multi-stage complex argument systems where each step of the protocol could be rewinded to extract part of the witness so that more accurate definition of protocol security is achieved despite \textit{special soundness} implies building the whole knowledge extractor which might has non-polynomial running time for multi-stage protocols. + +Here we briefly describe a knowledge extractor $\mathcal{E}$ for a witness $(\mathbf{a}^{(k)}, \mathbf{b}^{(k)})$ or non-trivial discrete logarithm relation for $(\mathbf{G, H}, Q)$ for one stage of protocol. +\begin{enumerate} + \item $\mathcal{E}$ runs $\mathcal{P}$ to the last stage to obtain $(L, R) \gets (L_1, R_1)$ + \item $\mathcal{E}$ rewinds the $\Pi'_{ip}$ last stage four times to obtain four challenges $x_1, x_2, x_3, x_4$, so that the following equations holds from expressing $P_{1}$: + \begin{align} + [x_1^2] L_k + P_k + [x_1^{-2}] R_k = \langle \mathbf{a}^{(k-1)}, \mathbf{G}^{(k-1)} \rangle + \langle \mathbf{b}^{(k-1)}, \mathbf{H}^{(k-1)} \rangle + [\langle \mathbf{a}^{(k-1)}, \mathbf{b}^{(k-1)} \rangle]Q + \end{align} + \item +\end{enumerate} \subsection{Range proofs} \subsection{Arithmetic circuits proofs} From a8bdb34e43e9c018f9665a7ec43d2ca6558809d6 Mon Sep 17 00:00:00 2001 From: Yevhen Hrubiian Date: Mon, 14 Jul 2025 09:31:29 +0300 Subject: [PATCH 05/25] improvs. --- lectures/2-9-bulletproofs.tex | 13 +++++++++---- 1 file changed, 9 insertions(+), 4 deletions(-) diff --git a/lectures/2-9-bulletproofs.tex b/lectures/2-9-bulletproofs.tex index b13089e..55cbc48 100644 --- a/lectures/2-9-bulletproofs.tex +++ b/lectures/2-9-bulletproofs.tex @@ -1,4 +1,5 @@ \documentclass[../lecture-notes-148x210.tex]{subfiles} +\usepackage{systeme} \begin{document} \subsection{Introduction} @@ -204,12 +205,16 @@ \subsubsection{Proving $\mathcal{R}'_{ip}$} Here we briefly describe a knowledge extractor $\mathcal{E}$ for a witness $(\mathbf{a}^{(k)}, \mathbf{b}^{(k)})$ or non-trivial discrete logarithm relation for $(\mathbf{G, H}, Q)$ for one stage of protocol. \begin{enumerate} - \item $\mathcal{E}$ runs $\mathcal{P}$ to the last stage to obtain $(L, R) \gets (L_1, R_1)$ - \item $\mathcal{E}$ rewinds the $\Pi'_{ip}$ last stage four times to obtain four challenges $x_1, x_2, x_3, x_4$, so that the following equations holds from expressing $P_{1}$: + \item $\mathcal{E}$ runs $\mathcal{P}$ to the last stage to obtain $(L, R) \gets (L_1, R_1)$. and $(a,b) \gets $ + \item $\mathcal{E}$ rewinds the $\Pi'_{ip}$ final stage four times to obtain four challenges $x_1, x_2, x_3, x_4$ and responses $(\mathbf{a}_1^{(0)}, \mathbf{b}_1^{(0)}), (\mathbf{a}_2^{(0)}, \mathbf{b}_2^{(0)}), (\mathbf{a}_3^{(0)}, \mathbf{b}_3^{(0)}), (\mathbf{a}_4^{(0)}, \mathbf{b}_4^{(0)})$ so that the following equation holds for $i \in \{ 1,2,3,4 \}$: \begin{align} - [x_1^2] L_k + P_k + [x_1^{-2}] R_k = \langle \mathbf{a}^{(k-1)}, \mathbf{G}^{(k-1)} \rangle + \langle \mathbf{b}^{(k-1)}, \mathbf{H}^{(k-1)} \rangle + [\langle \mathbf{a}^{(k-1)}, \mathbf{b}^{(k-1)} \rangle]Q + [x_i^2] L_1 + P_1 + [x_i^{-2}] R_1 = \langle \mathbf{a}_i^{(0)}, \mathbf{G}^{(0)} \rangle + \langle \mathbf{b}_i^{(0)}, \mathbf{H}^{(0)} \rangle + [\langle \mathbf{a}_i^{(0)}, \mathbf{b}_i^{(0)} \rangle]Q \end{align} - \item + \item $\mathcal{E}$ choses $(v_1, v_2, v_3) \in \mathbb{F}_p^3 $ as a solution for the system of linear equations: + $$\systeme{x_1^2 v_1 + x_2^2 v_2 + x_3^2 v_3=0, v_1+v_2+v_3=1, x_1^{-2} v_1 + x_2^{-2} v_2 + x_3^{-2} v_3=0}$$ + \label{eq:extractor_eq} + \item $\mathcal{E}$ takes linear combination of \ref{eq:extractor_eq} with coefficients $v_i$ for $i \in \{ 1,2,3 \}$: + $$ \end{enumerate} \subsection{Range proofs} From d0f1e7bae6691cc20d8e9e1d4cc63cc995031c97 Mon Sep 17 00:00:00 2001 From: Yevhen Hrubiian Date: Mon, 14 Jul 2025 19:55:36 +0300 Subject: [PATCH 06/25] further work --- lectures/2-9-bulletproofs.tex | 171 +++++++++++++++++++++++++++++----- 1 file changed, 150 insertions(+), 21 deletions(-) diff --git a/lectures/2-9-bulletproofs.tex b/lectures/2-9-bulletproofs.tex index 55cbc48..007c665 100644 --- a/lectures/2-9-bulletproofs.tex +++ b/lectures/2-9-bulletproofs.tex @@ -4,17 +4,70 @@ \begin{document} \subsection{Introduction} -\textbf{Bulletproofs} is a non-interactive zero-knowledge proof protocol with logarithmically sized proofs without a trusted setup. Originally, \textbf{bulletproofs} was developed to provide efficient inner-product proofs for range proofs in application to confidential transactions, but it applies to arbitrary arithmetic circuit (possibly encoded in R1CS). Technically, the protocol is built in an interactive fashion (like $\Sigma$-protocols) and made non-interactive with a Fiat-Shamir transform. One key feature that differs it from $\Sigma$-protocols is the number of challenges from a verifier $\mathcal{V}$: in $\Sigma$-protocols there is only one challenge, while \textbf{bulletproofs} implies a logarithmic in circuit size number of queries. The main advantages of \textbf{bulletproofs} are an absence of a trusted setup, security that relies on the \textit{discrete-logarithm} assumption without any other auxiliary structures like bilinear pairings, however, the main disadvantage of \textbf{bulletproofs} is linear in circuit size verification time though still efiicient for small circuits. +\textbf{Bulletproofs} is a non-interactive zero-knowledge proof protocol with logarithmically sized proofs without a trusted setup. Originally, \textbf{bulletproofs} was developed to provide efficient inner-product proofs for range proofs in application to confidential transactions, but it applies to arbitrary arithmetic circuit (possibly encoded in R1CS). Technically, the protocol is built in an interactive fashion (like $\Sigma$-protocols) and made non-interactive with a Fiat-Shamir transform. One key feature that differs it from $\Sigma$-protocols is the number of challenges from a verifier $\mathcal{V}$: in $\Sigma$-protocols there is only one challenge, while \textbf{bulletproofs} implies a logarithmic in circuit size number of queries. The main advantages of \textbf{bulletproofs} are an absence of a trusted setup, security that relies on the \textit{discrete-logarithm} assumption without any other auxiliary structures like bilinear pairings, however, the main disadvantage of \textbf{bulletproofs} is linear in circuit size verification time though still efficient for small circuits. Hence, we will describe some preliminaries. +\subsection{Notation} +Let $\mathbb{G}$ - cyclic group of prime order $p$ written additively, $\mathbf{G} = (G_1, \dots, G_n), \mathbf{H} = (H_1, \dots, H_n) \in \mathbb{G}^n$ - vectors of independent generators. We denote by $\langle \mathbf{a,b} \rangle$ - inner product of vectors $\mathbf{a} = (a_1, \dots, a_n), \mathbf{b} = (b_1, \dots, b_n) \in \mathbb{F}_p^n$ and $\langle \mathbf{a,G} \rangle = \sum_{i=1}^n [a_i] G_i \in \mathbb{G}$ - inner product of vector $\mathbf{a}$ with vector of generators $\mathbf{G}$. -\subsection{Proving non-linear relations with $\Sigma$-protocols} +\subsection{Zero-knowledge multiplication} +Let $a,b,c \in \mathbb{F}_p$. Here we build a zero-knowledge protocol for relation $\mathcal{R}'_{mul} = \{ c; a,b: c=ab \}$. We use well-known $\Sigma$-protocol framework for that, but firstly we make very useful generalization that could allow us to prove much larger class of relations. -\subsection{Inner-product argument} +Consider the first-degree polynomials $l(x) = a + s_L x, r(x) = b + s_R x \in \mathbb{F}_p[x]$. Let $t(x) = l(x)r(x)$ and relation +$$\mathcal{R}_{mul} = \{ (;l(x),r(x),t(x))\in (\emptyset \times \mathbb{F}_p^3) \vert t(x) = l(x)r(x)\}$$ +Firstly, observe that proving $t(x) = l(x)r(x)$ may be reduced to evaluation check at some challenge point $u \in \mathbb{F}_p$: $t(u) = l(u)r(u)$, due to the \textit{Schwartz-Zippel lemma}: +$$\mathsf{Pr}[l(u)r(u) = t(u) \vert l(x)r(x) \neq t(x)] \le \frac{max(\deg(l(x)r(x)), \deg(t(x)))}{p} = \frac{2}{p}$$ +is typycally negligible function from security level which makes this check \textit{sound}. + +Let's describe naїve unoptimized version \textbf{polynomial multiplication} protocol $\Pi'_{mul} = (\mathsf{Setup},\mathcal{P,V})$ for relation $\mathcal{R}_{mul}$. +During $\mathsf{Setup}$ parties agree on group elements $G,B \in \mathbb{G}$. After that parties involve in the following protocol: +\begin{itemize} + \item Prover $\mathcal{P}$ computes: + $$t(x) = l(x)r(x) = (a+s_L x)(a+s_R x) = ab + (as_R + bs_L) + s_L s_R x^2$$ + \item Prover $\mathcal{P}$ draws blinding factors $\alpha_0, \alpha_1, \beta_0, \beta_1, \tau_0, \tau_1, \tau_2 \xleftarrow{R} \mathbb{F}_p$ forming blinding polynomials + $$\alpha(x) = \alpha_0 + \alpha_1 x, \beta(x) = \beta_0 + \beta_1 x, \tau(x) = \tau_0 + \tau_1 x + \tau_2 x^2$$ + and sends to $\mathcal{V}$ Pedersen commitments for each coefficient of $l(x), r(x), t(x)$: + \begin{equation} + \begin{aligned} + L_0 &= [a]G + [\alpha_0]B \\ + L_1 &= [s_L]G + [\alpha_1]B \\ + &\\ + R_0 &= [b]G + [\beta_0]B \\ + R_1 &= [s_R]G + [\beta_1]B \\ + &\\ + T_0 &= [ab]G + [\tau_0]B \\ + T_1 &= [as_R + bs_L]G + [\tau_1]B \\ + T_2 &= [s_L s_R]G + [\tau_2]B + \end{aligned} + \end{equation} + \item Verifier $\mathcal{V}$ samples and sends to $\mathcal{P}$ random evaluation point $u \xleftarrow{R} \mathbb{F}_p$ + \item Prover $\mathcal{P}$ evaluates $l(x),r(x),t(x)$ and $\alpha(x), \beta(x), \tau(x)$ at $u$: + \begin{equation} + \begin{aligned} + l_u & = a + s_L u\\ + r_u &= b + s_R u\\ + t_u &= l_u r_u\\ + \alpha_u &= \alpha_0 + \alpha_1 u\\ + \beta_u &= \beta_0 + \beta_1 u\\ + \tau_u &= \tau_0 + \tau_1 u + \tau_2 u^2 + \end{aligned} + \end{equation} + and sends $(l_u, r_u, t_u, \alpha_u, \beta_u, \tau_u)$ to $\mathcal{V}$. + \item Verifier $\mathcal{V}$ performs checks: + \begin{equation} + \begin{aligned} + [l_u]G + [\alpha_u]B &\stackrel{\text{?}}{=} L_0 + [u]L_1 \\ + [r_u]G + [\beta_u]B &\stackrel{\text{?}}{=} R_0 + [u]R_1 \\ + [t_u]G + [\tau_u]B &\stackrel{\text{?}}{=} T_0 + [u]T_1 + [u^2]T_2 \\ + t_u &\stackrel{\text{?}}{=} l_u r_u + \end{aligned} + \end{equation} +\end{itemize} -Firstly, we describe the protocol for the \textbf{inner-product argument} - core component of the \textbf{bulletproofs} protocol. After that we will apply it to range proofs and arithmetic circuits. We have already seen that inner-products are the main ingridients for R1CS language because any R1CS relation could be seen as a batch of inner-products though it's not the most efficient representation and we'll see how to amortize all the constraints into inner-products more efficiently. -Let $\mathbb{G}$ - cyclic group of prime order $p$ written additively, $\mathbf{G} = (G_1, \dots, G_n), \mathbf{H} = (H_1, \dots, H_n) \in \mathbb{G}^n$ - vectors of independent generators. We denote by $\langle \mathbf{a,b} \rangle$ - inner product of vectors $\mathbf{a} = (a_1, \dots, a_n), \mathbf{b} = (b_1, \dots, b_n) \in \mathbb{F}_p^n$ and $\langle \mathbf{a,G} \rangle = \sum_{i=1}^n [a_i] G_i \in \mathbb{G}$ - inner product of vector $\mathbf{a}$ with vector of generators $\mathbf{G}$. +\subsection{Inner-product argument} + +Here we describe the further generalization of $\Pi_{mul}$ -- protocol for the \textbf{inner-product argument} - core component of the \textbf{bulletproofs} protocol. After that we will apply it to range proofs and arithmetic circuits. We have already seen that inner-products are the main ingridients for R1CS language because any R1CS relation could be seen as a batch of inner-products though it's not the most efficient representation and we'll see how to amortize all the constraints into inner-products more efficiently. The \textbf{inner-product argument} allows to prove that two vectors $\mathbf{a,b} \in \mathbb{F}_p^n$ satisfy the relation: $$\mathcal{R}_{ip} = \{ (\mathbf{G,H}, P, c; \mathbf{a,b}) \vert P = \langle \mathbf{a,G} \rangle + \langle \mathbf{b,H} \rangle \wedge \langle \mathbf{a,b} \rangle = c \}$$ @@ -52,8 +105,19 @@ \subsubsection{Inner-product compression} \mathbf{H}^{(k-1)} &= \mathbf{H_{lo}} \cdot u_k + u_k^{-1} \cdot \mathbf{H_{hi}} \end{align*} -Define $P_k \gets P' = \langle \mathbf{a,G} \rangle + \langle \mathbf{b,H} \rangle + [\langle \mathbf{a,b} \rangle]Q$ and define $P_{k-1}$ using compressed vectors to have the same form as $P_k$: -$$P_{k-1} = \langle \mathbf{a}^{(k-1)}, \mathbf{G}^{(k-1)} \rangle + \langle \mathbf{b}^{(k-1)}, \mathbf{H}^{(k-1)} \rangle + [\langle \mathbf{a}^{(k-1)}, \mathbf{b}^{(k-1)} \rangle]Q $$ +Define $P_k \gets P' = \langle \mathbf{a,G} \rangle + \langle \mathbf{b,H} \rangle + [\langle \mathbf{a,b} \rangle]Q$ and define $P_{k-1}$ using compressed vectors to have the same form as $P_k$, but in new basis $(\mathbf{G}^{(k-1)}, \mathbf{H}^{(k-1)})$: + +\begin{equation} + P_{k-1} = \langle \mathbf{a}^{(k-1)}, \mathbf{G}^{(k-1)} \rangle + \langle \mathbf{b}^{(k-1)}, \mathbf{H}^{(k-1)} \rangle + [\langle \mathbf{a}^{(k-1)}, \mathbf{b}^{(k-1)} \rangle]Q + \label{eq:p_k_1_from_new_basis} +\end{equation} + +Or alternatively, expressing $P_{k-1}$ in old basis $(\mathbf{G}^{(k)}, \mathbf{H}^{(k)})$ we get: +\begin{align} + P_{k-1} &= \langle u_k^{-1} \cdot \mathbf{a}^{(k-1)}, \mathbf{G_{lo}}^{(k)} \rangle + \langle u_k \cdot \mathbf{a}^{(k-1)}, \mathbf{G_{hi}}^{(k)} \rangle + \langle u_k \cdot \mathbf{b}^{(k-1)}, \mathbf{H_{lo}}^{(k)} \rangle \\ + &+ \langle u_k^{-1} \cdot \mathbf{b}^{(k-1)}, \mathbf{H_{hi}}^{(k)} \rangle + [\langle \mathbf{a}^{(k-1)}, \mathbf{b}^{(k-1)} \rangle]Q + \label{eq:p_k_1_from_old_basis} +\end{align} Subsituting compressed vectors and applying bilinearity property of inner product we get: \begin{align*} @@ -90,8 +154,8 @@ \subsubsection{Proving $\mathcal{R}'_{ip}$} \begin{definition} The \textbf{inner-product} protocol $\Pi'_{ip} = (\mathcal{P}, \mathcal{V})$ for relation $\mathcal{R}'_{ip} = \{ (\mathbf{G,H}, Q, P'; \mathbf{a,b}) \vert P' = \langle \mathbf{a,G} \rangle + \langle \mathbf{b,H} \rangle + [\langle \mathbf{a,b} \rangle]Q \}$, where all vectors have length $n=2^d$ with prover $\mathcal{P}$, verifier $\mathcal{V}$ is defined as follows: \begin{itemize} - \item Prover $\mathcal{P}$ sets $$(k, \mathbf{a}^{(k)}, \mathbf{b}^{(k)}, \mathbf{G}^{(k)}, \mathbf{H}^{(k)}, P_k) \gets (n, \mathbf{a,b,G,H},P')$$ - \item Verifier $\mathcal{V}$ sets $$(k, \mathbf{G}^{(k)}, \mathbf{H}^{(k)}, P_k) \gets (n, \mathbf{G,H},P')$$ + \item Prover $\mathcal{P}$ sets $$(k, \mathbf{a}^{(k)}, \mathbf{b}^{(k)}, \mathbf{G}^{(k)}, \mathbf{H}^{(k)}, P_k) \gets (d, \mathbf{a,b,G,H},P')$$ + \item Verifier $\mathcal{V}$ sets $$(k, \mathbf{G}^{(k)}, \mathbf{H}^{(k)}, P_k) \gets (d, \mathbf{G,H},P')$$ \item While $k > 0$ then: \begin{itemize} \item Prover $\mathcal{P}$ computes @@ -195,27 +259,92 @@ \subsubsection{Proving $\mathcal{R}'_{ip}$} \label{fig:interactive_ip} \end{figure} -As we can see, overall communication complexity of $\Pi_{ip}$ is $2\log_2 n \mathbb{G} + 2 \mathbb{F}_p$ so we come up with logarithmic proof size for our inner-product relation $\mathcal{R}_{ip}$. Now one might ask whether $\Pi_{ip}$ has any desired properties such as completeness, soundness and zero-knowledge. It appears that the first two holds under some generalizations needed for security proofs but not zero-knowledge (indeed, if $n=1$ then $\mathcal{P}$ sends witness pair $a,b$ directly). +As we can see, overall communication complexity of $\Pi_{ip}$ is $2\log_2 n \mathbb{G} + 2 \mathbb{F}_p$ so we come up with logarithmic proof size for our inner-product relation $\mathcal{R}_{ip}$. Now one might ask whether $\Pi_{ip}$ has any desired properties such as completeness, soundness and zero-knowledge. It appears that the first two holds under some generalizations needed for security proofs but not \textit{zero-knowledge} (indeed, if $n=1$ then $\mathcal{P}$ sends witness pair $a,b$ directly). We'll make further applications of \textit{inner-product argument} for range checks or arithmetic circuits \textit{zero-knowledge}. \begin{theorem}[Inner-Product Argument] The argument system $\Pi_{ip}$ for relation $\mathcal{R}_{ip}$ has \textit{perfect completeness and statistical witness-extended emulation} for either extracting a non-trivial discrete logarithm relation between $\mathbf{G,H}, Q$ or extracting valid witness $\mathbf{a,b}$. \end{theorem} -\textbf{Proof idea}. \textit{Perfect completeness} of $Pi_{ip}$ follows because $Pi_{ip}$ converts instance of $\mathcal{R}_{ip}$ to instance of $\mathcal{P}'_{ip}$ and $\Pi'_{ip}$ is trivially complete by construction due to \Cref{eq:ip-final-compressed}. Notation \textit{statistical witness-extended emulation} generalizes special soundness in the way applicable for multi-stage complex argument systems where each step of the protocol could be rewinded to extract part of the witness so that more accurate definition of protocol security is achieved despite \textit{special soundness} implies building the whole knowledge extractor which might has non-polynomial running time for multi-stage protocols. +\textbf{Proof idea}. \textit{Perfect completeness} of $Pi_{ip}$ follows because $Pi_{ip}$ converts instance of $\mathcal{R}_{ip}$ to instance of $\mathcal{P}'_{ip}$ and $\Pi'_{ip}$ is trivially complete by construction due to \Cref{eq:ip-final-compressed}. Notation \textit{statistical witness-extended emulation} generalizes \textit{special soundness} in the way applicable for multi-stage complex argument systems where each step of the protocol could be rewinded to extract part of the witness so that more accurate definition of protocol security is achieved despite \textit{special soundness} implies building the whole knowledge extractor which might has non-polynomial running time for multi-stage protocols. -Here we briefly describe a knowledge extractor $\mathcal{E}$ for a witness $(\mathbf{a}^{(k)}, \mathbf{b}^{(k)})$ or non-trivial discrete logarithm relation for $(\mathbf{G, H}, Q)$ for one stage of protocol. +Here we briefly describe a knowledge extractor $\mathcal{E}'_{ip}$ for a witness $(\mathbf{a}, \mathbf{b})$ or non-trivial discrete logarithm relation for $(\mathbf{G, H}, Q)$ for $\Pi'_{ip}$. \begin{enumerate} - \item $\mathcal{E}$ runs $\mathcal{P}$ to the last stage to obtain $(L, R) \gets (L_1, R_1)$. and $(a,b) \gets $ - \item $\mathcal{E}$ rewinds the $\Pi'_{ip}$ final stage four times to obtain four challenges $x_1, x_2, x_3, x_4$ and responses $(\mathbf{a}_1^{(0)}, \mathbf{b}_1^{(0)}), (\mathbf{a}_2^{(0)}, \mathbf{b}_2^{(0)}), (\mathbf{a}_3^{(0)}, \mathbf{b}_3^{(0)}), (\mathbf{a}_4^{(0)}, \mathbf{b}_4^{(0)})$ so that the following equation holds for $i \in \{ 1,2,3,4 \}$: - \begin{align} - [x_i^2] L_1 + P_1 + [x_i^{-2}] R_1 = \langle \mathbf{a}_i^{(0)}, \mathbf{G}^{(0)} \rangle + \langle \mathbf{b}_i^{(0)}, \mathbf{H}^{(0)} \rangle + [\langle \mathbf{a}_i^{(0)}, \mathbf{b}_i^{(0)} \rangle]Q - \end{align} - \item $\mathcal{E}$ choses $(v_1, v_2, v_3) \in \mathbb{F}_p^3 $ as a solution for the system of linear equations: - $$\systeme{x_1^2 v_1 + x_2^2 v_2 + x_3^2 v_3=0, v_1+v_2+v_3=1, x_1^{-2} v_1 + x_2^{-2} v_2 + x_3^{-2} v_3=0}$$ + \item $\mathcal{E}'_{ip}$ runs $\mathcal{P}$ to the last stage to obtain $(L, R) \gets (L_1, R_1)$. + \item $\mathcal{E}'_{ip}$ rewinds the $\Pi'_{ip}$ final stage four times to obtain four challenges $x_1, x_2, x_3, x_4$ and responses $(\mathbf{a}_1^{(0)}, \mathbf{b}_1^{(0)}), (\mathbf{a}_2^{(0)}, \mathbf{b}_2^{(0)}), (\mathbf{a}_3^{(0)}, \mathbf{b}_3^{(0)}), (\mathbf{a}_4^{(0)}, \mathbf{b}_4^{(0)})$ so that the following equation holds for $i \in \{ 1,2,3,4 \}$ (\ref{eq:p_k_1_from_old_basis}): + \begin{equation} + \begin{aligned} + & [x_i^2]L + P_1 + [x_i^{-2}]R = \langle x_i^{-1} \cdot \mathbf{a}_i^{(0)}, \mathbf{G_{lo}}^{(1)} \rangle + \langle x_i \cdot \mathbf{a}_i^{(0)}, \mathbf{G_{hi}}^{(1)} \rangle \\ + &+ \langle x_i \cdot \mathbf{b}_i^{(0)}, \mathbf{H_{lo}}^{(1)} \rangle + \langle x_i^{-1} \cdot \mathbf{b}_i^{(0)}, \mathbf{H_{hi}}^{(1)} \rangle + [\langle \mathbf{a}_i^{(0)}, \mathbf{b}_i^{(0)} \rangle]Q + \end{aligned} \label{eq:extractor_eq} - \item $\mathcal{E}$ takes linear combination of \ref{eq:extractor_eq} with coefficients $v_i$ for $i \in \{ 1,2,3 \}$: - $$ + \end{equation} + + \item $\mathcal{E}'_{ip}$ choses $(v_1, v_2, v_3) \in \mathbb{F}_p^3 $ as a solution for the system of linear equations: + \begin{equation} + \begin{cases} + x_1^2 v_1 + x_2^2 v_2 + x_3^2 v_3 = 0 \\ + v_1 + v_2 + v_3 = 1 \\ + x_1^{-2} v_1 + x_2^{-2} v_2 + x_3^{-2} v_3 = 0 + \end{cases} + \label{eq:extractor_v} + \end{equation} + \item $\mathcal{E}'_{ip}$ takes linear combination of (\ref{eq:extractor_eq}) with coefficients $v_i$ for $i \in \{ 1,2,3 \}$: + \begin{equation*} + \begin{aligned} + & \sum_{i=1}^3 [v_i x_i^2]L + [v_i]P_1 + [v_i x_i^{-2}]R = \sum_{i=1}^3 \langle v_i x_i^{-1} \cdot \mathbf{a}_i^{(0)}, \mathbf{G_{lo}}^{(1)} \rangle + \langle v_i x_i \cdot \mathbf{a}_i^{(0)}, \mathbf{G_{hi}}^{(1)} \rangle \\ + &+ \langle v_i x_i \cdot \mathbf{b}_i^{(0)}, \mathbf{H_{lo}}^{(1)} \rangle + \langle v_i x_i^{-1} \cdot \mathbf{b}_i^{(0)}, \mathbf{H_{hi}}^{(1)} \rangle + [v_i \langle \mathbf{a}_i^{(0)}, \mathbf{b}_i^{(0)} \rangle]Q + \end{aligned} + \end{equation*} + Simplifying equation above we get: + \begin{equation} + \begin{aligned} + & P_1 = \langle \sum_{i=1}^3 v_i x_i^{-1} \cdot \mathbf{a}_i^{(0)} + v_i x_i \cdot \mathbf{a}_i^{(0)}, \mathbf{G}^{(1)} \rangle +\\ + & \langle \sum_{i=1}^3 v_i x_i \cdot \mathbf{b}_i^{(0)} + v_i x_i^{-1} \cdot \mathbf{b}_i^{(0)}, \mathbf{H}^{(1)} \rangle + [\sum_{i=1}^3 v_i \langle \mathbf{a}_i^{(0)}, \mathbf{b}_i^{(0)} \rangle]Q + \end{aligned} + \label{eq:extractor_p1} + \end{equation} + On the other hand we have an expression for $P_1$ from (\ref{eq:p_k_1_from_new_basis}): + \begin{equation} + P_{1} = \langle \mathbf{a}^{(1)}, \mathbf{G}^{(1)} \rangle + \langle \mathbf{b}^{(1)}, \mathbf{H}^{(1)} \rangle + [\langle \mathbf{a}^{(1)}, \mathbf{b}^{(1)} \rangle]Q + \label{eq:extractor_p1_alt} + \end{equation} + \item Asserting equality (\ref{eq:extractor_p1})$=$(\ref{eq:extractor_p1_alt}) the extractor $\mathcal{E}'_{ip}$ sets: + \begin{align} + \mathbf{a}^{(1)} &= \sum_{i=1}^3 v_i x_i^{-1} \cdot \mathbf{a}_i^{(0)} + v_i x_i \cdot \mathbf{a}_i^{(0)} \\ + \mathbf{b}^{(1)} &= \sum_{i=1}^3 v_i x_i \cdot \mathbf{b}_i^{(0)} + v_i x_i^{-1} \cdot \mathbf{b}_i^{(0)} + \end{align} + + We need the fourth rewinding to assert equality of inner product: + $$\langle \mathbf{a}^{(1)}, \mathbf{b}^{(1)} \rangle = \sum_{i=1}^3 v_i \langle \mathbf{a}_i^{(0)}, \mathbf{b}_i^{(0)} \rangle$$ + We won't describe it fully since it takes some unwiedly technical details and refer a reder to the original \textit{bulletproofs} paper, where the full proof of extraction is described in Theorem 1. + \item The extractor $\mathcal{E}'_{ip}$ recursively extracts $\mathbf{a}^{(k+1)}, \mathbf{b}^{(k+1)}$ from $ \mathbf{a}^{(k)}, \mathbf{b}^{(k)}$ using the method described in steps 2-5 until it reaches the final witness $\mathbf{a}^{(d)}, \mathbf{b}^{(d)} = \mathbf{a}, \mathbf{b}$ for which holds relation $\mathcal{R}'_{ip}$: + $$P' = \langle \mathbf{a,G} \rangle + \langle \mathbf{b,H} \rangle + [\langle \mathbf{a,b} \rangle]Q$$ \end{enumerate} +Building the extractor $\mathcal{E}_{ip}$ for $\Pi_{ip}$ is quite simple relatively to what we've done by now: +\begin{enumerate} + \item $\mathcal{E}_{ip}$ runs $\Pi_{ip}$ to the end and applies the extractor $\mathcal{E}'_{ip}$ for $Pi'_{ip}$ to extract the witness $\mathbf{a,b}$ such that the following holds for $\mathcal{V}$'s challenge $r_1 \in \mathbb{F}_p$: + \begin{equation} + P + [r_1c]B = \langle \mathbf{a,G} \rangle + \langle \mathbf{b,H} \rangle + [r_1 \cdot \langle \mathbf{a,b} \rangle]B + \label{eq:extractor_r1} + \end{equation} + \item $\mathcal{E}_{ip}$ rewinds the prover $\mathcal{P}$ to obtain another challenge from $\mathcal{V}$: $r_2 \in \mathbb{F}_p$ and yield another witness $\mathbf{a',b'}$ from $\mathcal{E}'_{ip}$: + \begin{equation} + P + [r_2c]B = \langle \mathbf{a',G} \rangle + \langle \mathbf{b',H} \rangle + [r_2 \cdot \langle \mathbf{a',b'} \rangle]B + \label{eq:extractor_r2} + \end{equation} + + \item $\mathcal{E}_{ip}$ substitute (\ref{eq:extractor_r1}) from (\ref{eq:extractor_r2}) to get: + \begin{equation} + [c(r_1 - r_2)]B = \langle \mathbf{a-a',G} \rangle + \langle \mathbf{b-b',H} \rangle + [r_1 \cdot \langle \mathbf{a,b} \rangle - r_2 \cdot \langle \mathbf{a',b'} \rangle]B + \end{equation} + Unless $a = a'$ and $b = b'$ we get a non-trivial discrete log relation between $\mathbf{G,H}$ and $B$, otherwise if equality of witnesses holds we get: + \begin{equation} + [(r_1 - r_2)c]B = [(r_1 - r_2) \langle \mathbf{a,b} \rangle]B + \end{equation} + Hence $c = \langle \mathbf{a,b} \rangle$. +\end{enumerate} + +To formally finalize a proof of \textit{witness-extended emulation} we also need to apply so-called \textit{the forking lemma}, we again refer a reader to the original \textit{bulletproofs} paper $\quad \square$. \subsection{Range proofs} \subsection{Arithmetic circuits proofs} From 8f2475ba52aa8088e7b373d956f5df8c6f0c183a Mon Sep 17 00:00:00 2001 From: Yevhen Hrubiian Date: Tue, 15 Jul 2025 18:00:11 +0300 Subject: [PATCH 07/25] added zk-multiplication section --- lectures/2-9-bulletproofs.tex | 174 +++++++++++++++++++++++++++++----- 1 file changed, 150 insertions(+), 24 deletions(-) diff --git a/lectures/2-9-bulletproofs.tex b/lectures/2-9-bulletproofs.tex index 007c665..de4e46e 100644 --- a/lectures/2-9-bulletproofs.tex +++ b/lectures/2-9-bulletproofs.tex @@ -4,14 +4,17 @@ \begin{document} \subsection{Introduction} -\textbf{Bulletproofs} is a non-interactive zero-knowledge proof protocol with logarithmically sized proofs without a trusted setup. Originally, \textbf{bulletproofs} was developed to provide efficient inner-product proofs for range proofs in application to confidential transactions, but it applies to arbitrary arithmetic circuit (possibly encoded in R1CS). Technically, the protocol is built in an interactive fashion (like $\Sigma$-protocols) and made non-interactive with a Fiat-Shamir transform. One key feature that differs it from $\Sigma$-protocols is the number of challenges from a verifier $\mathcal{V}$: in $\Sigma$-protocols there is only one challenge, while \textbf{bulletproofs} implies a logarithmic in circuit size number of queries. The main advantages of \textbf{bulletproofs} are an absence of a trusted setup, security that relies on the \textit{discrete-logarithm} assumption without any other auxiliary structures like bilinear pairings, however, the main disadvantage of \textbf{bulletproofs} is linear in circuit size verification time though still efficient for small circuits. +\textbf{Bulletproofs} is a zero-knowledge proof protocol with logarithmically sized proofs without a trusted setup. Originally, \textbf{bulletproofs} was developed to provide efficient range proofs in application to confidential transactions, but it applies also to arbitrary arithmetic circuit (possibly encoded in R1CS). In the heart of protocol lays \textbf{inner-product argument} which we describe in details. Technically, the protocol is built in an interactive fashion (like $\Sigma$-protocols), but one could make it non-interactive with a Fiat-Shamir transform. One key feature that differs it from $\Sigma$-protocols is the number of challenges from a verifier $\mathcal{V}$ -- in $\Sigma$-protocols there is only one challenge, while \textbf{bulletproofs} implies a logarithmic in circuit size number of queries. + +Also, \textbf{bulletproofs}' \textbf{inner-product argument} could be used to build various polynomial commitment schemes -- crucial building block of proving systems built with \textit{IOP} framework (\textit{Halo, Nova, etc}). + +The main advantages of \textbf{bulletproofs} are an absence of a trusted setup, security that relies on the \textit{discrete-logarithm} assumption without any other auxiliary structures like bilinear pairings, quite fast prover for small circuits. However, the main disadvantage of \textbf{bulletproofs} is linear in circuit size verification time though still efficient for small circuits. -Hence, we will describe some preliminaries. \subsection{Notation} -Let $\mathbb{G}$ - cyclic group of prime order $p$ written additively, $\mathbf{G} = (G_1, \dots, G_n), \mathbf{H} = (H_1, \dots, H_n) \in \mathbb{G}^n$ - vectors of independent generators. We denote by $\langle \mathbf{a,b} \rangle$ - inner product of vectors $\mathbf{a} = (a_1, \dots, a_n), \mathbf{b} = (b_1, \dots, b_n) \in \mathbb{F}_p^n$ and $\langle \mathbf{a,G} \rangle = \sum_{i=1}^n [a_i] G_i \in \mathbb{G}$ - inner product of vector $\mathbf{a}$ with vector of generators $\mathbf{G}$. +Let $\mathbb{G}$ - cyclic group of prime order $p$ written additively, $\mathbf{G} = (G_1, \dots, G_n), \mathbf{H} = (H_1, \dots, H_n) \in \mathbb{G}^n$ - vectors of independent generators. We denote by $\langle \mathbf{a,b} \rangle$ - inner product of vectors $\mathbf{a} = (a_1, \dots, a_n), \mathbf{b} = (b_1, \dots, b_n) \in \mathbb{F}_p^n$ and $\langle \mathbf{a,G} \rangle = \sum_{i=1}^n [a_i] G_i \in \mathbb{G}$ - inner product of vector $\mathbf{a}$ with vector of generators $\mathbf{G}$. Denote by $\mathbf{k}^n$ vector of $k$'s first $n$ powers: $\mathbf{k}^n = (1, k, k^2, \dots, k^{n-1})$, for example $\mathbf{0}^n, \mathbf{1}^n$ represents vectors of zeros and ones respectively, while $\mathbf{2}^{n} = (1, 2, 4, \dots, 2^{n-1})$ \subsection{Zero-knowledge multiplication} -Let $a,b,c \in \mathbb{F}_p$. Here we build a zero-knowledge protocol for relation $\mathcal{R}'_{mul} = \{ c; a,b: c=ab \}$. We use well-known $\Sigma$-protocol framework for that, but firstly we make very useful generalization that could allow us to prove much larger class of relations. +Let $a,b,c \in \mathbb{F}_p$. Here we build a zero-knowledge protocol for relation $\mathcal{R}_{abc} = \{ ;c,a,b: c=ab \}$. We use well-known $\Sigma$-protocol framework for that, but firstly we make very useful generalization that could allow us to prove much larger class of relations. Consider the first-degree polynomials $l(x) = a + s_L x, r(x) = b + s_R x \in \mathbb{F}_p[x]$. Let $t(x) = l(x)r(x)$ and relation $$\mathcal{R}_{mul} = \{ (;l(x),r(x),t(x))\in (\emptyset \times \mathbb{F}_p^3) \vert t(x) = l(x)r(x)\}$$ @@ -19,37 +22,35 @@ \subsection{Zero-knowledge multiplication} $$\mathsf{Pr}[l(u)r(u) = t(u) \vert l(x)r(x) \neq t(x)] \le \frac{max(\deg(l(x)r(x)), \deg(t(x)))}{p} = \frac{2}{p}$$ is typycally negligible function from security level which makes this check \textit{sound}. -Let's describe naїve unoptimized version \textbf{polynomial multiplication} protocol $\Pi'_{mul} = (\mathsf{Setup},\mathcal{P,V})$ for relation $\mathcal{R}_{mul}$. +\subsubsection{Naїve polynomial multiplication protocol} +Let's describe naїve unoptimized version of \textbf{polynomial multiplication} protocol $\Pi'_{mul} = (\mathsf{Setup},\mathcal{P,V})$ for relation $\mathcal{R}_{mul}$. During $\mathsf{Setup}$ parties agree on group elements $G,B \in \mathbb{G}$. After that parties involve in the following protocol: \begin{itemize} \item Prover $\mathcal{P}$ computes: - $$t(x) = l(x)r(x) = (a+s_L x)(a+s_R x) = ab + (as_R + bs_L) + s_L s_R x^2$$ + $$t(x) = l(x)r(x) = (a+s_L x)(b+s_R x) = ab + (as_R + bs_L) + s_L s_R x^2$$ \item Prover $\mathcal{P}$ draws blinding factors $\alpha_0, \alpha_1, \beta_0, \beta_1, \tau_0, \tau_1, \tau_2 \xleftarrow{R} \mathbb{F}_p$ forming blinding polynomials $$\alpha(x) = \alpha_0 + \alpha_1 x, \beta(x) = \beta_0 + \beta_1 x, \tau(x) = \tau_0 + \tau_1 x + \tau_2 x^2$$ and sends to $\mathcal{V}$ Pedersen commitments for each coefficient of $l(x), r(x), t(x)$: \begin{equation} \begin{aligned} - L_0 &= [a]G + [\alpha_0]B \\ - L_1 &= [s_L]G + [\alpha_1]B \\ - &\\ - R_0 &= [b]G + [\beta_0]B \\ - R_1 &= [s_R]G + [\beta_1]B \\ - &\\ - T_0 &= [ab]G + [\tau_0]B \\ - T_1 &= [as_R + bs_L]G + [\tau_1]B \\ - T_2 &= [s_L s_R]G + [\tau_2]B + L_0 &= [a]G + [\alpha_0]B & R_0 &= [b]G + [\beta_0]B\\ + L_1 &= [s_L]G + [\alpha_1]B & R_1 &= [s_R]G + [\beta_1]B\\ + &&&\\ % todo: centering + && T_0 &= [ab]G + [\tau_0]B \\ + && T_1 &= [as_R + bs_L]G + [\tau_1]B \\ + && T_2 &= [s_L s_R]G + [\tau_2]B \end{aligned} \end{equation} + \begin{remark} + Each commitment could be also seen as a Pedersen commitment to a reciprocal blinding polynomial coefficient as well. + \end{remark} \item Verifier $\mathcal{V}$ samples and sends to $\mathcal{P}$ random evaluation point $u \xleftarrow{R} \mathbb{F}_p$ \item Prover $\mathcal{P}$ evaluates $l(x),r(x),t(x)$ and $\alpha(x), \beta(x), \tau(x)$ at $u$: \begin{equation} \begin{aligned} - l_u & = a + s_L u\\ - r_u &= b + s_R u\\ - t_u &= l_u r_u\\ - \alpha_u &= \alpha_0 + \alpha_1 u\\ - \beta_u &= \beta_0 + \beta_1 u\\ - \tau_u &= \tau_0 + \tau_1 u + \tau_2 u^2 + l_u & = a + s_L u & \alpha_u &= \alpha_0 + \alpha_1 u\\ + r_u &= b + s_R u & \beta_u &= \beta_0 + \beta_1 u\\ + t_u &= l_u r_u & \tau_u &= \tau_0 + \tau_1 u + \tau_2 u^2 \end{aligned} \end{equation} and sends $(l_u, r_u, t_u, \alpha_u, \beta_u, \tau_u)$ to $\mathcal{V}$. @@ -63,7 +64,111 @@ \subsection{Zero-knowledge multiplication} \end{aligned} \end{equation} \end{itemize} +\begin{theorem} + Naїve \textbf{polynomial multiplication} protocol $\Pi'_{mul}$ has \textit{perfect completeness, special soundness, perfect honest-verifier zero-knowledge} + \label{th:poly_mul_naive} +\end{theorem} +\textbf{Proof idea.} \textit{Perfect completeness} holds due to: +\begin{equation*} + \begin{aligned} % todo: fix overflow + [l_u]G + [\alpha_u]B &= [a + s_L u]G + [\alpha_0 + \alpha_1 u]B \\ + L_0 + [u]L_1 &= [a]G + [\alpha_0]B + [s_L u]G + [\alpha_1 u]B = [a + s_L u]G + [\alpha_0 + \alpha_1 u]B \\ + [r_u]G + [\beta_u]B &= [b + s_R u]G + [\beta_0 + \beta_1 u]B \\ + R_0 + [u]R_1 &= [b]G + [\beta_0]B + [s_R u]G + [\beta_1 u]B = [b + s_R u]G + [\beta_0 + \beta_1 u]B\\ + [t_u]G + [\tau_u]B &= [l_u r_u]G + [\tau_0 + \tau_1 u + \tau_2 u^2]B \\ + T_0 + [u]T_1 + [u^2]T_2 &= [ab]G + [\tau_0]B + [u(as_R + bs_L)]G + [u]\tau_1 B + [u^2]s_L s_R G + [u^2]\tau_2 B \\ + &= [ab + (as_R + bs_L)u + s_L s_R u^2]G + [\tau_0 + \tau_1 u + \tau_2 u^2]B \\ + &= [t_u]G + [\tau_u]B + \end{aligned} +\end{equation*} + +Proving \textit{honest-verifier zero-knowledge} is a bit complicated due to proper building of a simulator and proving indistinguishability of distributions, so we briefly describe the idea behind it: each commitment sent in the first phase by $\mathcal{P}$ is a Pedersen commitment which is hiding by design, every second phase response of $\mathcal{P}$ is an evaluation of some first or second degree polynomial at chosen point so there's not enough information for interpolation and polynomial reconstruction, moreover it could be easily simulated. + +To prove \textit{special soundness} we need to build a knowledge extractor $\mathcal{E}$: +\begin{enumerate} + \item $\mathcal{E}$ runs $\mathcal{P}$ to the end and rewinds back the second phase of $\mathcal{P}$, getting two non-equal challenges $u_1, u_2 \in \mathbb{F}_p$ and two prover responses $(l_{u_1}, r_{u_1}), (l_{u_2}, r_{u_2})$ + \item $\mathcal{E}$ solves the following systems of linear equations: + \begin{equation*} + \begin{cases} + l_{u_1} = a + s_L u_1 \\ + l_{u_2} = a + s_L u_2 + \end{cases} + \qquad + \begin{cases} + r_{u_1} = b + s_R u_1 \\ + r_{u_2} = b + s_R u_2 + \end{cases} + \end{equation*} + and gets the coefficients of witness polynomials: $(a, s_L, b, s_R) \quad \square$ +\end{enumerate} + +\subsubsection{Optimized polynomial multiplication protocol} +We could optimize our \textbf{polynomial multiplication protocol} furthermore. Note that we could simply apply vector Pedersen commitment for constant and linear terms using one more group element $H \in \mathbb{G}$. + +\begin{definition} + The \textbf{polynomial multiplication protocol} $\Pi_{mul} = (\mathsf{Setup}, \mathcal{P,V})$ for the relation $\mathcal{R}_{mul} = \{ (;l(x),r(x),t(x)) \vert t(x) = l(x)r(x)\}$ with prover $\mathcal{P}$ and verifier $\mathcal{V}$ is defined as follows: + \begin{itemize} + \item $\mathsf{Setup}$ returns triple of group generators with unknown discrete log relations $G, H, B \in \mathbb{G}$ + \item Parties $\mathcal{P, V}$ run the following protocol: + \begin{itemize} + \item Prover $\mathcal{P}$ computes: + $$t(x) = l(x)r(x) = (a+s_L x)(b+s_R x) = ab + (as_R + bs_L) + s_L s_R x^2$$ + \item Prover $\mathcal{P}$ draws blinding factors $\alpha, \beta, \tau_0, \tau_1, \tau_2 \xleftarrow{R} \mathbb{F}_p$ and sends to $\mathcal{V}$ the following commitments for coefficients of $l(x), r(x), t(x)$: + \begin{equation} + \begin{aligned} + A &= [a]G + [b]H + [\alpha]B\\ + S &= [s_L]G + [s_R]H + [\beta]B\\ + &\\ % todo: centering + T_0 &= [ab]G + [\tau_0]B \\ + T_1 &= [as_R + bs_L]G + [\tau_1]B \\ + T_2 &= [s_L s_R]G + [\tau_2]B + \end{aligned} + \end{equation} + \item Verifier $\mathcal{V}$ samples and sends to $\mathcal{P}$ random evaluation point $u \xleftarrow{R} \mathbb{F}_p$ + \item Prover $\mathcal{P}$ evaluates polynomials at $u$: + \begin{equation} + \begin{aligned} + l_u & = a + s_L u & \alpha_u &= \alpha + \beta u\\ + r_u &= b + s_R u & \tau_u &= \tau_0 + \tau_1 u + \tau_2 u^2\\ + t_u &= l_u r_u & + \end{aligned} + \end{equation} + and sends $(l_u, r_u, t_u, \alpha_u, \tau_u)$ to $\mathcal{V}$. + \item Verifier $\mathcal{V}$ performs checks: + \begin{equation} + \begin{aligned} + A + [u]S &\stackrel{\text{?}}{=} [l_u]G + [r_u]H + [\alpha_u]B \\ + [t_u]G + [\tau_u]B &\stackrel{\text{?}}{=} T_0 + [u]T_1 + [u^2]T_2 \\ + t_u &\stackrel{\text{?}}{=} l_u r_u + \end{aligned} + \end{equation} + \end{itemize} + \end{itemize} +\end{definition} + +\begin{theorem} + The \textbf{polynomial multiplication} protocol $\Pi_{mul}$ has \textit{perfect completeness, special soundness, perfect honest-verifier zero-knowledge} + \label{th:poly_mul} +\end{theorem} +\textbf{Proof}. We left to a reader proof of the theorem in the sake of brevity because it's very similar to the proof of \Cref{th:poly_mul_naive} $\quad \square$ + + +Finally, We could easily build the protocol for the zk-multiplication relation: $$\mathcal{R}_{abc} = \{ ;c,a,b \vert c=ab \}$$ + +\begin{definition} + The \textbf{multiplication protocol} $\Pi_{abc} = (\mathsf{Setup}, \mathcal{P,V})$ for the relation $\mathcal{R}_{abc}$ with prover $\mathcal{P}$ and verifier $\mathcal{V}$ is defined as follows: + \begin{itemize} + \item $\mathsf{Setup}$ returns triple of group generators with unknown discrete log relations $G, H, B \in \mathbb{G}$ + \item Parties $\mathcal{P, V}$ run the following protocol: + \begin{itemize} + \item Prover $\mathcal{P}$ draws random $s_L, s_R \xleftarrow{R} \mathbb{F}_p$ and defines polynomials: $$l(x)=a+s_L x,\quad r(x)=b+s_R x,\quad t(x)=l(x)r(x)$$ + \item Parties run $\Pi_{mul}$ on inputs $(l(x),r(x),t(x))$ + \end{itemize} + \end{itemize} +\end{definition} +The protocol obviously has \textit{perfect completeness, special soundness, perfect honest-verifier zero-knowledge} due to the \Cref{th:poly_mul}. +\subsubsection{Vector polynomial multiplication protocol} \subsection{Inner-product argument} @@ -317,7 +422,7 @@ \subsubsection{Proving $\mathcal{R}'_{ip}$} We need the fourth rewinding to assert equality of inner product: $$\langle \mathbf{a}^{(1)}, \mathbf{b}^{(1)} \rangle = \sum_{i=1}^3 v_i \langle \mathbf{a}_i^{(0)}, \mathbf{b}_i^{(0)} \rangle$$ We won't describe it fully since it takes some unwiedly technical details and refer a reder to the original \textit{bulletproofs} paper, where the full proof of extraction is described in Theorem 1. - \item The extractor $\mathcal{E}'_{ip}$ recursively extracts $\mathbf{a}^{(k+1)}, \mathbf{b}^{(k+1)}$ from $ \mathbf{a}^{(k)}, \mathbf{b}^{(k)}$ using the method described in steps 2-5 until it reaches the final witness $\mathbf{a}^{(d)}, \mathbf{b}^{(d)} = \mathbf{a}, \mathbf{b}$ for which holds relation $\mathcal{R}'_{ip}$: + \item The extractor $\mathcal{E}'_{ip}$ recursively extracts $\mathbf{a}^{(k+1)}, \mathbf{b}^{(k+1)}$ from $ \mathbf{a}^{(k)}, \mathbf{b}^{(k)}$ using the method described in steps 2-5 until it reaches the final witness $\mathbf{a}^{(d)}, \mathbf{b}^{(d)} = \mathbf{a}, \mathbf{b}$ for which the relation $\mathcal{R}'_{ip}$ holds: $$P' = \langle \mathbf{a,G} \rangle + \langle \mathbf{b,H} \rangle + [\langle \mathbf{a,b} \rangle]Q$$ \end{enumerate} Building the extractor $\mathcal{E}_{ip}$ for $\Pi_{ip}$ is quite simple relatively to what we've done by now: @@ -345,17 +450,38 @@ \subsubsection{Proving $\mathcal{R}'_{ip}$} \end{enumerate} To formally finalize a proof of \textit{witness-extended emulation} we also need to apply so-called \textit{the forking lemma}, we again refer a reader to the original \textit{bulletproofs} paper $\quad \square$. + +\subsection{Inner-product based polynomial commitment scheme} + +Here we describe one of the main theoretical applications of the \textbf{inner-product argument} -- \textbf{polynomial commitment scheme}. \subsection{Range proofs} -\subsection{Arithmetic circuits proofs} +Let's consider the relation $\mathcal{R}_{rp} = \{ (w; n) \vert v \in [0, 2^n) \}$. This relation is often called the \textbf{range proof} relation. It asserts that prescribed value $v$ lays in the interval $[0, 2^n)$. Range proofs have very significant applications in various privacy \textit{blockchain} protocols since them usually imply proving that transaction inputs or outputs are valid, e.g. have positive value or satisfy other relations between them. +For the first view it seems very inconspicuous why \textbf{inner-product argument} is useful for proving the range proof relation, but we'll show it ab initio. +Firstly, write $v$ in base-2 representation: $v = \sum_{i=0}^{\lfloor \log_2 v \rfloor} 2^i v_i$ and $\mathbf{a_L} = (v_0, v_1, \dots, v_{n-1})$ be the vector of bits padded with zeroes to length $n$, so the range validation that $v$ lays in $[0, 2^n)$ imlpies two checks: +\begin{itemize} + \item Each bit $v_i$ must be either $0$ or $1$ + \item The following inner-product equality holds: $\langle \mathbf{a_L}, \mathbf{2} \rangle = v$ +\end{itemize} -\subsection*{Acknowledgements} +We already know how to prove the second one inner product equality -- simply by taking evaluation point $u \gets 2$ in \textit{inner-product based polynomial commitment scheme}. +The first relation is a bit more tricky to check algebraically, but still we'll manage to do that, note that binary check for $v_i$ takes form $v_i(v_i - 1) = 0$, or in vector form: +\begin{align*} + \mathbf{a_R} \gets \mathbf{a_L} - \mathbf{1}^n \\ + \mathbf{a_L} \circ \mathbf{a_R} = \mathbf{0}^n +\end{align*} + +\subsection{Arithmetic circuits proofs} + +\subsection*{Acknowledgements} +This section was heavily inspired by: \begin{itemize} \item \href{https://doc-internal.dalek.rs/bulletproofs/index.html}{dalek's crate bulletproofs} \item \href{https://eprint.iacr.org/2017/1066.pdf}{bulletproofs whitepaper} + \item \href{https://rareskills.io/post/zk-multiplication}{RareSkills explanation} \end{itemize} \end{document} From 83f471f261133a670fe7eb5fbdb3889b9209420d Mon Sep 17 00:00:00 2001 From: Yevhen Hrubiian Date: Wed, 16 Jul 2025 18:47:23 +0300 Subject: [PATCH 08/25] add ipa commitment --- lectures/2-9-bulletproofs.tex | 95 +++++++++++++++++++++++++++++++++-- 1 file changed, 90 insertions(+), 5 deletions(-) diff --git a/lectures/2-9-bulletproofs.tex b/lectures/2-9-bulletproofs.tex index de4e46e..67ad2d2 100644 --- a/lectures/2-9-bulletproofs.tex +++ b/lectures/2-9-bulletproofs.tex @@ -166,19 +166,90 @@ \subsubsection{Optimized polynomial multiplication protocol} \end{itemize} \end{itemize} \end{definition} -The protocol obviously has \textit{perfect completeness, special soundness, perfect honest-verifier zero-knowledge} due to the \Cref{th:poly_mul}. +Here we refer to $A$ as a Pedersen commitment to $a,b$ and $T_0$ as a Pedersen commitment to their product which could be given to verifier before start of the protocol. The protocol obviously has \textit{perfect completeness, special soundness, perfect honest-verifier zero-knowledge} due to the \Cref{th:poly_mul}. -\subsubsection{Vector polynomial multiplication protocol} +\begin{remark} %todo: add reference for Chaum-Pedersen + Now curious reader may wonder why build so overwhelmingly complicated protocol for simple multiplication and not just use classic \textit{Chaum-Pedersen protocol} for DH-triplets? Indeed, it definitely could establish that for given group elements $[a]G, [b]G, [c]G$ equality $c=ab$ holds, but unfortunatelly commitments $[a]G, [b]G, [c]G$ do not have a \textit{binding} property (though preserving hiding property) so they're almost useless for building more complex protocols where the prover usually needs to bind to specific values without having an ability to silently modify them. +\end{remark} + +Also, there's a folklore version of very similar protocol for establishing product relationship between Pedersen committed values described in \href{https://people.cs.georgetown.edu/jthaler/ProofsArgsAndZK.pdf}{section 12.3 of Thaler's book} + +\subsubsection{Zero-knowledge inner-product protocol} + +We could extend our $\Pi_{mul}$ protocol even further to provide zero-knowledge proof for the inner-product of vectors: $\langle \mathbf{a, b} \rangle = v$. The main trick is to substitude polynomials $l(x), r(x) \in \mathbb{F}_p[x]$ by vector polynomials $\mathbf{l}(x), \mathbf{r}(x) \in \mathbb{F}_p^n[x]$ where constant terms are equal to $\mathbf{a}$ and $\mathbf{b}$ respectively, taking inner-product $\langle \mathbf{l}(x), \mathbf{r}(x) \rangle$ results in polynomial with scalar coefficients where constant term is equal to $\langle \mathbf{a, b} \rangle$. + +\begin{example} +Let $\mathbf{a} = (a_1, a_2)$ and $\mathbf{b} = (b_1, b_2)$ be vectors in $\mathbb{F}_p^2$. Consider vector polynomials with vector coefficients: +\[ +\mathbf{l}(x) = \mathbf{a} + \mathbf{s}_L x = (a_1, a_2) + (s_{L,1}, s_{L,2}) x +\] +\[ +\mathbf{r}(x) = \mathbf{b} + \mathbf{s}_R x = (b_1, b_2) + (s_{R,1}, s_{R,2}) x +\] +where $\mathbf{s}_L = (s_{L,1}, s_{L,2})$, $\mathbf{s}_R = (s_{R,1}, s_{R,2})$. + +Their inner-product is a degree two scalar polynomial: +\begin{align*} +t(x) &= \langle \mathbf{l}(x), \mathbf{r}(x) \rangle \\ + &= \langle \mathbf{a} + \mathbf{s}_L x,\; \mathbf{b} + \mathbf{s}_R x \rangle \\ + &= \langle \mathbf{a}, \mathbf{b} \rangle + \langle \mathbf{a}, \mathbf{s}_R \rangle x + \langle \mathbf{s}_L, \mathbf{b} \rangle x + \langle \mathbf{s}_L, \mathbf{s}_R \rangle x^2 +\end{align*} +So the constant term is the inner-product $\langle \mathbf{a}, \mathbf{b} \rangle$. +\end{example} + +\begin{definition} + The \textbf{zero-knowledge inner-product protocol} $\Pi_{zkip} = (\mathcal{P,V})$ for the relation + \begin{align*} + \mathcal{R}_{zkip} = \{ (\mathbf{G,H},G,B,A,V;\mathbf{a,b},\alpha, \gamma) \vert & A = \langle \mathbf{a,G} \rangle + \langle \mathbf{b,H} \rangle + [\alpha] B, \\ &V = [\langle \mathbf{a,b} \rangle]G + [\gamma]B \} + \end{align*} where $\mathbf{G,H} \in \mathbb{G}^n, G,B \in \mathbb{G}$ -- independent group generators with prover $\mathcal{P}$ and verifier $\mathcal{V}$ is defined as follows: + + \begin{itemize} + \item Prover $\mathcal{P}$ choses blinding vectors $\mathbf{s}_L, \mathbf{s}_R \in \mathbb{F}_p^n$ and computes polynomials: + \begin{align*} + \mathbf{l}(x) &= \mathbf{a} + \mathbf{s}_L x \\ + \mathbf{r}(x) &= \mathbf{b} + \mathbf{s}_R x \\ + t(x) &= \langle \mathbf{l}(x), \mathbf{r}(x) \rangle = \langle \mathbf{a}, \mathbf{b} \rangle + (\langle \mathbf{a}, \mathbf{s}_R \rangle + \langle \mathbf{s}_L, \mathbf{b} \rangle) x + \langle \mathbf{s}_L, \mathbf{s}_R \rangle x^2 + \end{align*} + \item Prover $\mathcal{P}$ draws blinding factors $\beta, \tau_1, \tau_2 \xleftarrow{R} \mathbb{F}_p$ and sends to $\mathcal{V}$ the following commitments for coefficients of $\mathbf{l}(x), \mathbf{r}(x), \mathbf{t}(x)$: + \begin{equation} + \begin{aligned} + S &= \langle \mathbf{s}_L, \mathbf{G} \rangle + \langle \mathbf{s}_R, \mathbf{H} \rangle + [\beta]B\\ + T_1 &= [\langle \mathbf{a}, \mathbf{s}_R \rangle + \langle \mathbf{s}_L, \mathbf{b} \rangle]G + [\tau_1]B \\ + T_2 &= [\langle \mathbf{s}_L \mathbf{s}_R \rangle]G + [\tau_2]B + \end{aligned} + \end{equation} + \item Verifier $\mathcal{V}$ samples and sends to $\mathcal{P}$ random evaluation point $u \xleftarrow{R} \mathbb{F}_p$ + \item Prover $\mathcal{P}$ evaluates polynomials at $u$: + \begin{equation} + \begin{aligned} + \mathbf{l}_u & = \mathbf{a} + \mathbf{s}_L u & \alpha_u &= \alpha + \beta u\\ + \mathbf{r}_u &= \mathbf{b} + \mathbf{s}_R u & \tau_u &= \tau_0 + \tau_1 u + \tau_2 u^2\\ + t_u &= \langle \mathbf{l}_u \mathbf{r}_u \rangle & + \end{aligned} + \end{equation} + and sends $(\mathbf{l}_u, \mathbf{r}_u, t_u, \alpha_u, \tau_u)$ to $\mathcal{V}$. + \item Verifier $\mathcal{V}$ performs checks: + \begin{equation} + \begin{aligned} + A + [u]S &\stackrel{\text{?}}{=} \langle \mathbf{l}_u, \mathbf{G} \rangle + \langle \mathbf{r}_u, \mathbf{H} \rangle + [\alpha_u]B \\ + [t_u]G + [\tau_u]B &\stackrel{\text{?}}{=} V + [u]T_1 + [u^2]T_2 \\ + t_u &\stackrel{\text{?}}{=} \langle \mathbf{l}_u \mathbf{r}_u \rangle + \end{aligned} + \end{equation} + \end{itemize} +\end{definition} + +The protocol also has \textit{perfect completeness, special soundness, perfect honest-verifier zero-knowledge} due to the \Cref{th:poly_mul}, however building the extractor needs some vector equations we omit for the sake of brevity. Also note that transcript size is linear in size of vectors $\mathbf{l}_u, \mathbf{r}_u$ which is extremly inefficient when vectors are large. So in the next section we present so called \textbf{inner-product argument} which is summoned to reduce conversational complexity to logarithmic in vector length making the last check $t_u \stackrel{\text{?}}{=} \langle \mathbf{l}_u \mathbf{r}_u \rangle$ quite efficient. \subsection{Inner-product argument} -Here we describe the further generalization of $\Pi_{mul}$ -- protocol for the \textbf{inner-product argument} - core component of the \textbf{bulletproofs} protocol. After that we will apply it to range proofs and arithmetic circuits. We have already seen that inner-products are the main ingridients for R1CS language because any R1CS relation could be seen as a batch of inner-products though it's not the most efficient representation and we'll see how to amortize all the constraints into inner-products more efficiently. +Here we describe the further generalization of $\Pi_{mul}$ -- efficient protocol for the \textbf{inner-product argument} - core component of the \textbf{bulletproofs} protocol. After that we will apply it to range proofs and arithmetic circuits. We have already seen that inner-products are the main ingridients for R1CS language because any R1CS relation could be seen as a batch of inner-products though it's not the most efficient representation and we'll see how to amortize all the constraints into inner-products more efficiently. The \textbf{inner-product argument} allows to prove that two vectors $\mathbf{a,b} \in \mathbb{F}_p^n$ satisfy the relation: $$\mathcal{R}_{ip} = \{ (\mathbf{G,H}, P, c; \mathbf{a,b}) \vert P = \langle \mathbf{a,G} \rangle + \langle \mathbf{b,H} \rangle \wedge \langle \mathbf{a,b} \rangle = c \}$$ We refer to $P \in \mathbb{G}$ as a binding Pedersen vector commitment to $\mathbf{a,b}$. -Trivial way to prove the relation is to send $\mathbf{a,b}$ to the verifier $\mathcal{V}$, but it is not a zero-knowledge proof nor efficient due to linear in $n$ size of the proof. We want to build a zero-knowledge proof of the relation $\mathcal{R}_{ip}$ with logarithmic in $n$ size of the proof. +One way to prove the relation is to use $\Pi_{zkip}$, but as we've seen it's not efficient due to linear in $n$ size of the proof. We want to build an argument system for the relation $\mathcal{R}_{ip}$ with logarithmic in $n$ size of the proof. Firtsly, let's combine statements $P = \langle \mathbf{a,G} \rangle + \langle \mathbf{b,H} \rangle \wedge \langle \mathbf{a,b} \rangle = c$ into a single statement by multiplying the second one by a random $r \in \mathbb{F}_p$ and some orthogonal generator $B \in \mathbb{G}$, summing up: $$ @@ -364,7 +435,7 @@ \subsubsection{Proving $\mathcal{R}'_{ip}$} \label{fig:interactive_ip} \end{figure} -As we can see, overall communication complexity of $\Pi_{ip}$ is $2\log_2 n \mathbb{G} + 2 \mathbb{F}_p$ so we come up with logarithmic proof size for our inner-product relation $\mathcal{R}_{ip}$. Now one might ask whether $\Pi_{ip}$ has any desired properties such as completeness, soundness and zero-knowledge. It appears that the first two holds under some generalizations needed for security proofs but not \textit{zero-knowledge} (indeed, if $n=1$ then $\mathcal{P}$ sends witness pair $a,b$ directly). We'll make further applications of \textit{inner-product argument} for range checks or arithmetic circuits \textit{zero-knowledge}. +As we can see, overall communication complexity of $\Pi_{ip}$ is $2\log_2 n$ group elements plus $2$ field elements so we come up with logarithmic proof size for our inner-product relation $\mathcal{R}_{ip}$. Now one might ask whether $\Pi_{ip}$ has any desired properties such as completeness, soundness and zero-knowledge. It appears that the first two holds under some generalizations needed for security proofs but not \textit{zero-knowledge} (indeed, if $n=1$ then $\mathcal{P}$ sends witness pair $a,b$ directly). We'll combile efficient \textbf{inner-product argument} $\Pi_{ip}$ with zero-knowledge $\Pi_{zkip}$ to achieve efficient zero-knowledge proofs for range proofs and arithmetic circuits. \begin{theorem}[Inner-Product Argument] The argument system $\Pi_{ip}$ for relation $\mathcal{R}_{ip}$ has \textit{perfect completeness and statistical witness-extended emulation} for either extracting a non-trivial discrete logarithm relation between $\mathbf{G,H}, Q$ or extracting valid witness $\mathbf{a,b}$. @@ -454,6 +525,20 @@ \subsubsection{Proving $\mathcal{R}'_{ip}$} \subsection{Inner-product based polynomial commitment scheme} Here we describe one of the main theoretical applications of the \textbf{inner-product argument} -- \textbf{polynomial commitment scheme}. + +\begin{definition} + The inner-product polynomial commitment scheme $\mathcal{C}_{ip} = (\mathsf{Commit, Open, VerifyOpen})$ is defined as follows. Let $f(x) = \sum_{i=0}^{n-1} a_i x^i \in \mathbb{F}_p[x]$ be a polynomial of degree $n-1$ and let $\mathbf{G} = (G_1, \dots, G_n)$ be independent group generators. + \begin{itemize} + \item $\mathsf{Commit}$ returns polynomial commitment $\mathsf{Com}(f) = \langle \mathbf{f}, \mathbf{G} \rangle$ where $\mathbf{f} = (f_0, \dots, f_{n-1})$ + \item $\mathsf{Open}$ given evaluation point $u \in \mathbb{F}_p$ computes $\mathbf{u^n} = (1, u, u^2, \dots, u^{n-1})$, obtains $f(u) = \langle \mathbf{f, u^n} \rangle$ and runs \textit{inner-product argument} $\Pi_{ip}$ non-interactively setting $\mathbf{a} = \mathbf{f}, \mathbf{b} = \mathbf{u^n}, P = \mathsf{Com}(f), c = f(u)$ to produce an evaluation proof $\pi_{ip}$. + \item $\mathsf{VerifyOpen}$ validates proof $\pi_{ip}$ running the non-interactive verifier $\mathcal{V}$ of \textit{inner-product} argument. + \end{itemize} +\end{definition} + +\begin{remark} + As the second vector $\mathbf{b} = \mathbf{u^n}$ is known to the verifier, the prover don't have to commit to it using vector $\mathbf{H}$, so the parties might adjust all the steps eliminating vector $\mathbf{H}$ and $\mathbf{b}$ vector compression as well. The full scheme is described \href{https://www.zkdocs.com/docs/zkdocs/commitments/ipa-pcs/}{here}. +\end{remark} + \subsection{Range proofs} Let's consider the relation $\mathcal{R}_{rp} = \{ (w; n) \vert v \in [0, 2^n) \}$. This relation is often called the \textbf{range proof} relation. It asserts that prescribed value $v$ lays in the interval $[0, 2^n)$. Range proofs have very significant applications in various privacy \textit{blockchain} protocols since them usually imply proving that transaction inputs or outputs are valid, e.g. have positive value or satisfy other relations between them. From d2479172fe5d46f45c785a8f94eff8ff7f75e7de Mon Sep 17 00:00:00 2001 From: Yevhen Hrubiian Date: Thu, 17 Jul 2025 20:07:33 +0300 Subject: [PATCH 09/25] begin range proofs --- lectures/2-9-bulletproofs.tex | 134 ++++++++++++++++++++++++++++++++-- 1 file changed, 126 insertions(+), 8 deletions(-) diff --git a/lectures/2-9-bulletproofs.tex b/lectures/2-9-bulletproofs.tex index 67ad2d2..3a76606 100644 --- a/lectures/2-9-bulletproofs.tex +++ b/lectures/2-9-bulletproofs.tex @@ -8,7 +8,7 @@ \subsection{Introduction} Also, \textbf{bulletproofs}' \textbf{inner-product argument} could be used to build various polynomial commitment schemes -- crucial building block of proving systems built with \textit{IOP} framework (\textit{Halo, Nova, etc}). -The main advantages of \textbf{bulletproofs} are an absence of a trusted setup, security that relies on the \textit{discrete-logarithm} assumption without any other auxiliary structures like bilinear pairings, quite fast prover for small circuits. However, the main disadvantage of \textbf{bulletproofs} is linear in circuit size verification time though still efficient for small circuits. +The main advantages of \textbf{bulletproofs} are an absence of a trusted setup and security against eavesdropping that relies on the \textit{discrete-logarithm} assumption without any other auxiliary structures like bilinear pairings. Also it has quite fast prover for small circuits making it practically usefull for client-side proving. However, the main disadvantage of \textbf{bulletproofs} is linear in circuit size verification time though still efficient for small circuits. \subsection{Notation} Let $\mathbb{G}$ - cyclic group of prime order $p$ written additively, $\mathbf{G} = (G_1, \dots, G_n), \mathbf{H} = (H_1, \dots, H_n) \in \mathbb{G}^n$ - vectors of independent generators. We denote by $\langle \mathbf{a,b} \rangle$ - inner product of vectors $\mathbf{a} = (a_1, \dots, a_n), \mathbf{b} = (b_1, \dots, b_n) \in \mathbb{F}_p^n$ and $\langle \mathbf{a,G} \rangle = \sum_{i=1}^n [a_i] G_i \in \mathbb{G}$ - inner product of vector $\mathbf{a}$ with vector of generators $\mathbf{G}$. Denote by $\mathbf{k}^n$ vector of $k$'s first $n$ powers: $\mathbf{k}^n = (1, k, k^2, \dots, k^{n-1})$, for example $\mathbf{0}^n, \mathbf{1}^n$ represents vectors of zeros and ones respectively, while $\mathbf{2}^{n} = (1, 2, 4, \dots, 2^{n-1})$ @@ -524,14 +524,16 @@ \subsubsection{Proving $\mathcal{R}'_{ip}$} \subsection{Inner-product based polynomial commitment scheme} -Here we describe one of the main theoretical applications of the \textbf{inner-product argument} -- \textbf{polynomial commitment scheme}. +Here we describe one of the main theoretical applications of the \textit{inner-product argument} -- \textbf{polynomial commitment scheme} that relies only on \textit{discrete logarithm} assumption, while studied before \textit{KZG} commitment scheme needs bilinear pairings. \begin{definition} The inner-product polynomial commitment scheme $\mathcal{C}_{ip} = (\mathsf{Commit, Open, VerifyOpen})$ is defined as follows. Let $f(x) = \sum_{i=0}^{n-1} a_i x^i \in \mathbb{F}_p[x]$ be a polynomial of degree $n-1$ and let $\mathbf{G} = (G_1, \dots, G_n)$ be independent group generators. \begin{itemize} - \item $\mathsf{Commit}$ returns polynomial commitment $\mathsf{Com}(f) = \langle \mathbf{f}, \mathbf{G} \rangle$ where $\mathbf{f} = (f_0, \dots, f_{n-1})$ - \item $\mathsf{Open}$ given evaluation point $u \in \mathbb{F}_p$ computes $\mathbf{u^n} = (1, u, u^2, \dots, u^{n-1})$, obtains $f(u) = \langle \mathbf{f, u^n} \rangle$ and runs \textit{inner-product argument} $\Pi_{ip}$ non-interactively setting $\mathbf{a} = \mathbf{f}, \mathbf{b} = \mathbf{u^n}, P = \mathsf{Com}(f), c = f(u)$ to produce an evaluation proof $\pi_{ip}$. - \item $\mathsf{VerifyOpen}$ validates proof $\pi_{ip}$ running the non-interactive verifier $\mathcal{V}$ of \textit{inner-product} argument. + \item $\mathsf{Commit}$ returns polynomial commitment $\mathsf{Com}(f) = \langle \mathbf{f}, \mathbf{G} \rangle$ where $\mathbf{f} = (a_0, \dots, a_{n-1})$ + \item $\mathsf{Open}$ given evaluation point $u \in \mathbb{F}_p$ computes $\mathbf{u^n} = (1, u, u^2, \dots, u^{n-1})$, obtains $f(u) = \langle \mathbf{f, u^n} \rangle$ and runs \textit{inner-product argument} $\Pi_{ip}$ non-interactively setting + $$\mathbf{a} = \mathbf{f}, \mathbf{b} = \mathbf{u^n}, P = \mathsf{Com}(f), c = f(u)$$ + to produce an evaluation proof $\pi_{ip}$ + \item $\mathsf{VerifyOpen}$ given evaluation point $u \in \mathbb{F}_p$ and commitment $\mathsf{Com}(f)$ validates proof $\pi_{ip}$ running the non-interactive verifier $\mathcal{V}$ of \textit{inner-product} argument. \end{itemize} \end{definition} @@ -541,24 +543,140 @@ \subsection{Inner-product based polynomial commitment scheme} \subsection{Range proofs} -Let's consider the relation $\mathcal{R}_{rp} = \{ (w; n) \vert v \in [0, 2^n) \}$. This relation is often called the \textbf{range proof} relation. It asserts that prescribed value $v$ lays in the interval $[0, 2^n)$. Range proofs have very significant applications in various privacy \textit{blockchain} protocols since them usually imply proving that transaction inputs or outputs are valid, e.g. have positive value or satisfy other relations between them. +Let $G,B\in \mathbb{G}$ -- independent group generators. Let's consider the relation: +$$\mathcal{R}_{rp} = \{ (w; G, B, V, n) \vert V = [v]G + [\gamma]B, v \in [0, 2^n) \}$$ +This relation is often called the \textbf{range proof} relation. It asserts that committed value $v$ lays in the interval $[0, 2^n)$. Range proofs have very significant applications in various privacy \textit{blockchain} protocols since them usually imply proving that transaction inputs or outputs are valid, e.g. have positive value or satisfy other relations between them. For the first view it seems very inconspicuous why \textbf{inner-product argument} is useful for proving the range proof relation, but we'll show it ab initio. Firstly, write $v$ in base-2 representation: $v = \sum_{i=0}^{\lfloor \log_2 v \rfloor} 2^i v_i$ and $\mathbf{a_L} = (v_0, v_1, \dots, v_{n-1})$ be the vector of bits padded with zeroes to length $n$, so the range validation that $v$ lays in $[0, 2^n)$ imlpies two checks: \begin{itemize} \item Each bit $v_i$ must be either $0$ or $1$ - \item The following inner-product equality holds: $\langle \mathbf{a_L}, \mathbf{2} \rangle = v$ + \item The following inner-product equality holds: $\langle \mathbf{a_L}, \mathbf{2}^n \rangle = v$ \end{itemize} We already know how to prove the second one inner product equality -- simply by taking evaluation point $u \gets 2$ in \textit{inner-product based polynomial commitment scheme}. The first relation is a bit more tricky to check algebraically, but still we'll manage to do that, note that binary check for $v_i$ takes form $v_i(v_i - 1) = 0$, or in vector form: \begin{align*} - \mathbf{a_R} \gets \mathbf{a_L} - \mathbf{1}^n \\ + \mathbf{a_R} = \mathbf{a_L} - \mathbf{1}^n \Leftrightarrow \mathbf{a_L} - \mathbf{a_R} - \mathbf{1}^n = \mathbf{0}^n \\ \mathbf{a_L} \circ \mathbf{a_R} = \mathbf{0}^n \end{align*} +\begin{example} + Let $\mathbf{a_L} = (1, 0, 1, 0)$, $\mathbf{a_R} = (0, -1, 0, -1)$, then + $\mathbf{a_L} \circ \mathbf{a_R} = (0, 0, 0, 0)$ +\end{example} + +This two checks imply verification that some vector is zero vector, for that we use some challenge $y \in \mathbb{F}_p$ and check inner-product equalities $$\langle \mathbf{a_L} \circ \mathbf{a_R}, \mathbf{y}^n \rangle = 0 \text{ and } \langle \mathbf{a_L} - \mathbf{a_R} - \mathbf{1}^n, \mathbf{y}^n \rangle = 0$$ +This checks are sound because the prover doesn't know challenge $y$ in advance. + +Note that $\langle \mathbf{a_L} \circ \mathbf{a_R}, \mathbf{y}^n \rangle = \langle \mathbf{a_L}, \mathbf{a_R} \circ \mathbf{y}^n \rangle$ so the prover could commit to $\mathbf{a_L, a_R}$ and verifier will adjust commitment for $\mathbf{a_R}$ using modified generators $\mathbf{H} \circ \mathbf{y}^{-n}$. + +Here we came up with three inner-product checks: +\begin{enumerate} + \item $\langle \mathbf{a_L}, \mathbf{2}^n \rangle = v$ + \item $\langle \mathbf{a_L}, \mathbf{a_R} \circ \mathbf{y}^n \rangle = 0$ + \item $\langle \mathbf{a_L} - \mathbf{a_R} - \mathbf{1}^n, \mathbf{y}^n \rangle = 0$ +\end{enumerate} + +Here we could soundly combine all three checks into one using random linear combintation with some verifier-provided challenge $z \in \mathbb{F}_p$: +$$z^2 \cdot \langle \mathbf{a_L}, \mathbf{2}^n \rangle + z \cdot \langle \mathbf{a_L} - \mathbf{a_R} - \mathbf{1}^n, \mathbf{y}^n \rangle + \langle \mathbf{a_L}, \mathbf{a_R} \circ \mathbf{y}^n \rangle = z^2v$$ + +\begin{remark} + Naїve check $\langle \mathbf{a_L}, \mathbf{2}^n \rangle + \langle \mathbf{a_L}, \mathbf{a_R} \circ \mathbf{y}^n \rangle + \langle \mathbf{a_L} - \mathbf{a_R} - \mathbf{1}^n, \mathbf{y}^n \rangle = v$ is not sound as Prover could adjust vectors to be non-zero but still satisfy the check. +\end{remark} + +Now simplify this expression having only one inner-product check: +\begin{equation*} + \begin{aligned} + & z^2 \cdot \langle \mathbf{a_L}, \mathbf{2}^n \rangle + z \cdot \langle \mathbf{a_L} - \mathbf{a_R} - \mathbf{1}^n, \mathbf{y}^n \rangle + \langle \mathbf{a_L}, \mathbf{a_R} \circ \mathbf{y}^n \rangle = z^2v \\ + & z^2 \cdot \langle \mathbf{a_L}, \mathbf{2}^n \rangle + \boxed{z \cdot \langle \mathbf{a_L}, \mathbf{y}^n \rangle - z \cdot \langle \mathbf{a_R}, \mathbf{y}^n \rangle - z \cdot \langle \mathbf{1}^n, \mathbf{y}^n \rangle} + \langle \mathbf{a_L}, \mathbf{a_R} \circ \mathbf{y}^n \rangle = z^2v \\ + & z^2 \cdot \langle \mathbf{a_L}, \mathbf{2}^n \rangle + \boxed{\langle \mathbf{a_L}, z \cdot \mathbf{y}^n \rangle + \langle - \mathbf{a_R}, z \cdot \mathbf{y}^n \rangle + \langle - z \cdot \mathbf{1}^n, \mathbf{y}^n \rangle} + \langle \mathbf{a_L}, \mathbf{a_R} \circ \mathbf{y}^n \rangle = z^2v \\ + & z^2 \cdot \langle \mathbf{a_L}, \mathbf{2}^n \rangle + \langle \mathbf{a_L}, z \cdot \mathbf{y}^n \rangle + \boxed{\langle \mathbf{a_R} \circ \mathbf{y}^n, - z \cdot \mathbf{1}^n \rangle} + \langle \mathbf{a_L}, \mathbf{a_R} \circ \mathbf{y}^n \rangle = z^2v + \boxed{\langle z \cdot \mathbf{1}^n, \mathbf{y}^n \rangle} \\ + & \boxed{\langle \mathbf{a_L}, z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n \rangle} + \langle \mathbf{a_R} \circ \mathbf{y}^n, - z \cdot \mathbf{1}^n \rangle + \boxed{\langle\mathbf{a_L}, \mathbf{a_R} \circ \mathbf{y}^n \rangle} = z^2v + \langle z \cdot \mathbf{1}^n, \mathbf{y}^n \rangle \\ + & \boxed{\langle \mathbf{a_L}, z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n + \mathbf{a_R} \circ \mathbf{y}^n\rangle} + \langle \mathbf{a_R} \circ \mathbf{y}^n, - z \cdot \mathbf{1}^n \rangle = z^2v + \langle z \cdot \mathbf{1}^n, \mathbf{y}^n \rangle \\ + \end{aligned} +\end{equation*} + +Now adding to both sides $\langle z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n, - z \cdot \mathbf{1}^n \rangle$: +\begin{equation*} + \begin{aligned} + & \langle \mathbf{a_L}, z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n + \mathbf{a_R} \circ \mathbf{y}^n\rangle + \langle \mathbf{a_R} \circ \mathbf{y}^n, - z \cdot \mathbf{1}^n \rangle + \boxed{\langle z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n, - z \cdot \mathbf{1}^n \rangle} = \\ + & = z^2v + \langle z \cdot \mathbf{1}^n, \mathbf{y}^n \rangle + \boxed{\langle z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n, - z \cdot \mathbf{1}^n \rangle} \\ + & \langle \mathbf{a_L}, z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n + \mathbf{a_R} \circ \mathbf{y}^n\rangle + \boxed{\langle z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n + \mathbf{a_R} \circ \mathbf{y}^n, - z \cdot \mathbf{1}^n \rangle} = \\ + & = z^2v + \boxed{\langle z \cdot \mathbf{1}^n, \mathbf{y}^n \rangle + \langle z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n, - z \cdot \mathbf{1}^n \rangle} \\ + & \langle \mathbf{a_L} - z \cdot \mathbf{1}^n, z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n + \mathbf{a_R} \circ \mathbf{y}^n\rangle = z^2v + \boxed{\delta(y,z)} + \end{aligned} +\end{equation*} + +Where $\delta(y,z)$ could easily be computed by verifier: +$$\delta(y,z) = \langle z \cdot \mathbf{1}^n, \mathbf{y}^n \rangle + \langle z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n, - z \cdot \mathbf{1}^n \rangle = (z-z^2)\langle \mathbf{1}^n, \mathbf{y}^n \rangle - z^3 \langle \mathbf{1}^n, \mathbf{2}^n \rangle $$ +Now we have only one inner-product check left: + +\begin{equation} + \langle \mathbf{a_L} - z \cdot \mathbf{1}^n, z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n + \mathbf{a_R} \circ \mathbf{y}^n\rangle = z^2v + \delta(y,z) + \label{eq:inner_product_check} +\end{equation} + +We will use a technique presented in $\Pi_{zkip}$ to provide zero-knowledge and \textbf{inner-product argument} to achieve logarithmic size-proof. One key problem is that the verifier must adjust commitments to compensate auxiliary terms. + +\begin{definition} + The \textbf{range proof protocol} $\Pi_{rp} = (\mathsf{Setup}, \mathcal{P,V})$ for the relation $\mathcal{R}_{rp} = \{ (w; G, B, V, n) \vert V = [v]G + [\gamma]B, v \in [0, 2^n) \}$ with prover $\mathcal{P}$ and verifier $\mathcal{V}$ is defined as follows: + \begin{itemize} + \item $\mathsf{Setup}$ returns vectors of group generators with unknown discrete log relations $\mathbf{G, H} \in \mathbb{G}^n$ + \item Prover does bit decomposition of $v$ to obtain vectors $\mathbf{a_L} \gets \mathbf{v}, \mathbf{b_L} \gets \mathbf{a_L} - \mathbf{1}^n$ and choses blinding terms $\mathbf{s}_L, \mathbf{s}_R \in \mathbb{F}_p^n, \alpha, \beta \in \mathbb{F}_p$ computing and sending commitments: + \begin{align*} + A = \langle \mathbf{a_L}, \mathbf{G} \rangle + \langle \mathbf{b_L}, \mathbf{H} \rangle + [\alpha]B\\ + S = \langle \mathbf{s}_L, \mathbf{G} \rangle + \langle \mathbf{s}_R, \mathbf{H} \rangle + [\beta]B + \end{align*} + \item Verifier $\mathcal{V}$ samples challenges $y, z \xleftarrow{R} \mathbb{F}_p$ and sends them to $\mathcal{P}$ + \item Prover $\mathcal{P}$ combines the three inner products into one using provided challenges $y,z$ (\ref{eq:inner_product_check}) and reconstructs vector polynomials $\mathbf{l}(x), \mathbf{r}(x), \mathbf{t}(x)$: + \begin{equation*} + \begin{aligned} + \mathbf{l}(x) &= \mathbf{a_L} - z \cdot \mathbf{1}^n + \mathbf{s_L} x\\ + \mathbf{r}(x) &= z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n + \mathbf{a_R} \circ \mathbf{y}^n + \mathbf{s}_R \circ \mathbf{y}^n x\\ + t(x) &= \langle \mathbf{l}(x), \mathbf{r}(x) \rangle = t_0 + t_1 x + t_2 x^2 + \end{aligned} + \end{equation*} + Where + \begin{equation*} + \begin{aligned} + t_0 &= \langle \mathbf{a_L} - z \cdot \mathbf{1}^n, z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n + \mathbf{a_R} \circ \mathbf{y}^n\rangle\\ + t_1 &= \langle \mathbf{a_L} - z \cdot \mathbf{1}^n, \mathbf{y}^n \circ \mathbf{s}_R \rangle + \langle \mathbf{y}^n \circ (\mathbf{a_R} + z\cdot \mathbf{1}^n ) + z^2 \cdot \mathbf{2}^n, \mathbf{s}_L\rangle \\ + t_2 & = \langle \mathbf{s}_L, \mathbf{y}^n \circ \mathbf{s}_R \rangle + \end{aligned} + \end{equation*} + \item Prover $\mathcal{P}$ draws blinding factors $\tau_1, \tau_2 \xleftarrow{R} \mathbb{F}_p$ and sends to $\mathcal{V}$ the following commitments for coefficients of $\mathbf{t}(x)$: + \begin{equation} + \begin{aligned} + T_1 &= [t_1]G + [\tau_1]B \\ + T_2 &= [t_2]G + [\tau_2]B + \end{aligned} + \end{equation} + \textbf{Note:} prover does not have to send commitment to $t_0$ as it's the inner-product we want to prove and it could be computed from high-level commitment $V$. + \item Verifier $\mathcal{V}$ samples and sends to $\mathcal{P}$ random evaluation point $u \xleftarrow{R} \mathbb{F}_p$ + \item Prover $\mathcal{P}$ evaluates polynomials at $u$: + \begin{equation} + \begin{aligned} + \mathbf{l}_u & = \mathbf{l}(u) & \alpha_u &= \alpha + \beta u\\ + \mathbf{r}_u &= \mathbf{r}(u) & \tau_u &= z^2\gamma + \tau_1 u + \tau_2 u^2\\ + t_u &= t(u) = \langle \mathbf{l}_u \mathbf{r}_u \rangle & + \end{aligned} + \end{equation} + and sends $(\mathbf{l}_u, \mathbf{r}_u, t_u, \alpha_u, \tau_u)$ to $\mathcal{V}$. + \item Verifier $\mathcal{V}$ performs checks: + \begin{equation} + \begin{aligned} + A + [u]S + \langle -z \cdot \mathbf{1}^n, \mathbf{G} \rangle &+ \langle z \cdot \mathbf{y}^n + z^2 \cdot \mathbf{2}^n, \mathbf{y}^{-n} \circ \mathbf{H} \rangle \\ + &\stackrel{\text{?}}{=} \langle \mathbf{l}_u, \mathbf{G} \rangle + \langle \mathbf{r}_u, \mathbf{H} \rangle + [\alpha_u]B \\ + [t_u]G + [\tau_u]B &\stackrel{\text{?}}{=} V + [u]T_1 + [u^2]T_2 \\ + t_u &\stackrel{\text{?}}{=} \langle \mathbf{l}_u \mathbf{r}_u \rangle + \end{aligned} + \end{equation} + \end{itemize} +\end{definition} + \subsection{Arithmetic circuits proofs} \subsection*{Acknowledgements} From 513ed2ecb74fa1959774ded1a802c2f3fe34ee08 Mon Sep 17 00:00:00 2001 From: Yevhen Hrubiian Date: Fri, 18 Jul 2025 09:58:38 +0300 Subject: [PATCH 10/25] work on range proofs --- lectures/2-9-bulletproofs.tex | 21 ++++++++++++++++++++- 1 file changed, 20 insertions(+), 1 deletion(-) diff --git a/lectures/2-9-bulletproofs.tex b/lectures/2-9-bulletproofs.tex index 3a76606..126e4c9 100644 --- a/lectures/2-9-bulletproofs.tex +++ b/lectures/2-9-bulletproofs.tex @@ -621,6 +621,15 @@ \subsection{Range proofs} We will use a technique presented in $\Pi_{zkip}$ to provide zero-knowledge and \textbf{inner-product argument} to achieve logarithmic size-proof. One key problem is that the verifier must adjust commitments to compensate auxiliary terms. +Firstly, construct the blinding polynomials for $\mathbf{a_L}$ and $\mathbf{a_R}$ with substitution: +\begin{align*} + \mathbf{a_L}' &\gets \mathbf{a_L} + \mathbf{s_L} x \quad + \mathbf{a_R}' \gets \mathbf{a_R} + \mathbf{s_R} x \\ + \mathbf{l}(x) &= \mathbf{a_L}' - z \cdot \mathbf{1}^n = (\mathbf{a_L} + \mathbf{s_L} x) - z \cdot \mathbf{1}^n\\ + \mathbf{r}(x) &= z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n + \mathbf{a_R}' \circ \mathbf{y}^n = z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n + (\mathbf{a_R} + \mathbf{s_R} x) \circ \mathbf{y}^n \\ + & = z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n + \mathbf{a_R} \circ \mathbf{y}^n + \mathbf{s_R} \circ \mathbf{y}^n x +\end{align*} + \begin{definition} The \textbf{range proof protocol} $\Pi_{rp} = (\mathsf{Setup}, \mathcal{P,V})$ for the relation $\mathcal{R}_{rp} = \{ (w; G, B, V, n) \vert V = [v]G + [\gamma]B, v \in [0, 2^n) \}$ with prover $\mathcal{P}$ and verifier $\mathcal{V}$ is defined as follows: \begin{itemize} @@ -669,7 +678,7 @@ \subsection{Range proofs} \begin{equation} \begin{aligned} A + [u]S + \langle -z \cdot \mathbf{1}^n, \mathbf{G} \rangle &+ \langle z \cdot \mathbf{y}^n + z^2 \cdot \mathbf{2}^n, \mathbf{y}^{-n} \circ \mathbf{H} \rangle \\ - &\stackrel{\text{?}}{=} \langle \mathbf{l}_u, \mathbf{G} \rangle + \langle \mathbf{r}_u, \mathbf{H} \rangle + [\alpha_u]B \\ + &\stackrel{\text{?}}{=} \langle \mathbf{l}_u, \mathbf{G} \rangle + \langle \mathbf{r}_u, \mathbf{y}^{-n} \circ \mathbf{H} \rangle + [\alpha_u]B \\ [t_u]G + [\tau_u]B &\stackrel{\text{?}}{=} V + [u]T_1 + [u^2]T_2 \\ t_u &\stackrel{\text{?}}{=} \langle \mathbf{l}_u \mathbf{r}_u \rangle \end{aligned} @@ -677,6 +686,16 @@ \subsection{Range proofs} \end{itemize} \end{definition} +\begin{remark} + The last two steps of $\Pi_{rp}$ could be substituted with an inner-product argument $\Pi_{ip}$ to provide logarithmic size-proof with the following steps: + \begin{itemize} + \item $\mathcal{P}$ sends $t_u, \alpha_u, \tau_u$ to $\mathcal{V}$ and computes commitment: + $$P = \langle \mathbf{l}_u, \mathbf{G} \rangle + \langle \mathbf{r}_u, \mathbf{y}^{-n} \circ \mathbf{H} \rangle$$ + \item $\mathcal{V}$ performs check $[t_u]G + [\tau_u]B \stackrel{\text{?}}{=} V + [u]T_1 + [u^2]T_2$, halts if it fails and reconstructs commitment $P$ otherwise: + $$P = A + [u]S + \langle -z \cdot \mathbf{1}^n, \mathbf{G} \rangle + \langle z \cdot \mathbf{y}^n + z^2 \cdot \mathbf{2}^n, \mathbf{y}^{-n} \circ \mathbf{H} \rangle - [\alpha_u]B$$ + \item Parties run $\Pi_{ip}$ on inputs: $(\mathbf{G},\mathbf{y}^{-n} \circ \mathbf{H}, P, t_u; \mathbf{l}_u, \mathbf{r}_u)$ + \end{itemize} +\end{remark} \subsection{Arithmetic circuits proofs} \subsection*{Acknowledgements} From 32e18bd6599031216a1968653433dc97e44cda86 Mon Sep 17 00:00:00 2001 From: Yevhen Hrubiian Date: Fri, 18 Jul 2025 19:15:30 +0300 Subject: [PATCH 11/25] finalize rp --- lectures/2-9-bulletproofs.tex | 110 ++++++++++++++++++++++++++++------ 1 file changed, 91 insertions(+), 19 deletions(-) diff --git a/lectures/2-9-bulletproofs.tex b/lectures/2-9-bulletproofs.tex index 126e4c9..2921f63 100644 --- a/lectures/2-9-bulletproofs.tex +++ b/lectures/2-9-bulletproofs.tex @@ -71,14 +71,17 @@ \subsubsection{Naїve polynomial multiplication protocol} \textbf{Proof idea.} \textit{Perfect completeness} holds due to: \begin{equation*} \begin{aligned} % todo: fix overflow - [l_u]G + [\alpha_u]B &= [a + s_L u]G + [\alpha_0 + \alpha_1 u]B \\ - L_0 + [u]L_1 &= [a]G + [\alpha_0]B + [s_L u]G + [\alpha_1 u]B = [a + s_L u]G + [\alpha_0 + \alpha_1 u]B \\ - [r_u]G + [\beta_u]B &= [b + s_R u]G + [\beta_0 + \beta_1 u]B \\ - R_0 + [u]R_1 &= [b]G + [\beta_0]B + [s_R u]G + [\beta_1 u]B = [b + s_R u]G + [\beta_0 + \beta_1 u]B\\ - [t_u]G + [\tau_u]B &= [l_u r_u]G + [\tau_0 + \tau_1 u + \tau_2 u^2]B \\ - T_0 + [u]T_1 + [u^2]T_2 &= [ab]G + [\tau_0]B + [u(as_R + bs_L)]G + [u]\tau_1 B + [u^2]s_L s_R G + [u^2]\tau_2 B \\ + [l_u]G + [\alpha_u]B &= \textcolor{RoyalBlue}{[a + s_L u]G + [\alpha_0 + \alpha_1 u]B} \\ + L_0 + [u]L_1 &= [a]G + [\alpha_0]B + [s_L u]G + [\alpha_1 u]B = \\ + &= \textcolor{RoyalBlue}{[a + s_L u]G + [\alpha_0 + \alpha_1 u]B} \\ + [r_u]G + [\beta_u]B &= \textcolor{OliveGreen}{[b + s_R u]G + [\beta_0 + \beta_1 u]B} \\ + R_0 + [u]R_1 &= [b]G + [\beta_0]B + [s_R u]G + [\beta_1 u]B = \\ + &= \textcolor{RoyalBlue}{[b + s_R u]G + [\beta_0 + \beta_1 u]B}\\ + [t_u]G + [\tau_u]B &= \textcolor{Plum}{[l_u r_u]G + [\tau_0 + \tau_1 u + \tau_2 u^2]B} \\ + T_0 + [u]T_1 + [u^2]T_2 &= [ab]G + [\tau_0]B + [u(as_R + bs_L)]G + \\ + &+[u]\tau_1 B + [u^2]s_L s_R G + [u^2]\tau_2 B \\ &= [ab + (as_R + bs_L)u + s_L s_R u^2]G + [\tau_0 + \tau_1 u + \tau_2 u^2]B \\ - &= [t_u]G + [\tau_u]B + &= \textcolor{Plum}{[l_u r_u]G + [\tau_0 + \tau_1 u + \tau_2 u^2]B} \end{aligned} \end{equation*} @@ -324,6 +327,10 @@ \subsubsection{Inner-product compression} \label{eq:ip-final-compressed} \end{equation} +\begin{remark} + We take $n=2^d$ without loss of generality, since one could always pad the vectors with zeroes to make their length a power of two and inner-product compression would shrink the size of the vectors by a factor of two per compression step down to one element. +\end{remark} + \subsubsection{Proving $\mathcal{R}'_{ip}$} Let's describe the \textbf{inner-product} protocol $\Pi'_{ip}$ for relation $\mathcal{R}'_{ip}$. @@ -544,7 +551,7 @@ \subsection{Inner-product based polynomial commitment scheme} \subsection{Range proofs} Let $G,B\in \mathbb{G}$ -- independent group generators. Let's consider the relation: -$$\mathcal{R}_{rp} = \{ (w; G, B, V, n) \vert V = [v]G + [\gamma]B, v \in [0, 2^n) \}$$ +$$\mathcal{R}_{rp} = \{ (G, B, V, n; v, \gamma) \vert V = [v]G + [\gamma]B, v \in [0, 2^n) \}$$ This relation is often called the \textbf{range proof} relation. It asserts that committed value $v$ lays in the interval $[0, 2^n)$. Range proofs have very significant applications in various privacy \textit{blockchain} protocols since them usually imply proving that transaction inputs or outputs are valid, e.g. have positive value or satisfy other relations between them. For the first view it seems very inconspicuous why \textbf{inner-product argument} is useful for proving the range proof relation, but we'll show it ab initio. @@ -622,16 +629,16 @@ \subsection{Range proofs} We will use a technique presented in $\Pi_{zkip}$ to provide zero-knowledge and \textbf{inner-product argument} to achieve logarithmic size-proof. One key problem is that the verifier must adjust commitments to compensate auxiliary terms. Firstly, construct the blinding polynomials for $\mathbf{a_L}$ and $\mathbf{a_R}$ with substitution: +$$\mathbf{a_L}' \gets \mathbf{a_L} + \mathbf{s_L} x \quad \mathbf{a_R}' \gets \mathbf{a_R} + \mathbf{s_R} x$$ +Compute polynomials $\mathbf{l}(x) = \mathbf{l}_0 + \mathbf{l}_1 x, \quad \mathbf{r}(x) = \mathbf{r}_0 + \mathbf{r}_1 x$: \begin{align*} - \mathbf{a_L}' &\gets \mathbf{a_L} + \mathbf{s_L} x \quad - \mathbf{a_R}' \gets \mathbf{a_R} + \mathbf{s_R} x \\ - \mathbf{l}(x) &= \mathbf{a_L}' - z \cdot \mathbf{1}^n = (\mathbf{a_L} + \mathbf{s_L} x) - z \cdot \mathbf{1}^n\\ + \mathbf{l}(x) &= \mathbf{a_L}' - z \cdot \mathbf{1}^n = (\mathbf{a_L} + \mathbf{s_L} x) - z \cdot \mathbf{1}^n = \mathbf{a_L} - z \cdot \mathbf{1}^n + \mathbf{s_L} x\\ \mathbf{r}(x) &= z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n + \mathbf{a_R}' \circ \mathbf{y}^n = z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n + (\mathbf{a_R} + \mathbf{s_R} x) \circ \mathbf{y}^n \\ & = z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n + \mathbf{a_R} \circ \mathbf{y}^n + \mathbf{s_R} \circ \mathbf{y}^n x \end{align*} - +So that $\langle \mathbf{l}_0, \mathbf{r}_0 \rangle = z^2v + \delta(y,z)$ -- inner product that we want to prove using a bit modified $\Pi_{zkip}$. \begin{definition} - The \textbf{range proof protocol} $\Pi_{rp} = (\mathsf{Setup}, \mathcal{P,V})$ for the relation $\mathcal{R}_{rp} = \{ (w; G, B, V, n) \vert V = [v]G + [\gamma]B, v \in [0, 2^n) \}$ with prover $\mathcal{P}$ and verifier $\mathcal{V}$ is defined as follows: + The \textbf{range proof protocol} $\Pi_{rp} = (\mathsf{Setup}, \mathcal{P,V})$ for the relation $\mathcal{R}_{rp} = \{ (G, B, V, n; v, \gamma) \vert V = [v]G + [\gamma]B, v \in [0, 2^n) \}$ with prover $\mathcal{P}$ and verifier $\mathcal{V}$ is defined as follows: \begin{itemize} \item $\mathsf{Setup}$ returns vectors of group generators with unknown discrete log relations $\mathbf{G, H} \in \mathbb{G}^n$ \item Prover does bit decomposition of $v$ to obtain vectors $\mathbf{a_L} \gets \mathbf{v}, \mathbf{b_L} \gets \mathbf{a_L} - \mathbf{1}^n$ and choses blinding terms $\mathbf{s}_L, \mathbf{s}_R \in \mathbb{F}_p^n, \alpha, \beta \in \mathbb{F}_p$ computing and sending commitments: @@ -651,7 +658,7 @@ \subsection{Range proofs} Where \begin{equation*} \begin{aligned} - t_0 &= \langle \mathbf{a_L} - z \cdot \mathbf{1}^n, z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n + \mathbf{a_R} \circ \mathbf{y}^n\rangle\\ + t_0 &= \langle \mathbf{a_L} - z \cdot \mathbf{1}^n, z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n + \mathbf{a_R} \circ \mathbf{y}^n\rangle = z^2v + \delta(y,z)\\ t_1 &= \langle \mathbf{a_L} - z \cdot \mathbf{1}^n, \mathbf{y}^n \circ \mathbf{s}_R \rangle + \langle \mathbf{y}^n \circ (\mathbf{a_R} + z\cdot \mathbf{1}^n ) + z^2 \cdot \mathbf{2}^n, \mathbf{s}_L\rangle \\ t_2 & = \langle \mathbf{s}_L, \mathbf{y}^n \circ \mathbf{s}_R \rangle \end{aligned} @@ -670,7 +677,7 @@ \subsection{Range proofs} \begin{aligned} \mathbf{l}_u & = \mathbf{l}(u) & \alpha_u &= \alpha + \beta u\\ \mathbf{r}_u &= \mathbf{r}(u) & \tau_u &= z^2\gamma + \tau_1 u + \tau_2 u^2\\ - t_u &= t(u) = \langle \mathbf{l}_u \mathbf{r}_u \rangle & + t_u &= t(u) = z^2v + \delta(y,z) + t_1u + t_2u^2 & \end{aligned} \end{equation} and sends $(\mathbf{l}_u, \mathbf{r}_u, t_u, \alpha_u, \tau_u)$ to $\mathcal{V}$. @@ -679,7 +686,7 @@ \subsection{Range proofs} \begin{aligned} A + [u]S + \langle -z \cdot \mathbf{1}^n, \mathbf{G} \rangle &+ \langle z \cdot \mathbf{y}^n + z^2 \cdot \mathbf{2}^n, \mathbf{y}^{-n} \circ \mathbf{H} \rangle \\ &\stackrel{\text{?}}{=} \langle \mathbf{l}_u, \mathbf{G} \rangle + \langle \mathbf{r}_u, \mathbf{y}^{-n} \circ \mathbf{H} \rangle + [\alpha_u]B \\ - [t_u]G + [\tau_u]B &\stackrel{\text{?}}{=} V + [u]T_1 + [u^2]T_2 \\ + [t_u]G + [\tau_u]B &\stackrel{\text{?}}{=} [z^2]V + [\delta(y,z)]G + [u]T_1 + [u^2]T_2 \\ t_u &\stackrel{\text{?}}{=} \langle \mathbf{l}_u \mathbf{r}_u \rangle \end{aligned} \end{equation} @@ -689,15 +696,80 @@ \subsection{Range proofs} \begin{remark} The last two steps of $\Pi_{rp}$ could be substituted with an inner-product argument $\Pi_{ip}$ to provide logarithmic size-proof with the following steps: \begin{itemize} - \item $\mathcal{P}$ sends $t_u, \alpha_u, \tau_u$ to $\mathcal{V}$ and computes commitment: + \item After $\mathcal{P}$ evaluates polynomials at $u$ he sends $(t_u, \alpha_u, \tau_u$ to $\mathcal{V})$ and computes commitment: $$P = \langle \mathbf{l}_u, \mathbf{G} \rangle + \langle \mathbf{r}_u, \mathbf{y}^{-n} \circ \mathbf{H} \rangle$$ - \item $\mathcal{V}$ performs check $[t_u]G + [\tau_u]B \stackrel{\text{?}}{=} V + [u]T_1 + [u^2]T_2$, halts if it fails and reconstructs commitment $P$ otherwise: + \item $\mathcal{V}$ performs check $[t_u]G + [\tau_u]B \stackrel{\text{?}}{=} [z^2]V + [\delta(y,z)]G + [u]T_1 + [u^2]T_2$, halts if it fails and reconstructs commitment $P$ otherwise: $$P = A + [u]S + \langle -z \cdot \mathbf{1}^n, \mathbf{G} \rangle + \langle z \cdot \mathbf{y}^n + z^2 \cdot \mathbf{2}^n, \mathbf{y}^{-n} \circ \mathbf{H} \rangle - [\alpha_u]B$$ - \item Parties run $\Pi_{ip}$ on inputs: $(\mathbf{G},\mathbf{y}^{-n} \circ \mathbf{H}, P, t_u; \mathbf{l}_u, \mathbf{r}_u)$ + \item Parties run inner-product argument $\Pi_{ip}$ on $(\mathbf{G},\mathbf{y}^{-n} \circ \mathbf{H}, P, t_u; \mathbf{l}_u, \mathbf{r}_u)$ \end{itemize} \end{remark} + +\begin{theorem} + The \textbf{range proof protocol} $\Pi_{rp}$ has \textit{perfect completeness, computational extended witness emulation, perfect honest-verifier zero-knowledge} + \label{th:range_proof} +\end{theorem} +\textbf{Proof idea}. \textit{Perfect completeness} + +Expand LHS of the commitment consistency check: +\begin{equation*} + \begin{aligned} + &\textcolor{teal}{A} + \textcolor{olive}{[u]S} + \langle -z \cdot \mathbf{1}^n, \mathbf{G} \rangle + \langle z \cdot \mathbf{y}^n + z^2 \cdot \mathbf{2}^n, \mathbf{y}^{-n} \circ \mathbf{H} \rangle = \\ + &\textcolor{teal}{\langle \mathbf{a_L}, \mathbf{G} \rangle + \langle \mathbf{b_L}, \mathbf{H} \rangle + [\alpha]B} + \textcolor{olive}{u \cdot \langle \mathbf{s}_L, \mathbf{G} \rangle + u \cdot \langle \mathbf{s}_R, \mathbf{H} \rangle + [u\beta]B} + \\ + & \langle -z \cdot \mathbf{1}^n, \mathbf{G} \rangle + \langle z \cdot \mathbf{y}^n + z^2 \cdot \mathbf{2}^n, \mathbf{y}^{-n} \circ \mathbf{H} \rangle = \\ + &\langle \mathbf{a_L}, \mathbf{G} \rangle + \langle \mathbf{b_L}, \mathbf{H} \rangle + u \cdot \langle \mathbf{s}_L, \mathbf{G} \rangle + u \cdot \langle \mathbf{s}_R, \mathbf{H} \rangle + \\ + &\langle -z \cdot \mathbf{1}^n, \mathbf{G} \rangle + \langle z \cdot \mathbf{1}^n, \mathbf{H} \rangle + \langle z^2 \cdot \mathbf{2}^n, \mathbf{y}^{-n} \circ \mathbf{H} \rangle + [\alpha + u\beta]B + \end{aligned} +\end{equation*} + +Expand RHS of the commitment consistency check: +\begin{equation*} + \begin{aligned} + &\textcolor{Violet}{\langle \mathbf{l}_u, \mathbf{G} \rangle} + \textcolor{Sepia}{\langle \mathbf{r}_u, \mathbf{y}^{-n} \circ \mathbf{H} \rangle} + [\textcolor{Peach}{\alpha_u}]B = \\ + &\textcolor{Violet}{\langle \mathbf{a_L} - z \cdot \mathbf{1}^n + \mathbf{s_L} \cdot u, \mathbf{G} \rangle} + \textcolor{Sepia}{\langle z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n + \mathbf{a_R} \circ \mathbf{y}^n + \mathbf{s}_R \circ \mathbf{y}^n \cdot u, \mathbf{y}^{-n} \circ \mathbf{H} \rangle} \\ + &+ [\textcolor{Peach}{\alpha + u\beta}]B = \\ + & \langle \mathbf{a}_L, \mathbf{G} \rangle + \langle -z \cdot \mathbf{1}^n, \mathbf{G} \rangle + u \cdot \langle \mathbf{s}_L, \mathbf{G} \rangle + \langle z^2 \cdot \mathbf{2}^n, \mathbf{y}^{-n} \circ \mathbf{H} \rangle + \langle z \cdot \mathbf{1}^n, \mathbf{H} \rangle + \\ + & \langle \mathbf{a_R}, \mathbf{H} \rangle + u\cdot \langle \mathbf{s}_R, \mathbf{H} \rangle + [\alpha + u\beta]B + \end{aligned} +\end{equation*} +We could see that \textit{LHS} is equal to \textit{RHS} so the first check pass. + +Taking the polynomial evaluation check: +\begin{equation*} + \begin{aligned} + [t_u]G + [\tau_u]B \stackrel{\text{?}}{=}& [z^2]V + \delta(y,z) + [u]T_1 + [u^2]T_2\\ + [t_u]G + \cancel{[z^2\gamma + \tau_1 u + \tau_2 u^2]B} \stackrel{?}{=}& [z^2 v]G + \cancel{[z^2 \gamma]B} +[\delta(y,z)]G + \\ + & [ut_1]G + \cancel{[u\tau_1]B} + [t_2 u^2] G + \cancel{[\tau_2 u^2]B} \\ + [t_u]G \stackrel{?}{=}& [z^2v + \delta(y,z) + t_1u + t_2u^2]G + \end{aligned} +\end{equation*} +Which holds since $t_u = t(u) = z^2v + \delta(y,z) + t_1u + t_2u^2$. The third inner-product also holds as $t_u = \langle \mathbf{l}_u \mathbf{r}_u \rangle$. + +\textit{Perfect honest-verifier zero-knowledge} follows from the zero-knowledge construction of $\Pi_{zkip}$ protocol as Vertifier learns no information about $\mathbf{a}_L, \mathbf{a}_R$. + +\textit{Computational extended witness emulation} implies building extractor that combines extractors for two subprotocols: $\mathcal{E}_{ip}$ extracts witness ($\mathbf{l}_u, \mathbf{r}_u$) from $\Pi_{ip}$ than the extractor $\mathcal{E}_{zkip}$ extracts high-level witness ($\mathbf{a}_L, \mathbf{a}_R$) from $\Pi_{zkip} \quad \square$ + +The proof size of \textbf{range-proof} protocol is $2 \log_2n +4$ group $\mathbb{G}$ elements and $5$ field $\mathbb{F}_p$ elements. +\begin{remark} + Range proofs could be efficiently aggregated: e.g. one could prove the relation using slighly modified range proof protocol $\Pi_{rp}$ + $$\mathcal{R}_{rpm} = \{ (G, B, \vec{V}, n; \vec{v}, \vec{\gamma}) \vert \forall i \in 1..m: V_i = [v_i]G + [\gamma_i]B, v_i \in [0, 2^n) \}$$ + Where $\vec{v} = (v_1, v_2, \dots, v_m)$ and $\vec{\gamma} = (\gamma_1, \gamma_2, \dots, \gamma_m)$ -- respectively secrets and blinding factors. Detail explanation of aggregation protocol could be found in \href{https://eprint.iacr.org/2017/1066.pdf}{original bulletroofs paper} +\end{remark} + +\begin{example} + One of the most famous \textit{NP-complete} problems is the \textbf{subset-sum problem}: given a set of numbers presented as vector $\mathbf{s}$ and number $v \in \mathbb{N}$, does a some subset sums up to $v$. It turns out that we could use our \textbf{range-proof} protocol for this problem. One could simply replace first inner-product check $\langle \mathbf{a_L}, \mathbf{2}^n \rangle = v$ with $\langle \mathbf{a_L}, \mathbf{s} \rangle = v$ where $\mathbf{a_L}$ is the secret vector of bits that encode positions of $\mathbf{s}$ that sum up to $v$. + + For example take $\mathbf{s} = (6,8,2,3)$ and $v = 14$. Then setting $\mathbf{a_L} = (1,1,0,0)$ we could use $\Pi_{rp}$ to prove that there exists a subset of $\mathbf{s}$ that sums up to $v=14$ without disclosing that subset. + + Therefore, \textbf{bulletproofs range proof} protocol is capable to prove a knowledge of witness to any $NP$-problem as they all could be reduced to the $\textbf{subset-sum problem}$ +\end{example} \subsection{Arithmetic circuits proofs} +\textbf{Bulletproofs} presents not only range proofs, but also efficient proofs for arithmetic circuits satisfiability. As we could see before, inner-product relation is quite powerful tool and could be used to prove a knowledge of witness to any $NP$-problem. But here we present a more convinient way to compile arithmetic circuits into inner-product relation. + +\subsubsection{Arithmetization} + +\textbf{Bulletproofs} arithmetization slightly differs from the classic R1CS we studied before, however it could be transformed vice-versa easily. Also \textbf{bulletproofs} arithmetization is more convinient and human-friendly for encoding most of the arithmetic circuits than the R1CS. + \subsection*{Acknowledgements} This section was heavily inspired by: \begin{itemize} From feee17deb3206f017545eaef079d163d2cae6b13 Mon Sep 17 00:00:00 2001 From: Yevhen Hrubiian Date: Tue, 22 Jul 2025 10:06:28 +0300 Subject: [PATCH 12/25] work on arithmetic circuits --- lectures/2-9-bulletproofs.tex | 296 +++++++++++++++++++++++++++++++++- 1 file changed, 295 insertions(+), 1 deletion(-) diff --git a/lectures/2-9-bulletproofs.tex b/lectures/2-9-bulletproofs.tex index 2921f63..3fd6d3b 100644 --- a/lectures/2-9-bulletproofs.tex +++ b/lectures/2-9-bulletproofs.tex @@ -2,13 +2,25 @@ \usepackage{systeme} \begin{document} + +% --- Annotation: Snark Poem --- +\begin{quote} + \small "Just the place for a Snark!" the Bellman cried,\\ + As he landed his crew with care;\\ + Supporting each man on the top of the tide\\ + By a finger entwined in his hair. +\end{quote} +\begin{flushright} +\small \textit{-- Lewis Carroll, "The Hunting of the Snark"} +\end{flushright} + \subsection{Introduction} \textbf{Bulletproofs} is a zero-knowledge proof protocol with logarithmically sized proofs without a trusted setup. Originally, \textbf{bulletproofs} was developed to provide efficient range proofs in application to confidential transactions, but it applies also to arbitrary arithmetic circuit (possibly encoded in R1CS). In the heart of protocol lays \textbf{inner-product argument} which we describe in details. Technically, the protocol is built in an interactive fashion (like $\Sigma$-protocols), but one could make it non-interactive with a Fiat-Shamir transform. One key feature that differs it from $\Sigma$-protocols is the number of challenges from a verifier $\mathcal{V}$ -- in $\Sigma$-protocols there is only one challenge, while \textbf{bulletproofs} implies a logarithmic in circuit size number of queries. Also, \textbf{bulletproofs}' \textbf{inner-product argument} could be used to build various polynomial commitment schemes -- crucial building block of proving systems built with \textit{IOP} framework (\textit{Halo, Nova, etc}). -The main advantages of \textbf{bulletproofs} are an absence of a trusted setup and security against eavesdropping that relies on the \textit{discrete-logarithm} assumption without any other auxiliary structures like bilinear pairings. Also it has quite fast prover for small circuits making it practically usefull for client-side proving. However, the main disadvantage of \textbf{bulletproofs} is linear in circuit size verification time though still efficient for small circuits. +The main advantages of \textbf{bulletproofs} are an absence of a trusted setup and security against eavesdropping that relies on the \textit{discrete-logarithm} assumption without any other auxiliary structures like bilinear pairings. Also it has quite fast prover for small circuits making it practically usefull for client-side proving. However, the main disadvantage of \textbf{bulletproofs} is that it isn't a classic \textit{SNARK} due to linear in circuit size verification time, however still very efficient for small circuits. \subsection{Notation} Let $\mathbb{G}$ - cyclic group of prime order $p$ written additively, $\mathbf{G} = (G_1, \dots, G_n), \mathbf{H} = (H_1, \dots, H_n) \in \mathbb{G}^n$ - vectors of independent generators. We denote by $\langle \mathbf{a,b} \rangle$ - inner product of vectors $\mathbf{a} = (a_1, \dots, a_n), \mathbf{b} = (b_1, \dots, b_n) \in \mathbb{F}_p^n$ and $\langle \mathbf{a,G} \rangle = \sum_{i=1}^n [a_i] G_i \in \mathbb{G}$ - inner product of vector $\mathbf{a}$ with vector of generators $\mathbf{G}$. Denote by $\mathbf{k}^n$ vector of $k$'s first $n$ powers: $\mathbf{k}^n = (1, k, k^2, \dots, k^{n-1})$, for example $\mathbf{0}^n, \mathbf{1}^n$ represents vectors of zeros and ones respectively, while $\mathbf{2}^{n} = (1, 2, 4, \dots, 2^{n-1})$ @@ -770,6 +782,288 @@ \subsubsection{Arithmetization} \textbf{Bulletproofs} arithmetization slightly differs from the classic R1CS we studied before, however it could be transformed vice-versa easily. Also \textbf{bulletproofs} arithmetization is more convinient and human-friendly for encoding most of the arithmetic circuits than the R1CS. +There are two types of variables in \textit{bulletproofs} constraint system: \textit{low-level} and \textit{high-level}. Typycally \textit{high-level} variables are provided with Pedersen commitments $\mathbf{V} \in \mathbb{G}^m$ as the private witness inputs $\mathbf{v} \in \mathbb{F}_p^m$ to the circuit, while \textit{low-level} variables $\mathbf{a_L}, \mathbf{a_R}, \mathbf{a_O} \in \mathbb{F}_p^n$ are the intermediate witness values of computation. We will define circuit as a set of multiplication constraints operating with \textit{low-level} variables and set of linear constraints which links \textit{low-level} variables between each other and \textit{high-level} variables as well. + +Multiplication constraints are defined with one vector equation: +$$ \mathbf{a_L} \circ \mathbf{a_R} = \mathbf{a_O} $$ + +Linear constraints are defined via: +$$ \mathbf{W}_L \cdot \mathbf{a_L} + \mathbf{W}_R \cdot \mathbf{a_R} + \mathbf{W}_O \cdot \mathbf{a_O} = \mathbf{W}_V \cdot \mathbf{v} + \mathbf{c} $$ + +Where $\mathbf{a_L, a_R, a_O}$ -- vectors of left and right inputs for multiplication gates and output values (all of them are low-level variables). $\mathbf{W_L, W_R, W_O} \in \mathbb{F}_p^{q \times n}, \mathbf{W}_V \in \mathbb{F}_p^{q \times m}$ -- public matrices of weights for linear constraints(obviously known to verifier). $\mathbf{c} \in \mathbb{F}_p^q$ -- public vector of constants. Typycally they encode wiring of the circuit and other linear relations between variables. + +\begin{example} + Consider the following elliptic curve membership circuit. Here witness $(v_1, v_2)$ should satisfy elliptic curve equation: + \begin{equation*} + y^2 = x^3 + ax + b + \end{equation*} + + The arithmetization for this circuit is as follows: + + \textbf{Low-level variables:} + \begin{equation*} + \mathbf{a_L} = \begin{bmatrix} x \\ x \\ y \end{bmatrix}, \quad + \mathbf{a_R} = \begin{bmatrix} x \\ x^2 \\ y \end{bmatrix}, \quad + \mathbf{a_O} = \begin{bmatrix} x^2\\x^3 \\ y^2 \end{bmatrix} + \end{equation*} + + \textbf{High-level variables:} + \begin{equation*} + \mathbf{v} = \begin{bmatrix} v_1 \\ v_2 \end{bmatrix} + \end{equation*} + + \textbf{Multiplication constraints:} + \begin{equation*} + \mathbf{a_L} \circ \mathbf{a_R} = \mathbf{a_O} \Rightarrow \begin{bmatrix} + x \cdot x = x^2 \\ + x \cdot x^2 = x^3 \\ + y \cdot y = y^2 + \end{bmatrix} + \end{equation*} + + \textbf{Linear constraints:} + + \begin{equation*} + \begin{aligned} + \mathbf{a_L}^{(1)} &= v_1 \\ + \mathbf{a_R}^{(1)} &= v_1 \\ + \mathbf{a_L}^{(2)} - \mathbf{a_L}^{(1)} &= 0 \\ + \mathbf{a_R}^{(2)} - \mathbf{a_O}^{(1)} &= 0 \\ + \mathbf{a_L}^{(3)} &= v_2 \\ + \mathbf{a_R}^{(3)} &= v_2 \\ + \mathbf{a_O}^{(3)} - \mathbf{a_O}^{(2)} - a \cdot \mathbf{a_L}^{(1)} &= b + \end{aligned} + \end{equation*} + + \begin{equation*} + \mathbf{W}_L \cdot \mathbf{a_L} + \mathbf{W}_R \cdot \mathbf{a_R} + \mathbf{W}_O \cdot \mathbf{a_O} = \mathbf{W}_V \cdot \mathbf{v} + \mathbf{c} + \end{equation*} + + Where: + \begin{align*} + \mathbf{W}_L = \begin{bmatrix} + 1 & 0 & 0 \\ + 0 & 0 & 0 \\ + -1 & 1 & 0 \\ + 0 & 0 & 0 \\ + 0 & 0 & 1 \\ + 0 & 0 & 0 \\ + -a & 0 & 0 + \end{bmatrix},\quad + \mathbf{W}_R = \begin{bmatrix} + 0 & 0 & 0 \\ + 1 & 0 & 0 \\ + 0 & 0 & 0 \\ + 0 & 1 & 0 \\ + 0 & 0 & 0 \\ + 0 & 0 & 1 \\ + 0 & 0 & 0 + \end{bmatrix},\quad + \mathbf{W}_O = \begin{bmatrix} + 0 & 0 & 0 \\ + 0 & 0 & 0 \\ + 0 & 0 & 0 \\ + -1 & 0 & 0 \\ + 0 & 0 & 0 \\ + 0 & 0 & 0 \\ + 0 & -1 & 1 + \end{bmatrix},\\ + \mathbf{W}_V = \begin{bmatrix} + 1 & 0 \\ + 1 & 0 \\ + 0 & 0 \\ + 0 & 0 \\ + 0 & 1 \\ + 0 & 1 \\ + 0 & 0 + \end{bmatrix},\quad + \mathbf{c} = \begin{bmatrix} + 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ b + \end{bmatrix} +\end{align*} + + + + +\textbf{Classic R1CS arithmetization:} + +We can also represent the same circuit using the classic R1CS formalism: +\[ +(A \cdot \mathbf{w}) \circ (B \cdot \mathbf{w}) = (C \cdot \mathbf{w}) +\] +where $\mathbf{w}$ is extended witness vector $\mathbf{w} = (1, x, y, x^2, x^3, y^2)$ + +The R1CS constraints for the circuit $y^2 = x^3 + a x + b$ are: +\[ +\begin{array}{rl} +\text{Constraint 1:} & x \times x = x^2 \\ +\text{Constraint 2:} & x^2 \times x = x^3 \\ +\text{Constraint 3:} & y \times y = y^2 \\ +\text{Constraint 4:} & x^3 + a x + b = y^2 +\end{array} +\] + +Explicitly, the R1CS matrices are: +\[ +A = \begin{bmatrix} +0 & 1 & 0 & 0 & 0 & 0 \\ +0 & 0 & 0 & 1 & 0 & 0 \\ +0 & 0 & 1 & 0 & 0 & 0 \\ +0 & 0 & 0 & 0 & 1 & 0 \\ +\end{bmatrix} +\] +\[ +B = \begin{bmatrix} +0 & 1 & 0 & 0 & 0 & 0 \\ +0 & 1 & 0 & 0 & 0 & 0 \\ +0 & 0 & 1 & 0 & 0 & 0 \\ +1 & a & 0 & 0 & 0 & 0 \\ +\end{bmatrix} +\] +\[ +C = \begin{bmatrix} +0 & 0 & 0 & 1 & 0 & 0 \\ +0 & 0 & 0 & 0 & 1 & 0 \\ +0 & 0 & 0 & 0 & 0 & 1 \\ +0 & 0 & 0 & 0 & 0 & 1 \\ +\end{bmatrix} +\] +\end{example} + +We could use similar to \textit{range-proofs} technique to transform constraints of the circuit into inner-product relation. For multiplicative constraints take random $y \in \mathbb{F}_p$ and apply zero check: +$$ \langle \mathbf{a_L} \circ \mathbf{a_R} - \mathbf{a_O}, \mathbf{y}^n \rangle = 0$$ + +Do the same for linear constraints, but for different randomness $z \in \mathbb{F}_p$: +$$ \langle \mathbf{W}_L \cdot \mathbf{a_L} + \mathbf{W}_R \cdot \mathbf{a_R} + \mathbf{W}_O \cdot \mathbf{a_O} - \mathbf{W}_V \cdot \mathbf{v} - \mathbf{c}, \mathbf{z}^q \rangle = 0$$ + +Combine this two checks to one using the same randomness $z$: +\begin{equation*} + \begin{aligned} + \langle \mathbf{a_L} \circ \mathbf{a_R} - \mathbf{a_O}, \mathbf{y}^n \rangle + z \cdot \langle \mathbf{W}_L \cdot \mathbf{a_L} + \mathbf{W}_R \cdot \mathbf{a_R} + \mathbf{W}_O \cdot \mathbf{a_O} - \mathbf{W}_V \cdot \mathbf{v} - \mathbf{c}, \mathbf{z}^q \rangle = 0 \\ + \langle \mathbf{a_L} \circ \mathbf{a_R} - \mathbf{a_O}, \mathbf{y}^n \rangle + \langle z \cdot \mathbf{z}^q, \mathbf{W}_L \cdot \mathbf{a_L} + \mathbf{W}_R \cdot \mathbf{a_R} + \mathbf{W}_O \cdot \mathbf{a_O} - \mathbf{W}_V \cdot \mathbf{v} - \mathbf{c} \rangle = 0 + \end{aligned} +\end{equation*} + +This check is sound as typically a prover could not control values of $y,z$ before he commits to $\mathbf{a_L, a_R, a_O}$ and $\mathbf{v}$. + +Then split the second inner product and factor-out public terms to RHS: +\begin{equation*} + \begin{aligned} + \langle \mathbf{a_L} \circ \mathbf{a_R} - \mathbf{a_O}, \mathbf{y}^n \rangle + \langle z \cdot \mathbf{z}^q, \mathbf{W}_L \cdot \mathbf{a_L} \rangle + \langle z \cdot \mathbf{z}^q, \mathbf{W}_R \cdot \mathbf{a_R} \rangle + \langle z \cdot \mathbf{z}^q, \mathbf{W}_O \cdot \mathbf{a_O} \rangle \\ + - \langle z \cdot \mathbf{z}^q, \mathbf{W}_V \cdot \mathbf{v} \rangle - \langle z \cdot \mathbf{z}^q, \mathbf{c} \rangle = 0 \\ + \langle \mathbf{a_L} \circ \mathbf{a_R} - \mathbf{a_O}, \mathbf{y}^n \rangle + \langle z \cdot \mathbf{z}^q, \mathbf{W}_L \cdot \mathbf{a_L} \rangle + \langle z \cdot \mathbf{z}^q, \mathbf{W}_R \cdot \mathbf{a_R} \rangle + \langle z \cdot \mathbf{z}^q, \mathbf{W}_O \cdot \mathbf{a_O} \rangle \\ + - \langle z \cdot \mathbf{z}^q, \mathbf{W}_V \cdot \mathbf{v} \rangle = \langle z \cdot \mathbf{z}^q, \mathbf{c} \rangle + \end{aligned} +\end{equation*} + +Applying conjugation rule(if $A$ -- linear operator and $A^T$ -- its transpose(conjugate) then $\langle \mathbf{a}, A \mathbf{b} \rangle = \langle A^T \mathbf{a}, \mathbf{b} \rangle$) to the second inner product we get: +\begin{equation*} + \begin{aligned} + \langle \mathbf{a_L} \circ \mathbf{a_R} - \mathbf{a_O}, \mathbf{y}^n \rangle + \langle \mathbf{W}_L^T \cdot (z \cdot \mathbf{z}^q), \mathbf{a_L} \rangle + \langle \mathbf{W}_R^T \cdot (z \cdot \mathbf{z}^q), \mathbf{a_R} \rangle + \\ + \langle \mathbf{W}_O^T \cdot (z \cdot \mathbf{z}^q), \mathbf{a_O} \rangle - \langle \mathbf{W}_V^T \cdot (z \cdot \mathbf{z}^q), \mathbf{v} \rangle = \langle z \cdot \mathbf{z}^q, \mathbf{c} \rangle + \end{aligned} +\end{equation*} + +Denote $w_c = \langle z \cdot \mathbf{z}^q, \mathbf{c} \rangle$ and flattened linear constraints(still public and easily computed by verifier): +\begin{equation*} + \begin{aligned} + \mathbf{w}_L = \mathbf{W}_L^T \cdot (z \cdot \mathbf{z}^q) \\ + \mathbf{w}_R = \mathbf{W}_R^T \cdot (z \cdot \mathbf{z}^q) \\ + \mathbf{w}_O = \mathbf{W}_O^T \cdot (z \cdot \mathbf{z}^q) \\ + \mathbf{w}_V = \mathbf{W}_V^T \cdot (z \cdot \mathbf{z}^q) \\ + \end{aligned} +\end{equation*} + +Now we could finally rewrite the equation as: +\begin{equation*} + \begin{aligned} + \langle \mathbf{a_L} \circ \mathbf{a_R} - \mathbf{a_O}, \mathbf{y}^n \rangle + \langle \mathbf{w}_L, \mathbf{a_L} \rangle + \langle \mathbf{w}_R, \mathbf{a_R} \rangle + \langle \mathbf{w}_O, \mathbf{a_O} \rangle - \langle \mathbf{w}_V, \mathbf{v} \rangle = w_c + \end{aligned} +\end{equation*} + +Then rearrange and combine terms so that $\mathbf{a_L}$, $\mathbf{a_O}$ be on the left side and $\mathbf{a_R}$ on the right side: +\begin{equation*} + \begin{aligned} + \langle \mathbf{a_L} \circ \mathbf{a_R}, \mathbf{y}^n \rangle - \langle \mathbf{a_O}, \mathbf{y}^n \rangle + \langle \mathbf{w}_L, \mathbf{a_L} \rangle + \langle \mathbf{w}_R, \mathbf{a_R} \rangle + \langle \mathbf{w}_O, \mathbf{a_O} \rangle = \langle \mathbf{w}_V, \mathbf{v} \rangle + w_c \\ + \langle \mathbf{a_L}, \mathbf{y}^n \circ \mathbf{a_R} \rangle + \langle \mathbf{a_O}, - \mathbf{y}^n + \mathbf{w}_O \rangle + \langle \mathbf{w}_L, \mathbf{a_L} \rangle + \langle \mathbf{w}_R, \mathbf{a_R} \rangle = \langle \mathbf{w}_V, \mathbf{v} \rangle + w_c \\ + \langle \mathbf{a_L}, \mathbf{y}^n \circ \mathbf{a_R} \rangle + \langle \mathbf{a_O}, - \mathbf{y}^n + \mathbf{w}_O \rangle + \langle \mathbf{w}_L, \mathbf{a_L} \rangle + \langle \mathbf{y}^n \circ \mathbf{a}_R, \mathbf{y}^{-n} \circ \mathbf{w_R} \rangle = \langle \mathbf{w}_V, \mathbf{v} \rangle + w_c \\ + \langle \mathbf{a_L} + \mathbf{y}^{-n} \circ \mathbf{w_R}, \mathbf{y}^n \circ \mathbf{a_R} \rangle + \langle \mathbf{a_O}, - \mathbf{y}^n + \mathbf{w}_O \rangle + \langle \mathbf{w}_L, \mathbf{a_L} \rangle = \langle \mathbf{w}_V, \mathbf{v} \rangle + w_c \\ + \end{aligned} +\end{equation*} + +Add $\delta(y, z) = \langle \mathbf{y}^{-n} \circ \mathbf{w}_R, \mathbf{w}_L \rangle $ to both sides: +$$ +\begin{aligned} +w_c + \langle \mathbf{w}_V, \mathbf{v} \rangle + +\delta(y, z) = \langle \mathbf{a}_L + \mathbf{y}^{-n} \circ \mathbf{w}_R, \mathbf{y}^n \circ \mathbf{a}_R \rangle + \langle \mathbf{a}_L, \mathbf{w}_L \rangle +\\ +\langle \mathbf{a}_O, -\mathbf{y}^n + \mathbf{w}_O \rangle + +\langle \mathbf{y}^{-n} \circ \mathbf{w}_R, \mathbf{w}_L \rangle\\ +w_c + \langle \mathbf{w}_V, \mathbf{v} \rangle + \delta(y, z) + = \langle \mathbf{a}_L + \mathbf{y}^{-n} \circ \mathbf{w}_R, + \mathbf{y}^n \circ \mathbf{a}_R \rangle + + \langle \mathbf{a}_L + \mathbf{y}^{-n} \circ \mathbf{w}_R, + \mathbf{w}_L \rangle + \\ + \langle \mathbf{a}_O, + -\mathbf{y}^n + \mathbf{w}_O \rangle \\ + w_c + \langle \mathbf{w}_V, \mathbf{v} \rangle + \delta(y, z) = +\langle \mathbf{a}_L + \mathbf{y}^{-n} \circ \mathbf{w}_R, +\mathbf{y}^n \circ \mathbf{a}_R + +\mathbf{w}_L \rangle + +\langle \mathbf{a}_O, +-\mathbf{y}^n + \mathbf{w}_O \rangle +\end{aligned} +$$ +Now it seems we are stuck as there is two separate inner products so one could not simply linearly blind witness parts, multiply corresponding polynomials and obtain desired inner product in constant term just as we did in the \textit{range-proof} case. But fortunately we could take polynomials of higher degree and obtain desired sum of inner-products as some coefficient of the product of polynomials: +\begin{equation} + \langle \mathbf{a}x + \mathbf{c}x^2, \mathbf{d} + \mathbf{b}x \rangle = s_1x + s_2x^2 + s_3x^3 = x \cdot \langle \mathbf{a}, \mathbf{d} \rangle + x^2 \cdot (\langle \mathbf{a}, \mathbf{b} \rangle + \langle \mathbf{c}, \mathbf{d} \rangle) + x^3 \cdot \langle \mathbf{c}, \mathbf{b} \rangle + \label{eq:inner-product-circuit-1} +\end{equation} +So take: +\begin{equation} + \begin{aligned} + &\mathbf{a} \gets \mathbf{a}_L + \mathbf{y}^{-n} \circ \mathbf{w}_R &&\mathbf{b} \gets \mathbf{y}^n \circ \mathbf{a}_R + \mathbf{w}_L \\ + &\mathbf{c} \gets \mathbf{a}_O &&\mathbf{d} \gets -\mathbf{y}^n + \mathbf{w}_O + \end{aligned} + \label{eq:inner-product-circuit-assignment} +\end{equation} +So than we could get desired sum of inner products as the second-degree coefficient $s_2$: +$$ w_c + \langle \mathbf{w}_V, \mathbf{v} \rangle + \delta(y, z) = s_2$$ +In order to obtain final polynimials $\mathbf{l}(x), \mathbf{r}(x)$ we must firstly blind $\mathbf{a_L}, \mathbf{a_R}$: +\begin{equation} + \begin{aligned} + \mathbf{a_L} \gets \mathbf{a_L} + \mathbf{s_L}x^2 && \mathbf{a_R} \gets \mathbf{a_R} + \mathbf{s_R}x^2 + \end{aligned} + \label{eq:inner-product-circuit-blinding} +\end{equation} +\begin{remark} + We multiplied blinders $\mathbf{s_L}, \mathbf{s_R}$ with the second power of challenge $x^2$ because we want the blinding terms to do not interfere with other parts of inner-product. + + $\mathbf{a_O}$ does not need separate blinding as it's located on the left side of the inner-product (\ref{eq:inner-product-circuit-1}) along with $\mathbf{a_L}$, which is already blinded by $\mathbf{s_L}$. +\end{remark} + +Now we could compute polynomials $\mathbf{l}(x), \mathbf{r}(x)$ from (\ref{eq:inner-product-circuit-1}) using assignments from (\ref{eq:inner-product-circuit-assignment}) and blindings from (\ref{eq:inner-product-circuit-blinding}): +\begin{equation} + \begin{aligned} + {\mathbf{l}}(x) &= (\mathbf{a}_L + \mathbf{s}_L \cdot x^2) \cdot x + \mathbf{y}^{-n} \circ \mathbf{w}_R \cdot x + \mathbf{a}_O \cdot x^2 \\ + &= \mathbf{a}_L \cdot x + \mathbf{s}_L \cdot x^3 + \mathbf{y}^{-n} \circ \mathbf{w}_R \cdot x + \mathbf{a}_O \cdot x^2 \\ + &= \mathbf{s}_L \cdot x^3 + \mathbf{a}_O \cdot x^2 + (\mathbf{a}_L + \mathbf{y}^{-n} \circ \mathbf{w}_R) \cdot x \\ + {\mathbf{r}}(x) &= \mathbf{y}^n \circ (\mathbf{a}_R + \mathbf{s}_R \cdot x^2) \cdot x + \mathbf{w}_L \cdot x - \mathbf{y}^n + \mathbf{w}_O \\ + &= \mathbf{y}^n \circ \mathbf{a}_R \cdot x + \mathbf{y}^n \circ \mathbf{s}_R \cdot x^3 + \mathbf{w}_L \cdot x - \mathbf{y}^n + \mathbf{w}_O \\ + &= \mathbf{y}^n \circ \mathbf{s}_R \cdot x^3 + (\mathbf{y}^n \circ \mathbf{a}_R + \mathbf{w}_L) \cdot x - \mathbf{y}^n + \mathbf{w}_O + \end{aligned} + \label{eq:inner-product-circuit-polynomials} +\end{equation} + +Now we could compute the polynomial $t(x) = \langle \mathbf{l}(x), \mathbf{r}(x) \rangle$: +\begin{equation} + \begin{aligned} + t(x) &= \langle \mathbf{l}(x), \mathbf{r}(x) \rangle = t_1 x + t_2 x^2 + t_3 x^3 + t_4 x^4 + t_5 x^5 + t_6 x^6 = \sum_{i=1}^{6} t_i x^i \\ + t_1 &= \langle \mathbf{a}_L + \mathbf{y}^{-n} \circ \mathbf{w}_R, - \mathbf{y}^n + \mathbf{w}_O \rangle + \end{aligned} + \label{eq:inner-product-circuit-polynomials-product} +\end{equation} + + \subsection*{Acknowledgements} This section was heavily inspired by: \begin{itemize} From d9aebb87db8d477fdde8ace1924e068a781f09b0 Mon Sep 17 00:00:00 2001 From: Yevhen Hrubiian Date: Tue, 22 Jul 2025 15:02:36 +0300 Subject: [PATCH 13/25] add t --- lectures/2-9-bulletproofs.tex | 13 +++++++------ 1 file changed, 7 insertions(+), 6 deletions(-) diff --git a/lectures/2-9-bulletproofs.tex b/lectures/2-9-bulletproofs.tex index 3fd6d3b..bc3d0c4 100644 --- a/lectures/2-9-bulletproofs.tex +++ b/lectures/2-9-bulletproofs.tex @@ -1006,11 +1006,7 @@ \subsubsection{Arithmetization} \langle \mathbf{a}_O, -\mathbf{y}^n + \mathbf{w}_O \rangle \\ w_c + \langle \mathbf{w}_V, \mathbf{v} \rangle + \delta(y, z) = -\langle \mathbf{a}_L + \mathbf{y}^{-n} \circ \mathbf{w}_R, -\mathbf{y}^n \circ \mathbf{a}_R + -\mathbf{w}_L \rangle + -\langle \mathbf{a}_O, --\mathbf{y}^n + \mathbf{w}_O \rangle +\langle \mathbf{a}_L + \mathbf{y}^{-n} \circ \mathbf{w}_R, \mathbf{y}^n \circ \mathbf{a}_R + \mathbf{w}_L \rangle + \langle \mathbf{a}_O, -\mathbf{y}^n + \mathbf{w}_O \rangle \end{aligned} $$ Now it seems we are stuck as there is two separate inner products so one could not simply linearly blind witness parts, multiply corresponding polynomials and obtain desired inner product in constant term just as we did in the \textit{range-proof} case. But fortunately we could take polynomials of higher degree and obtain desired sum of inner-products as some coefficient of the product of polynomials: @@ -1058,7 +1054,12 @@ \subsubsection{Arithmetization} \begin{equation} \begin{aligned} t(x) &= \langle \mathbf{l}(x), \mathbf{r}(x) \rangle = t_1 x + t_2 x^2 + t_3 x^3 + t_4 x^4 + t_5 x^5 + t_6 x^6 = \sum_{i=1}^{6} t_i x^i \\ - t_1 &= \langle \mathbf{a}_L + \mathbf{y}^{-n} \circ \mathbf{w}_R, - \mathbf{y}^n + \mathbf{w}_O \rangle + t_1 &= \langle \mathbf{a}_L + \mathbf{y}^{-n} \circ \mathbf{w}_R, - \mathbf{y}^n + \mathbf{w}_O \rangle \\ + t_2 &= \langle \mathbf{a}_L + \mathbf{y}^{-n} \circ \mathbf{w}_R, \mathbf{y}^n \circ \mathbf{a}_R + \mathbf{w}_L \rangle + \langle \mathbf{a}_O, -\mathbf{y}^n + \mathbf{w}_O \rangle = w_c + \langle \mathbf{w}_V, \mathbf{v} \rangle + \delta(y, z) \\ + t_3 &= \langle \mathbf{s}_L, - \mathbf{y}^n + \mathbf{w}_O \rangle + \langle \mathbf{a}_O, \mathbf{y}^n \circ \mathbf{a}_R + \mathbf{w}_L \rangle \\ + t_4 &= \langle \mathbf{s}_L, \mathbf{y}^n \circ \mathbf{a}_R + \mathbf{w}_L \rangle + \langle \mathbf{a}_L + \mathbf{y}^{-n} \circ \mathbf{w}_R, \mathbf{y}^n \circ \mathbf{s}_R \rangle \\ + t_5 &= \langle \mathbf{a}_O, \mathbf{y}^n \circ \mathbf{s}_R \rangle \\ + t_6 &= \langle \mathbf{s}_L, \mathbf{y}^n \circ \mathbf{s}_R \rangle \end{aligned} \label{eq:inner-product-circuit-polynomials-product} \end{equation} From 7f04aee194f8838636b35e0aca0991fa650de3af Mon Sep 17 00:00:00 2001 From: Yevhen Hrubiian Date: Wed, 23 Jul 2025 11:22:53 +0300 Subject: [PATCH 14/25] finilize the draft --- lectures/2-9-bulletproofs.tex | 318 +++++++++++++++++++++++++--------- 1 file changed, 240 insertions(+), 78 deletions(-) diff --git a/lectures/2-9-bulletproofs.tex b/lectures/2-9-bulletproofs.tex index bc3d0c4..fb65403 100644 --- a/lectures/2-9-bulletproofs.tex +++ b/lectures/2-9-bulletproofs.tex @@ -568,64 +568,64 @@ \subsection{Range proofs} For the first view it seems very inconspicuous why \textbf{inner-product argument} is useful for proving the range proof relation, but we'll show it ab initio. -Firstly, write $v$ in base-2 representation: $v = \sum_{i=0}^{\lfloor \log_2 v \rfloor} 2^i v_i$ and $\mathbf{a_L} = (v_0, v_1, \dots, v_{n-1})$ be the vector of bits padded with zeroes to length $n$, so the range validation that $v$ lays in $[0, 2^n)$ imlpies two checks: +Firstly, write $v$ in base-2 representation: $v = \sum_{i=0}^{\lfloor \log_2 v \rfloor} 2^i v_i$ and $\mathbf{a}_L = (v_0, v_1, \dots, v_{n-1})$ be the vector of bits padded with zeroes to length $n$, so the range validation that $v$ lays in $[0, 2^n)$ imlpies two checks: \begin{itemize} \item Each bit $v_i$ must be either $0$ or $1$ - \item The following inner-product equality holds: $\langle \mathbf{a_L}, \mathbf{2}^n \rangle = v$ + \item The following inner-product equality holds: $\langle \mathbf{a}_L, \mathbf{2}^n \rangle = v$ \end{itemize} We already know how to prove the second one inner product equality -- simply by taking evaluation point $u \gets 2$ in \textit{inner-product based polynomial commitment scheme}. The first relation is a bit more tricky to check algebraically, but still we'll manage to do that, note that binary check for $v_i$ takes form $v_i(v_i - 1) = 0$, or in vector form: \begin{align*} - \mathbf{a_R} = \mathbf{a_L} - \mathbf{1}^n \Leftrightarrow \mathbf{a_L} - \mathbf{a_R} - \mathbf{1}^n = \mathbf{0}^n \\ - \mathbf{a_L} \circ \mathbf{a_R} = \mathbf{0}^n + \mathbf{a}_R = \mathbf{a}_L - \mathbf{1}^n \Leftrightarrow \mathbf{a}_L - \mathbf{a}_R - \mathbf{1}^n = \mathbf{0}^n \\ + \mathbf{a}_L \circ \mathbf{a}_R = \mathbf{0}^n \end{align*} \begin{example} - Let $\mathbf{a_L} = (1, 0, 1, 0)$, $\mathbf{a_R} = (0, -1, 0, -1)$, then - $\mathbf{a_L} \circ \mathbf{a_R} = (0, 0, 0, 0)$ + Let $\mathbf{a}_L = (1, 0, 1, 0)$, $\mathbf{a}_R = (0, -1, 0, -1)$, then + $\mathbf{a}_L \circ \mathbf{a}_R = (0, 0, 0, 0)$ \end{example} -This two checks imply verification that some vector is zero vector, for that we use some challenge $y \in \mathbb{F}_p$ and check inner-product equalities $$\langle \mathbf{a_L} \circ \mathbf{a_R}, \mathbf{y}^n \rangle = 0 \text{ and } \langle \mathbf{a_L} - \mathbf{a_R} - \mathbf{1}^n, \mathbf{y}^n \rangle = 0$$ +This two checks imply verification that some vector is zero vector, for that we use some challenge $y \in \mathbb{F}_p$ and check inner-product equalities $$\langle \mathbf{a}_L \circ \mathbf{a}_R, \mathbf{y}^n \rangle = 0 \text{ and } \langle \mathbf{a}_L - \mathbf{a}_R - \mathbf{1}^n, \mathbf{y}^n \rangle = 0$$ This checks are sound because the prover doesn't know challenge $y$ in advance. -Note that $\langle \mathbf{a_L} \circ \mathbf{a_R}, \mathbf{y}^n \rangle = \langle \mathbf{a_L}, \mathbf{a_R} \circ \mathbf{y}^n \rangle$ so the prover could commit to $\mathbf{a_L, a_R}$ and verifier will adjust commitment for $\mathbf{a_R}$ using modified generators $\mathbf{H} \circ \mathbf{y}^{-n}$. +Note that $\langle \mathbf{a}_L \circ \mathbf{a}_R, \mathbf{y}^n \rangle = \langle \mathbf{a}_L, \mathbf{a}_R \circ \mathbf{y}^n \rangle$ so the prover could commit to $\mathbf{a_L, a_R}$ and verifier will adjust commitment for $\mathbf{a}_R$ using modified generators $\mathbf{H} \circ \mathbf{y}^{-n}$. Here we came up with three inner-product checks: \begin{enumerate} - \item $\langle \mathbf{a_L}, \mathbf{2}^n \rangle = v$ - \item $\langle \mathbf{a_L}, \mathbf{a_R} \circ \mathbf{y}^n \rangle = 0$ - \item $\langle \mathbf{a_L} - \mathbf{a_R} - \mathbf{1}^n, \mathbf{y}^n \rangle = 0$ + \item $\langle \mathbf{a}_L, \mathbf{2}^n \rangle = v$ + \item $\langle \mathbf{a}_L, \mathbf{a}_R \circ \mathbf{y}^n \rangle = 0$ + \item $\langle \mathbf{a}_L - \mathbf{a}_R - \mathbf{1}^n, \mathbf{y}^n \rangle = 0$ \end{enumerate} Here we could soundly combine all three checks into one using random linear combintation with some verifier-provided challenge $z \in \mathbb{F}_p$: -$$z^2 \cdot \langle \mathbf{a_L}, \mathbf{2}^n \rangle + z \cdot \langle \mathbf{a_L} - \mathbf{a_R} - \mathbf{1}^n, \mathbf{y}^n \rangle + \langle \mathbf{a_L}, \mathbf{a_R} \circ \mathbf{y}^n \rangle = z^2v$$ +$$z^2 \cdot \langle \mathbf{a}_L, \mathbf{2}^n \rangle + z \cdot \langle \mathbf{a}_L - \mathbf{a}_R - \mathbf{1}^n, \mathbf{y}^n \rangle + \langle \mathbf{a}_L, \mathbf{a}_R \circ \mathbf{y}^n \rangle = z^2v$$ \begin{remark} - Naїve check $\langle \mathbf{a_L}, \mathbf{2}^n \rangle + \langle \mathbf{a_L}, \mathbf{a_R} \circ \mathbf{y}^n \rangle + \langle \mathbf{a_L} - \mathbf{a_R} - \mathbf{1}^n, \mathbf{y}^n \rangle = v$ is not sound as Prover could adjust vectors to be non-zero but still satisfy the check. + Naїve check $\langle \mathbf{a}_L, \mathbf{2}^n \rangle + \langle \mathbf{a}_L, \mathbf{a}_R \circ \mathbf{y}^n \rangle + \langle \mathbf{a}_L - \mathbf{a}_R - \mathbf{1}^n, \mathbf{y}^n \rangle = v$ is not sound as Prover could adjust vectors to be non-zero but still satisfy the check. \end{remark} Now simplify this expression having only one inner-product check: \begin{equation*} \begin{aligned} - & z^2 \cdot \langle \mathbf{a_L}, \mathbf{2}^n \rangle + z \cdot \langle \mathbf{a_L} - \mathbf{a_R} - \mathbf{1}^n, \mathbf{y}^n \rangle + \langle \mathbf{a_L}, \mathbf{a_R} \circ \mathbf{y}^n \rangle = z^2v \\ - & z^2 \cdot \langle \mathbf{a_L}, \mathbf{2}^n \rangle + \boxed{z \cdot \langle \mathbf{a_L}, \mathbf{y}^n \rangle - z \cdot \langle \mathbf{a_R}, \mathbf{y}^n \rangle - z \cdot \langle \mathbf{1}^n, \mathbf{y}^n \rangle} + \langle \mathbf{a_L}, \mathbf{a_R} \circ \mathbf{y}^n \rangle = z^2v \\ - & z^2 \cdot \langle \mathbf{a_L}, \mathbf{2}^n \rangle + \boxed{\langle \mathbf{a_L}, z \cdot \mathbf{y}^n \rangle + \langle - \mathbf{a_R}, z \cdot \mathbf{y}^n \rangle + \langle - z \cdot \mathbf{1}^n, \mathbf{y}^n \rangle} + \langle \mathbf{a_L}, \mathbf{a_R} \circ \mathbf{y}^n \rangle = z^2v \\ - & z^2 \cdot \langle \mathbf{a_L}, \mathbf{2}^n \rangle + \langle \mathbf{a_L}, z \cdot \mathbf{y}^n \rangle + \boxed{\langle \mathbf{a_R} \circ \mathbf{y}^n, - z \cdot \mathbf{1}^n \rangle} + \langle \mathbf{a_L}, \mathbf{a_R} \circ \mathbf{y}^n \rangle = z^2v + \boxed{\langle z \cdot \mathbf{1}^n, \mathbf{y}^n \rangle} \\ - & \boxed{\langle \mathbf{a_L}, z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n \rangle} + \langle \mathbf{a_R} \circ \mathbf{y}^n, - z \cdot \mathbf{1}^n \rangle + \boxed{\langle\mathbf{a_L}, \mathbf{a_R} \circ \mathbf{y}^n \rangle} = z^2v + \langle z \cdot \mathbf{1}^n, \mathbf{y}^n \rangle \\ - & \boxed{\langle \mathbf{a_L}, z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n + \mathbf{a_R} \circ \mathbf{y}^n\rangle} + \langle \mathbf{a_R} \circ \mathbf{y}^n, - z \cdot \mathbf{1}^n \rangle = z^2v + \langle z \cdot \mathbf{1}^n, \mathbf{y}^n \rangle \\ + & z^2 \cdot \langle \mathbf{a}_L, \mathbf{2}^n \rangle + z \cdot \langle \mathbf{a}_L - \mathbf{a}_R - \mathbf{1}^n, \mathbf{y}^n \rangle + \langle \mathbf{a}_L, \mathbf{a}_R \circ \mathbf{y}^n \rangle = z^2v \\ + & z^2 \cdot \langle \mathbf{a}_L, \mathbf{2}^n \rangle + \boxed{z \cdot \langle \mathbf{a}_L, \mathbf{y}^n \rangle - z \cdot \langle \mathbf{a}_R, \mathbf{y}^n \rangle - z \cdot \langle \mathbf{1}^n, \mathbf{y}^n \rangle} + \langle \mathbf{a}_L, \mathbf{a}_R \circ \mathbf{y}^n \rangle = z^2v \\ + & z^2 \cdot \langle \mathbf{a}_L, \mathbf{2}^n \rangle + \boxed{\langle \mathbf{a}_L, z \cdot \mathbf{y}^n \rangle + \langle - \mathbf{a}_R, z \cdot \mathbf{y}^n \rangle + \langle - z \cdot \mathbf{1}^n, \mathbf{y}^n \rangle} + \langle \mathbf{a}_L, \mathbf{a}_R \circ \mathbf{y}^n \rangle = z^2v \\ + & z^2 \cdot \langle \mathbf{a}_L, \mathbf{2}^n \rangle + \langle \mathbf{a}_L, z \cdot \mathbf{y}^n \rangle + \boxed{\langle \mathbf{a}_R \circ \mathbf{y}^n, - z \cdot \mathbf{1}^n \rangle} + \langle \mathbf{a}_L, \mathbf{a}_R \circ \mathbf{y}^n \rangle = z^2v + \boxed{\langle z \cdot \mathbf{1}^n, \mathbf{y}^n \rangle} \\ + & \boxed{\langle \mathbf{a}_L, z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n \rangle} + \langle \mathbf{a}_R \circ \mathbf{y}^n, - z \cdot \mathbf{1}^n \rangle + \boxed{\langle\mathbf{a}_L, \mathbf{a}_R \circ \mathbf{y}^n \rangle} = z^2v + \langle z \cdot \mathbf{1}^n, \mathbf{y}^n \rangle \\ + & \boxed{\langle \mathbf{a}_L, z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n + \mathbf{a}_R \circ \mathbf{y}^n\rangle} + \langle \mathbf{a}_R \circ \mathbf{y}^n, - z \cdot \mathbf{1}^n \rangle = z^2v + \langle z \cdot \mathbf{1}^n, \mathbf{y}^n \rangle \\ \end{aligned} \end{equation*} Now adding to both sides $\langle z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n, - z \cdot \mathbf{1}^n \rangle$: \begin{equation*} \begin{aligned} - & \langle \mathbf{a_L}, z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n + \mathbf{a_R} \circ \mathbf{y}^n\rangle + \langle \mathbf{a_R} \circ \mathbf{y}^n, - z \cdot \mathbf{1}^n \rangle + \boxed{\langle z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n, - z \cdot \mathbf{1}^n \rangle} = \\ + & \langle \mathbf{a}_L, z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n + \mathbf{a}_R \circ \mathbf{y}^n\rangle + \langle \mathbf{a}_R \circ \mathbf{y}^n, - z \cdot \mathbf{1}^n \rangle + \boxed{\langle z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n, - z \cdot \mathbf{1}^n \rangle} = \\ & = z^2v + \langle z \cdot \mathbf{1}^n, \mathbf{y}^n \rangle + \boxed{\langle z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n, - z \cdot \mathbf{1}^n \rangle} \\ - & \langle \mathbf{a_L}, z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n + \mathbf{a_R} \circ \mathbf{y}^n\rangle + \boxed{\langle z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n + \mathbf{a_R} \circ \mathbf{y}^n, - z \cdot \mathbf{1}^n \rangle} = \\ + & \langle \mathbf{a}_L, z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n + \mathbf{a}_R \circ \mathbf{y}^n\rangle + \boxed{\langle z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n + \mathbf{a}_R \circ \mathbf{y}^n, - z \cdot \mathbf{1}^n \rangle} = \\ & = z^2v + \boxed{\langle z \cdot \mathbf{1}^n, \mathbf{y}^n \rangle + \langle z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n, - z \cdot \mathbf{1}^n \rangle} \\ - & \langle \mathbf{a_L} - z \cdot \mathbf{1}^n, z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n + \mathbf{a_R} \circ \mathbf{y}^n\rangle = z^2v + \boxed{\delta(y,z)} + & \langle \mathbf{a}_L - z \cdot \mathbf{1}^n, z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n + \mathbf{a}_R \circ \mathbf{y}^n\rangle = z^2v + \boxed{\delta(y,z)} \end{aligned} \end{equation*} @@ -634,44 +634,44 @@ \subsection{Range proofs} Now we have only one inner-product check left: \begin{equation} - \langle \mathbf{a_L} - z \cdot \mathbf{1}^n, z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n + \mathbf{a_R} \circ \mathbf{y}^n\rangle = z^2v + \delta(y,z) + \langle \mathbf{a}_L - z \cdot \mathbf{1}^n, z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n + \mathbf{a}_R \circ \mathbf{y}^n\rangle = z^2v + \delta(y,z) \label{eq:inner_product_check} \end{equation} We will use a technique presented in $\Pi_{zkip}$ to provide zero-knowledge and \textbf{inner-product argument} to achieve logarithmic size-proof. One key problem is that the verifier must adjust commitments to compensate auxiliary terms. -Firstly, construct the blinding polynomials for $\mathbf{a_L}$ and $\mathbf{a_R}$ with substitution: -$$\mathbf{a_L}' \gets \mathbf{a_L} + \mathbf{s_L} x \quad \mathbf{a_R}' \gets \mathbf{a_R} + \mathbf{s_R} x$$ +Firstly, construct the blinding polynomials for $\mathbf{a}_L$ and $\mathbf{a}_R$ with substitution: +$$\mathbf{a}_L' \gets \mathbf{a}_L + \mathbf{s}_L x \quad \mathbf{a}_R' \gets \mathbf{a}_R + \mathbf{s}_R x$$ Compute polynomials $\mathbf{l}(x) = \mathbf{l}_0 + \mathbf{l}_1 x, \quad \mathbf{r}(x) = \mathbf{r}_0 + \mathbf{r}_1 x$: \begin{align*} - \mathbf{l}(x) &= \mathbf{a_L}' - z \cdot \mathbf{1}^n = (\mathbf{a_L} + \mathbf{s_L} x) - z \cdot \mathbf{1}^n = \mathbf{a_L} - z \cdot \mathbf{1}^n + \mathbf{s_L} x\\ - \mathbf{r}(x) &= z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n + \mathbf{a_R}' \circ \mathbf{y}^n = z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n + (\mathbf{a_R} + \mathbf{s_R} x) \circ \mathbf{y}^n \\ - & = z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n + \mathbf{a_R} \circ \mathbf{y}^n + \mathbf{s_R} \circ \mathbf{y}^n x + \mathbf{l}(x) &= \mathbf{a}_L' - z \cdot \mathbf{1}^n = (\mathbf{a}_L + \mathbf{s}_L x) - z \cdot \mathbf{1}^n = \mathbf{a}_L - z \cdot \mathbf{1}^n + \mathbf{s}_L x\\ + \mathbf{r}(x) &= z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n + \mathbf{a}_R' \circ \mathbf{y}^n = z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n + (\mathbf{a}_R + \mathbf{s}_R x) \circ \mathbf{y}^n \\ + & = z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n + \mathbf{a}_R \circ \mathbf{y}^n + \mathbf{s}_R \circ \mathbf{y}^n x \end{align*} So that $\langle \mathbf{l}_0, \mathbf{r}_0 \rangle = z^2v + \delta(y,z)$ -- inner product that we want to prove using a bit modified $\Pi_{zkip}$. \begin{definition} The \textbf{range proof protocol} $\Pi_{rp} = (\mathsf{Setup}, \mathcal{P,V})$ for the relation $\mathcal{R}_{rp} = \{ (G, B, V, n; v, \gamma) \vert V = [v]G + [\gamma]B, v \in [0, 2^n) \}$ with prover $\mathcal{P}$ and verifier $\mathcal{V}$ is defined as follows: \begin{itemize} \item $\mathsf{Setup}$ returns vectors of group generators with unknown discrete log relations $\mathbf{G, H} \in \mathbb{G}^n$ - \item Prover does bit decomposition of $v$ to obtain vectors $\mathbf{a_L} \gets \mathbf{v}, \mathbf{b_L} \gets \mathbf{a_L} - \mathbf{1}^n$ and choses blinding terms $\mathbf{s}_L, \mathbf{s}_R \in \mathbb{F}_p^n, \alpha, \beta \in \mathbb{F}_p$ computing and sending commitments: + \item Prover does bit decomposition of $v$ to obtain vectors $\mathbf{a}_L \gets \mathbf{v}, \mathbf{b_L} \gets \mathbf{a}_L - \mathbf{1}^n$ and choses blinding terms $\mathbf{s}_L, \mathbf{s}_R \in \mathbb{F}_p^n, \alpha, \beta \in \mathbb{F}_p$ computing and sending commitments: \begin{align*} - A = \langle \mathbf{a_L}, \mathbf{G} \rangle + \langle \mathbf{b_L}, \mathbf{H} \rangle + [\alpha]B\\ + A = \langle \mathbf{a}_L, \mathbf{G} \rangle + \langle \mathbf{b_L}, \mathbf{H} \rangle + [\alpha]B\\ S = \langle \mathbf{s}_L, \mathbf{G} \rangle + \langle \mathbf{s}_R, \mathbf{H} \rangle + [\beta]B \end{align*} \item Verifier $\mathcal{V}$ samples challenges $y, z \xleftarrow{R} \mathbb{F}_p$ and sends them to $\mathcal{P}$ \item Prover $\mathcal{P}$ combines the three inner products into one using provided challenges $y,z$ (\ref{eq:inner_product_check}) and reconstructs vector polynomials $\mathbf{l}(x), \mathbf{r}(x), \mathbf{t}(x)$: \begin{equation*} \begin{aligned} - \mathbf{l}(x) &= \mathbf{a_L} - z \cdot \mathbf{1}^n + \mathbf{s_L} x\\ - \mathbf{r}(x) &= z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n + \mathbf{a_R} \circ \mathbf{y}^n + \mathbf{s}_R \circ \mathbf{y}^n x\\ + \mathbf{l}(x) &= \mathbf{a}_L - z \cdot \mathbf{1}^n + \mathbf{s}_L x\\ + \mathbf{r}(x) &= z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n + \mathbf{a}_R \circ \mathbf{y}^n + \mathbf{s}_R \circ \mathbf{y}^n x\\ t(x) &= \langle \mathbf{l}(x), \mathbf{r}(x) \rangle = t_0 + t_1 x + t_2 x^2 \end{aligned} \end{equation*} Where \begin{equation*} \begin{aligned} - t_0 &= \langle \mathbf{a_L} - z \cdot \mathbf{1}^n, z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n + \mathbf{a_R} \circ \mathbf{y}^n\rangle = z^2v + \delta(y,z)\\ - t_1 &= \langle \mathbf{a_L} - z \cdot \mathbf{1}^n, \mathbf{y}^n \circ \mathbf{s}_R \rangle + \langle \mathbf{y}^n \circ (\mathbf{a_R} + z\cdot \mathbf{1}^n ) + z^2 \cdot \mathbf{2}^n, \mathbf{s}_L\rangle \\ + t_0 &= \langle \mathbf{a}_L - z \cdot \mathbf{1}^n, z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n + \mathbf{a}_R \circ \mathbf{y}^n\rangle = z^2v + \delta(y,z)\\ + t_1 &= \langle \mathbf{a}_L - z \cdot \mathbf{1}^n, \mathbf{y}^n \circ \mathbf{s}_R \rangle + \langle \mathbf{y}^n \circ (\mathbf{a}_R + z\cdot \mathbf{1}^n ) + z^2 \cdot \mathbf{2}^n, \mathbf{s}_L\rangle \\ t_2 & = \langle \mathbf{s}_L, \mathbf{y}^n \circ \mathbf{s}_R \rangle \end{aligned} \end{equation*} @@ -722,30 +722,30 @@ \subsection{Range proofs} \end{theorem} \textbf{Proof idea}. \textit{Perfect completeness} -Expand LHS of the commitment consistency check: +Expand LHS of the polynomial correctness check: \begin{equation*} \begin{aligned} &\textcolor{teal}{A} + \textcolor{olive}{[u]S} + \langle -z \cdot \mathbf{1}^n, \mathbf{G} \rangle + \langle z \cdot \mathbf{y}^n + z^2 \cdot \mathbf{2}^n, \mathbf{y}^{-n} \circ \mathbf{H} \rangle = \\ - &\textcolor{teal}{\langle \mathbf{a_L}, \mathbf{G} \rangle + \langle \mathbf{b_L}, \mathbf{H} \rangle + [\alpha]B} + \textcolor{olive}{u \cdot \langle \mathbf{s}_L, \mathbf{G} \rangle + u \cdot \langle \mathbf{s}_R, \mathbf{H} \rangle + [u\beta]B} + \\ + &\textcolor{teal}{\langle \mathbf{a}_L, \mathbf{G} \rangle + \langle \mathbf{b_L}, \mathbf{H} \rangle + [\alpha]B} + \textcolor{olive}{u \cdot \langle \mathbf{s}_L, \mathbf{G} \rangle + u \cdot \langle \mathbf{s}_R, \mathbf{H} \rangle + [u\beta]B} + \\ & \langle -z \cdot \mathbf{1}^n, \mathbf{G} \rangle + \langle z \cdot \mathbf{y}^n + z^2 \cdot \mathbf{2}^n, \mathbf{y}^{-n} \circ \mathbf{H} \rangle = \\ - &\langle \mathbf{a_L}, \mathbf{G} \rangle + \langle \mathbf{b_L}, \mathbf{H} \rangle + u \cdot \langle \mathbf{s}_L, \mathbf{G} \rangle + u \cdot \langle \mathbf{s}_R, \mathbf{H} \rangle + \\ + &\langle \mathbf{a}_L, \mathbf{G} \rangle + \langle \mathbf{b_L}, \mathbf{H} \rangle + u \cdot \langle \mathbf{s}_L, \mathbf{G} \rangle + u \cdot \langle \mathbf{s}_R, \mathbf{H} \rangle + \\ &\langle -z \cdot \mathbf{1}^n, \mathbf{G} \rangle + \langle z \cdot \mathbf{1}^n, \mathbf{H} \rangle + \langle z^2 \cdot \mathbf{2}^n, \mathbf{y}^{-n} \circ \mathbf{H} \rangle + [\alpha + u\beta]B \end{aligned} \end{equation*} -Expand RHS of the commitment consistency check: +Expand RHS of the polynomial correctness check: \begin{equation*} \begin{aligned} &\textcolor{Violet}{\langle \mathbf{l}_u, \mathbf{G} \rangle} + \textcolor{Sepia}{\langle \mathbf{r}_u, \mathbf{y}^{-n} \circ \mathbf{H} \rangle} + [\textcolor{Peach}{\alpha_u}]B = \\ - &\textcolor{Violet}{\langle \mathbf{a_L} - z \cdot \mathbf{1}^n + \mathbf{s_L} \cdot u, \mathbf{G} \rangle} + \textcolor{Sepia}{\langle z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n + \mathbf{a_R} \circ \mathbf{y}^n + \mathbf{s}_R \circ \mathbf{y}^n \cdot u, \mathbf{y}^{-n} \circ \mathbf{H} \rangle} \\ + &\textcolor{Violet}{\langle \mathbf{a}_L - z \cdot \mathbf{1}^n + \mathbf{s}_L \cdot u, \mathbf{G} \rangle} + \textcolor{Sepia}{\langle z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n + \mathbf{a}_R \circ \mathbf{y}^n + \mathbf{s}_R \circ \mathbf{y}^n \cdot u, \mathbf{y}^{-n} \circ \mathbf{H} \rangle} \\ &+ [\textcolor{Peach}{\alpha + u\beta}]B = \\ & \langle \mathbf{a}_L, \mathbf{G} \rangle + \langle -z \cdot \mathbf{1}^n, \mathbf{G} \rangle + u \cdot \langle \mathbf{s}_L, \mathbf{G} \rangle + \langle z^2 \cdot \mathbf{2}^n, \mathbf{y}^{-n} \circ \mathbf{H} \rangle + \langle z \cdot \mathbf{1}^n, \mathbf{H} \rangle + \\ - & \langle \mathbf{a_R}, \mathbf{H} \rangle + u\cdot \langle \mathbf{s}_R, \mathbf{H} \rangle + [\alpha + u\beta]B + & \langle \mathbf{a}_R, \mathbf{H} \rangle + u\cdot \langle \mathbf{s}_R, \mathbf{H} \rangle + [\alpha + u\beta]B \end{aligned} \end{equation*} We could see that \textit{LHS} is equal to \textit{RHS} so the first check pass. -Taking the polynomial evaluation check: +Taking the $t_0$ correctness check: \begin{equation*} \begin{aligned} [t_u]G + [\tau_u]B \stackrel{\text{?}}{=}& [z^2]V + \delta(y,z) + [u]T_1 + [u^2]T_2\\ @@ -768,9 +768,9 @@ \subsection{Range proofs} \end{remark} \begin{example} - One of the most famous \textit{NP-complete} problems is the \textbf{subset-sum problem}: given a set of numbers presented as vector $\mathbf{s}$ and number $v \in \mathbb{N}$, does a some subset sums up to $v$. It turns out that we could use our \textbf{range-proof} protocol for this problem. One could simply replace first inner-product check $\langle \mathbf{a_L}, \mathbf{2}^n \rangle = v$ with $\langle \mathbf{a_L}, \mathbf{s} \rangle = v$ where $\mathbf{a_L}$ is the secret vector of bits that encode positions of $\mathbf{s}$ that sum up to $v$. + One of the most famous \textit{NP-complete} problems is the \textbf{subset-sum problem}: given a set of numbers presented as vector $\mathbf{s}$ and number $v \in \mathbb{N}$, does a some subset sums up to $v$. It turns out that we could use our \textbf{range-proof} protocol for this problem. One could simply replace first inner-product check $\langle \mathbf{a}_L, \mathbf{2}^n \rangle = v$ with $\langle \mathbf{a}_L, \mathbf{s} \rangle = v$ where $\mathbf{a}_L$ is the secret vector of bits that encode positions of $\mathbf{s}$ that sum up to $v$. - For example take $\mathbf{s} = (6,8,2,3)$ and $v = 14$. Then setting $\mathbf{a_L} = (1,1,0,0)$ we could use $\Pi_{rp}$ to prove that there exists a subset of $\mathbf{s}$ that sums up to $v=14$ without disclosing that subset. + For example take $\mathbf{s} = (6,8,2,3)$ and $v = 14$. Then setting $\mathbf{a}_L = (1,1,0,0)$ we could use $\Pi_{rp}$ to prove that there exists a subset of $\mathbf{s}$ that sums up to $v=14$ without disclosing that subset. Therefore, \textbf{bulletproofs range proof} protocol is capable to prove a knowledge of witness to any $NP$-problem as they all could be reduced to the $\textbf{subset-sum problem}$ \end{example} @@ -782,13 +782,13 @@ \subsubsection{Arithmetization} \textbf{Bulletproofs} arithmetization slightly differs from the classic R1CS we studied before, however it could be transformed vice-versa easily. Also \textbf{bulletproofs} arithmetization is more convinient and human-friendly for encoding most of the arithmetic circuits than the R1CS. -There are two types of variables in \textit{bulletproofs} constraint system: \textit{low-level} and \textit{high-level}. Typycally \textit{high-level} variables are provided with Pedersen commitments $\mathbf{V} \in \mathbb{G}^m$ as the private witness inputs $\mathbf{v} \in \mathbb{F}_p^m$ to the circuit, while \textit{low-level} variables $\mathbf{a_L}, \mathbf{a_R}, \mathbf{a_O} \in \mathbb{F}_p^n$ are the intermediate witness values of computation. We will define circuit as a set of multiplication constraints operating with \textit{low-level} variables and set of linear constraints which links \textit{low-level} variables between each other and \textit{high-level} variables as well. +There are two types of variables in \textit{bulletproofs} constraint system: \textit{low-level} and \textit{high-level}. Typycally \textit{high-level} variables are provided with Pedersen commitments $\mathbf{V} \in \mathbb{G}^m$ as the private witness inputs $\mathbf{v} \in \mathbb{F}_p^m$ to the circuit, while \textit{low-level} variables $\mathbf{a}_L, \mathbf{a}_R, \mathbf{a}_O \in \mathbb{F}_p^n$ are the intermediate witness values of computation. We will define circuit as a set of multiplication constraints operating with \textit{low-level} variables and set of linear constraints which links \textit{low-level} variables between each other and \textit{high-level} variables as well. Multiplication constraints are defined with one vector equation: -$$ \mathbf{a_L} \circ \mathbf{a_R} = \mathbf{a_O} $$ +$$ \mathbf{a}_L \circ \mathbf{a}_R = \mathbf{a}_O $$ Linear constraints are defined via: -$$ \mathbf{W}_L \cdot \mathbf{a_L} + \mathbf{W}_R \cdot \mathbf{a_R} + \mathbf{W}_O \cdot \mathbf{a_O} = \mathbf{W}_V \cdot \mathbf{v} + \mathbf{c} $$ +$$ \mathbf{W}_L \cdot \mathbf{a}_L + \mathbf{W}_R \cdot \mathbf{a}_R + \mathbf{W}_O \cdot \mathbf{a}_O = \mathbf{W}_V \cdot \mathbf{v} + \mathbf{c} $$ Where $\mathbf{a_L, a_R, a_O}$ -- vectors of left and right inputs for multiplication gates and output values (all of them are low-level variables). $\mathbf{W_L, W_R, W_O} \in \mathbb{F}_p^{q \times n}, \mathbf{W}_V \in \mathbb{F}_p^{q \times m}$ -- public matrices of weights for linear constraints(obviously known to verifier). $\mathbf{c} \in \mathbb{F}_p^q$ -- public vector of constants. Typycally they encode wiring of the circuit and other linear relations between variables. @@ -802,9 +802,9 @@ \subsubsection{Arithmetization} \textbf{Low-level variables:} \begin{equation*} - \mathbf{a_L} = \begin{bmatrix} x \\ x \\ y \end{bmatrix}, \quad - \mathbf{a_R} = \begin{bmatrix} x \\ x^2 \\ y \end{bmatrix}, \quad - \mathbf{a_O} = \begin{bmatrix} x^2\\x^3 \\ y^2 \end{bmatrix} + \mathbf{a}_L = \begin{bmatrix} x \\ x \\ y \end{bmatrix}, \quad + \mathbf{a}_R = \begin{bmatrix} x \\ x^2 \\ y \end{bmatrix}, \quad + \mathbf{a}_O = \begin{bmatrix} x^2\\x^3 \\ y^2 \end{bmatrix} \end{equation*} \textbf{High-level variables:} @@ -814,7 +814,7 @@ \subsubsection{Arithmetization} \textbf{Multiplication constraints:} \begin{equation*} - \mathbf{a_L} \circ \mathbf{a_R} = \mathbf{a_O} \Rightarrow \begin{bmatrix} + \mathbf{a}_L \circ \mathbf{a}_R = \mathbf{a}_O \Rightarrow \begin{bmatrix} x \cdot x = x^2 \\ x \cdot x^2 = x^3 \\ y \cdot y = y^2 @@ -825,18 +825,18 @@ \subsubsection{Arithmetization} \begin{equation*} \begin{aligned} - \mathbf{a_L}^{(1)} &= v_1 \\ - \mathbf{a_R}^{(1)} &= v_1 \\ - \mathbf{a_L}^{(2)} - \mathbf{a_L}^{(1)} &= 0 \\ - \mathbf{a_R}^{(2)} - \mathbf{a_O}^{(1)} &= 0 \\ - \mathbf{a_L}^{(3)} &= v_2 \\ - \mathbf{a_R}^{(3)} &= v_2 \\ - \mathbf{a_O}^{(3)} - \mathbf{a_O}^{(2)} - a \cdot \mathbf{a_L}^{(1)} &= b + \mathbf{a}_L^{(1)} &= v_1 \\ + \mathbf{a}_R^{(1)} &= v_1 \\ + \mathbf{a}_L^{(2)} - \mathbf{a}_L^{(1)} &= 0 \\ + \mathbf{a}_R^{(2)} - \mathbf{a}_O^{(1)} &= 0 \\ + \mathbf{a}_L^{(3)} &= v_2 \\ + \mathbf{a}_R^{(3)} &= v_2 \\ + \mathbf{a}_O^{(3)} - \mathbf{a}_O^{(2)} - a \cdot \mathbf{a}_L^{(1)} &= b \end{aligned} \end{equation*} \begin{equation*} - \mathbf{W}_L \cdot \mathbf{a_L} + \mathbf{W}_R \cdot \mathbf{a_R} + \mathbf{W}_O \cdot \mathbf{a_O} = \mathbf{W}_V \cdot \mathbf{v} + \mathbf{c} + \mathbf{W}_L \cdot \mathbf{a}_L + \mathbf{W}_R \cdot \mathbf{a}_R + \mathbf{W}_O \cdot \mathbf{a}_O = \mathbf{W}_V \cdot \mathbf{v} + \mathbf{c} \end{equation*} Where: @@ -930,17 +930,33 @@ \subsubsection{Arithmetization} \] \end{example} -We could use similar to \textit{range-proofs} technique to transform constraints of the circuit into inner-product relation. For multiplicative constraints take random $y \in \mathbb{F}_p$ and apply zero check: -$$ \langle \mathbf{a_L} \circ \mathbf{a_R} - \mathbf{a_O}, \mathbf{y}^n \rangle = 0$$ +\subsubsection{Proving a circuit satisfiability} +In this section we build an argument system for the following arithmetic circuit satisfiability relation: +\begin{equation} + \begin{aligned} + \mathcal{R}_{sat} = \left\{ + \begin{array}{l} + (G, B, \mathbf{V}, \mathbf{W}_L, \mathbf{W}_R, \mathbf{W}_O, \mathbf{W}_V, \mathbf{c}; \mathbf{a}_L, \mathbf{a}_R, \mathbf{a}_O, \mathbf{v}, \mathbf{r}) | \\ + \forall i=1..m: V_i = [v_i]G + [r_i]B \wedge \\ + \mathbf{a}_L \circ \mathbf{a}_R = \mathbf{a}_O \wedge \\ + \mathbf{W}_L \cdot \mathbf{a}_L + \mathbf{W}_R \cdot \mathbf{a}_R + \mathbf{W}_O \cdot \mathbf{a}_O = \mathbf{W}_V \cdot \mathbf{v} + \mathbf{c} + \end{array}\right\} + \end{aligned} +\end{equation} +Where $\mathbf{a}_L, \mathbf{a}_R, \mathbf{a}_O \in \mathbb{F}_p^n$, $\mathbf{v}, \mathbf{r} \in \mathbb{F}_p^m$, $\mathbf{W}_L, \mathbf{W}_R, \mathbf{W}_O \in \mathbb{F}_p^{q \times n}$, $\mathbf{W}_V \in \mathbb{F}_p^{q \times m}$, $\mathbf{c} \in \mathbb{F}_p^q$. +Informally this relation states that there exists a valid witness $\mathbf{v}$ that satisfies all constraints of the circuit. For the verifier witness is presented only as commitments vector $\mathbf{V}$. + +We could use similar to \textit{range-proofs} technique to compile constraints of the circuit into inner-product relation. For multiplicative constraints take random $y \in \mathbb{F}_p$ and apply zero check: +$$ \langle \mathbf{a}_L \circ \mathbf{a}_R - \mathbf{a}_O, \mathbf{y}^n \rangle = 0$$ Do the same for linear constraints, but for different randomness $z \in \mathbb{F}_p$: -$$ \langle \mathbf{W}_L \cdot \mathbf{a_L} + \mathbf{W}_R \cdot \mathbf{a_R} + \mathbf{W}_O \cdot \mathbf{a_O} - \mathbf{W}_V \cdot \mathbf{v} - \mathbf{c}, \mathbf{z}^q \rangle = 0$$ +$$ \langle \mathbf{W}_L \cdot \mathbf{a}_L + \mathbf{W}_R \cdot \mathbf{a}_R + \mathbf{W}_O \cdot \mathbf{a}_O - \mathbf{W}_V \cdot \mathbf{v} - \mathbf{c}, \mathbf{z}^q \rangle = 0$$ Combine this two checks to one using the same randomness $z$: \begin{equation*} \begin{aligned} - \langle \mathbf{a_L} \circ \mathbf{a_R} - \mathbf{a_O}, \mathbf{y}^n \rangle + z \cdot \langle \mathbf{W}_L \cdot \mathbf{a_L} + \mathbf{W}_R \cdot \mathbf{a_R} + \mathbf{W}_O \cdot \mathbf{a_O} - \mathbf{W}_V \cdot \mathbf{v} - \mathbf{c}, \mathbf{z}^q \rangle = 0 \\ - \langle \mathbf{a_L} \circ \mathbf{a_R} - \mathbf{a_O}, \mathbf{y}^n \rangle + \langle z \cdot \mathbf{z}^q, \mathbf{W}_L \cdot \mathbf{a_L} + \mathbf{W}_R \cdot \mathbf{a_R} + \mathbf{W}_O \cdot \mathbf{a_O} - \mathbf{W}_V \cdot \mathbf{v} - \mathbf{c} \rangle = 0 + \langle \mathbf{a}_L \circ \mathbf{a}_R - \mathbf{a}_O, \mathbf{y}^n \rangle + z \cdot \langle \mathbf{W}_L \cdot \mathbf{a}_L + \mathbf{W}_R \cdot \mathbf{a}_R + \mathbf{W}_O \cdot \mathbf{a}_O - \mathbf{W}_V \cdot \mathbf{v} - \mathbf{c}, \mathbf{z}^q \rangle = 0 \\ + \langle \mathbf{a}_L \circ \mathbf{a}_R - \mathbf{a}_O, \mathbf{y}^n \rangle + \langle z \cdot \mathbf{z}^q, \mathbf{W}_L \cdot \mathbf{a}_L + \mathbf{W}_R \cdot \mathbf{a}_R + \mathbf{W}_O \cdot \mathbf{a}_O - \mathbf{W}_V \cdot \mathbf{v} - \mathbf{c} \rangle = 0 \end{aligned} \end{equation*} @@ -949,9 +965,9 @@ \subsubsection{Arithmetization} Then split the second inner product and factor-out public terms to RHS: \begin{equation*} \begin{aligned} - \langle \mathbf{a_L} \circ \mathbf{a_R} - \mathbf{a_O}, \mathbf{y}^n \rangle + \langle z \cdot \mathbf{z}^q, \mathbf{W}_L \cdot \mathbf{a_L} \rangle + \langle z \cdot \mathbf{z}^q, \mathbf{W}_R \cdot \mathbf{a_R} \rangle + \langle z \cdot \mathbf{z}^q, \mathbf{W}_O \cdot \mathbf{a_O} \rangle \\ + \langle \mathbf{a}_L \circ \mathbf{a}_R - \mathbf{a}_O, \mathbf{y}^n \rangle + \langle z \cdot \mathbf{z}^q, \mathbf{W}_L \cdot \mathbf{a}_L \rangle + \langle z \cdot \mathbf{z}^q, \mathbf{W}_R \cdot \mathbf{a}_R \rangle + \langle z \cdot \mathbf{z}^q, \mathbf{W}_O \cdot \mathbf{a}_O \rangle \\ - \langle z \cdot \mathbf{z}^q, \mathbf{W}_V \cdot \mathbf{v} \rangle - \langle z \cdot \mathbf{z}^q, \mathbf{c} \rangle = 0 \\ - \langle \mathbf{a_L} \circ \mathbf{a_R} - \mathbf{a_O}, \mathbf{y}^n \rangle + \langle z \cdot \mathbf{z}^q, \mathbf{W}_L \cdot \mathbf{a_L} \rangle + \langle z \cdot \mathbf{z}^q, \mathbf{W}_R \cdot \mathbf{a_R} \rangle + \langle z \cdot \mathbf{z}^q, \mathbf{W}_O \cdot \mathbf{a_O} \rangle \\ + \langle \mathbf{a}_L \circ \mathbf{a}_R - \mathbf{a}_O, \mathbf{y}^n \rangle + \langle z \cdot \mathbf{z}^q, \mathbf{W}_L \cdot \mathbf{a}_L \rangle + \langle z \cdot \mathbf{z}^q, \mathbf{W}_R \cdot \mathbf{a}_R \rangle + \langle z \cdot \mathbf{z}^q, \mathbf{W}_O \cdot \mathbf{a}_O \rangle \\ - \langle z \cdot \mathbf{z}^q, \mathbf{W}_V \cdot \mathbf{v} \rangle = \langle z \cdot \mathbf{z}^q, \mathbf{c} \rangle \end{aligned} \end{equation*} @@ -959,8 +975,8 @@ \subsubsection{Arithmetization} Applying conjugation rule(if $A$ -- linear operator and $A^T$ -- its transpose(conjugate) then $\langle \mathbf{a}, A \mathbf{b} \rangle = \langle A^T \mathbf{a}, \mathbf{b} \rangle$) to the second inner product we get: \begin{equation*} \begin{aligned} - \langle \mathbf{a_L} \circ \mathbf{a_R} - \mathbf{a_O}, \mathbf{y}^n \rangle + \langle \mathbf{W}_L^T \cdot (z \cdot \mathbf{z}^q), \mathbf{a_L} \rangle + \langle \mathbf{W}_R^T \cdot (z \cdot \mathbf{z}^q), \mathbf{a_R} \rangle + \\ - \langle \mathbf{W}_O^T \cdot (z \cdot \mathbf{z}^q), \mathbf{a_O} \rangle - \langle \mathbf{W}_V^T \cdot (z \cdot \mathbf{z}^q), \mathbf{v} \rangle = \langle z \cdot \mathbf{z}^q, \mathbf{c} \rangle + \langle \mathbf{a}_L \circ \mathbf{a}_R - \mathbf{a}_O, \mathbf{y}^n \rangle + \langle \mathbf{W}_L^T \cdot (z \cdot \mathbf{z}^q), \mathbf{a}_L \rangle + \langle \mathbf{W}_R^T \cdot (z \cdot \mathbf{z}^q), \mathbf{a}_R \rangle + \\ + \langle \mathbf{W}_O^T \cdot (z \cdot \mathbf{z}^q), \mathbf{a}_O \rangle - \langle \mathbf{W}_V^T \cdot (z \cdot \mathbf{z}^q), \mathbf{v} \rangle = \langle z \cdot \mathbf{z}^q, \mathbf{c} \rangle \end{aligned} \end{equation*} @@ -977,17 +993,17 @@ \subsubsection{Arithmetization} Now we could finally rewrite the equation as: \begin{equation*} \begin{aligned} - \langle \mathbf{a_L} \circ \mathbf{a_R} - \mathbf{a_O}, \mathbf{y}^n \rangle + \langle \mathbf{w}_L, \mathbf{a_L} \rangle + \langle \mathbf{w}_R, \mathbf{a_R} \rangle + \langle \mathbf{w}_O, \mathbf{a_O} \rangle - \langle \mathbf{w}_V, \mathbf{v} \rangle = w_c + \langle \mathbf{a}_L \circ \mathbf{a}_R - \mathbf{a}_O, \mathbf{y}^n \rangle + \langle \mathbf{w}_L, \mathbf{a}_L \rangle + \langle \mathbf{w}_R, \mathbf{a}_R \rangle + \langle \mathbf{w}_O, \mathbf{a}_O \rangle - \langle \mathbf{w}_V, \mathbf{v} \rangle = w_c \end{aligned} \end{equation*} -Then rearrange and combine terms so that $\mathbf{a_L}$, $\mathbf{a_O}$ be on the left side and $\mathbf{a_R}$ on the right side: +Then rearrange and combine terms so that $\mathbf{a}_L$, $\mathbf{a}_O$ be on the left side and $\mathbf{a}_R$ on the right side: \begin{equation*} \begin{aligned} - \langle \mathbf{a_L} \circ \mathbf{a_R}, \mathbf{y}^n \rangle - \langle \mathbf{a_O}, \mathbf{y}^n \rangle + \langle \mathbf{w}_L, \mathbf{a_L} \rangle + \langle \mathbf{w}_R, \mathbf{a_R} \rangle + \langle \mathbf{w}_O, \mathbf{a_O} \rangle = \langle \mathbf{w}_V, \mathbf{v} \rangle + w_c \\ - \langle \mathbf{a_L}, \mathbf{y}^n \circ \mathbf{a_R} \rangle + \langle \mathbf{a_O}, - \mathbf{y}^n + \mathbf{w}_O \rangle + \langle \mathbf{w}_L, \mathbf{a_L} \rangle + \langle \mathbf{w}_R, \mathbf{a_R} \rangle = \langle \mathbf{w}_V, \mathbf{v} \rangle + w_c \\ - \langle \mathbf{a_L}, \mathbf{y}^n \circ \mathbf{a_R} \rangle + \langle \mathbf{a_O}, - \mathbf{y}^n + \mathbf{w}_O \rangle + \langle \mathbf{w}_L, \mathbf{a_L} \rangle + \langle \mathbf{y}^n \circ \mathbf{a}_R, \mathbf{y}^{-n} \circ \mathbf{w_R} \rangle = \langle \mathbf{w}_V, \mathbf{v} \rangle + w_c \\ - \langle \mathbf{a_L} + \mathbf{y}^{-n} \circ \mathbf{w_R}, \mathbf{y}^n \circ \mathbf{a_R} \rangle + \langle \mathbf{a_O}, - \mathbf{y}^n + \mathbf{w}_O \rangle + \langle \mathbf{w}_L, \mathbf{a_L} \rangle = \langle \mathbf{w}_V, \mathbf{v} \rangle + w_c \\ + \langle \mathbf{a}_L \circ \mathbf{a}_R, \mathbf{y}^n \rangle - \langle \mathbf{a}_O, \mathbf{y}^n \rangle + \langle \mathbf{w}_L, \mathbf{a}_L \rangle + \langle \mathbf{w}_R, \mathbf{a}_R \rangle + \langle \mathbf{w}_O, \mathbf{a}_O \rangle = \langle \mathbf{w}_V, \mathbf{v} \rangle + w_c \\ + \langle \mathbf{a}_L, \mathbf{y}^n \circ \mathbf{a}_R \rangle + \langle \mathbf{a}_O, - \mathbf{y}^n + \mathbf{w}_O \rangle + \langle \mathbf{w}_L, \mathbf{a}_L \rangle + \langle \mathbf{w}_R, \mathbf{a}_R \rangle = \langle \mathbf{w}_V, \mathbf{v} \rangle + w_c \\ + \langle \mathbf{a}_L, \mathbf{y}^n \circ \mathbf{a}_R \rangle + \langle \mathbf{a}_O, - \mathbf{y}^n + \mathbf{w}_O \rangle + \langle \mathbf{w}_L, \mathbf{a}_L \rangle + \langle \mathbf{y}^n \circ \mathbf{a}_R, \mathbf{y}^{-n} \circ \mathbf{w_R} \rangle = \langle \mathbf{w}_V, \mathbf{v} \rangle + w_c \\ + \langle \mathbf{a}_L + \mathbf{y}^{-n} \circ \mathbf{w_R}, \mathbf{y}^n \circ \mathbf{a}_R \rangle + \langle \mathbf{a}_O, - \mathbf{y}^n + \mathbf{w}_O \rangle + \langle \mathbf{w}_L, \mathbf{a}_L \rangle = \langle \mathbf{w}_V, \mathbf{v} \rangle + w_c \\ \end{aligned} \end{equation*} @@ -1024,17 +1040,17 @@ \subsubsection{Arithmetization} \end{equation} So than we could get desired sum of inner products as the second-degree coefficient $s_2$: $$ w_c + \langle \mathbf{w}_V, \mathbf{v} \rangle + \delta(y, z) = s_2$$ -In order to obtain final polynimials $\mathbf{l}(x), \mathbf{r}(x)$ we must firstly blind $\mathbf{a_L}, \mathbf{a_R}$: +In order to obtain final polynimials $\mathbf{l}(x), \mathbf{r}(x)$ we must firstly blind $\mathbf{a}_L, \mathbf{a}_R$: \begin{equation} \begin{aligned} - \mathbf{a_L} \gets \mathbf{a_L} + \mathbf{s_L}x^2 && \mathbf{a_R} \gets \mathbf{a_R} + \mathbf{s_R}x^2 + \mathbf{a}_L \gets \mathbf{a}_L + \mathbf{s}_Lx^2 && \mathbf{a}_R \gets \mathbf{a}_R + \mathbf{s}_Rx^2 \end{aligned} \label{eq:inner-product-circuit-blinding} \end{equation} \begin{remark} - We multiplied blinders $\mathbf{s_L}, \mathbf{s_R}$ with the second power of challenge $x^2$ because we want the blinding terms to do not interfere with other parts of inner-product. + We multiplied blinders $\mathbf{s}_L, \mathbf{s}_R$ with the second power of challenge $x^2$ because we want the blinding terms to do not interfere with other parts of inner-product. - $\mathbf{a_O}$ does not need separate blinding as it's located on the left side of the inner-product (\ref{eq:inner-product-circuit-1}) along with $\mathbf{a_L}$, which is already blinded by $\mathbf{s_L}$. + $\mathbf{a}_O$ does not need separate blinding as it's located on the left side of the inner-product (\ref{eq:inner-product-circuit-1}) along with $\mathbf{a}_L$, which is already blinded by $\mathbf{s}_L$. \end{remark} Now we could compute polynomials $\mathbf{l}(x), \mathbf{r}(x)$ from (\ref{eq:inner-product-circuit-1}) using assignments from (\ref{eq:inner-product-circuit-assignment}) and blindings from (\ref{eq:inner-product-circuit-blinding}): @@ -1055,7 +1071,9 @@ \subsubsection{Arithmetization} \begin{aligned} t(x) &= \langle \mathbf{l}(x), \mathbf{r}(x) \rangle = t_1 x + t_2 x^2 + t_3 x^3 + t_4 x^4 + t_5 x^5 + t_6 x^6 = \sum_{i=1}^{6} t_i x^i \\ t_1 &= \langle \mathbf{a}_L + \mathbf{y}^{-n} \circ \mathbf{w}_R, - \mathbf{y}^n + \mathbf{w}_O \rangle \\ - t_2 &= \langle \mathbf{a}_L + \mathbf{y}^{-n} \circ \mathbf{w}_R, \mathbf{y}^n \circ \mathbf{a}_R + \mathbf{w}_L \rangle + \langle \mathbf{a}_O, -\mathbf{y}^n + \mathbf{w}_O \rangle = w_c + \langle \mathbf{w}_V, \mathbf{v} \rangle + \delta(y, z) \\ + t_2 &= \langle \mathbf{a}_L + \mathbf{y}^{-n} \circ \mathbf{w}_R, \mathbf{y}^n \circ \mathbf{a}_R + \mathbf{w}_L \rangle + \langle \mathbf{a}_O, -\mathbf{y}^n + \mathbf{w}_O \rangle \\ + &= \langle \mathbf{a}_L \circ \mathbf{a}_R, \mathbf{y}^n \rangle - \langle \mathbf{a}_O, \mathbf{y}^n \rangle + \langle \mathbf{w}_L, \mathbf{a}_L \rangle + \langle \mathbf{w}_R, \mathbf{a}_R \rangle + \langle \mathbf{w}_O, \mathbf{a}_O \rangle + \delta(y, z) \\ + &= w_c + \langle \mathbf{w}_V, \mathbf{v} \rangle + \delta(y, z) \\ t_3 &= \langle \mathbf{s}_L, - \mathbf{y}^n + \mathbf{w}_O \rangle + \langle \mathbf{a}_O, \mathbf{y}^n \circ \mathbf{a}_R + \mathbf{w}_L \rangle \\ t_4 &= \langle \mathbf{s}_L, \mathbf{y}^n \circ \mathbf{a}_R + \mathbf{w}_L \rangle + \langle \mathbf{a}_L + \mathbf{y}^{-n} \circ \mathbf{w}_R, \mathbf{y}^n \circ \mathbf{s}_R \rangle \\ t_5 &= \langle \mathbf{a}_O, \mathbf{y}^n \circ \mathbf{s}_R \rangle \\ @@ -1063,8 +1081,152 @@ \subsubsection{Arithmetization} \end{aligned} \label{eq:inner-product-circuit-polynomials-product} \end{equation} +Our proving strategy is the same as in the \textit{range-proof} case: +\begin{itemize} + \item Prove that $\mathbf{l}(x), \mathbf{r}(x)$ are correct using binding commitments to their coefficients. + \item Prove that $t_2$ is correct (as it's the inner-product we want to prove) using evaluation at challenge point $u$. + \item Apply inner-product argument to compress the proof of evaluation of $t(x)$ at challenge point $u$. +\end{itemize} + +\begin{definition} + The \textbf{arithmetic circuit satisfiability} protocol $\Pi_{sat} = (\mathsf{Setup}, \mathcal{P}, \mathcal{V})$ for the relation $\mathcal{R}_{sat}$ with prover $\mathcal{P}$ and verifier $\mathcal{V}$ is defined as follows: + \begin{itemize} + \item $\mathsf{Setup}$: returns vector of group generators with unknown discrete log relations $\mathbf{G,H} \in \mathbb{G}^n$. + \item Prover $\mathcal{P}$ choses blinding factors $\alpha, \beta, \gamma \in \mathbb{F}_p, \mathbf{s}_L, \mathbf{s}_R \in \mathbb{F}_p^n$ and sends the following commitments to $\mathcal{V}$: + \begin{equation*} + \begin{aligned} + A_I &= \langle \mathbf{a}_L, \mathbf{G} \rangle + \langle \mathbf{a}_R, \mathbf{H} \rangle + [\alpha]B\\ + A_O &= \langle \mathbf{a}_O, \mathbf{G} \rangle + [\gamma]B\\ + S &= \langle \mathbf{s}_L, \mathbf{G} \rangle + \langle \mathbf{s}_R, \mathbf{H} \rangle + [\beta]B + \end{aligned} + \end{equation*} + \item Verifier samples challenges $y,z \xleftarrow{R} \mathbb{F}_p$ and sends them to $\mathcal{P}$. + \item Using provided challenges $y,z$ prover forms polynomials $\mathbf{l}(x), \mathbf{r}(x)$: + \begin{equation*} + \begin{aligned} + \mathbf{l}(x) &= \mathbf{s}_L \cdot x^3 + \mathbf{a}_O \cdot x^2 + (\mathbf{a}_L + \mathbf{y}^{-n} \circ \mathbf{w}_R) \cdot x \\ + \mathbf{r}(x) &= \mathbf{y}^n \circ \mathbf{s}_R \cdot x^3 + (\mathbf{y}^n \circ \mathbf{a}_R + \mathbf{w}_L) \cdot x - \mathbf{y}^n + \mathbf{w}_O + \end{aligned} + \end{equation*} + computes $$t(x) = \langle \mathbf{l}(x), \mathbf{r}(x) \rangle = t_1 x + t_2 x^2 + t_3 x^3 + t_4 x^4 + t_5 x^5 + t_6 x^6$$ + choses random blinding factors $\tau_1, \tau_3, \tau_4, \tau_5, \tau_6 \in \mathbb{F}_p$ and sends to $\mathcal{V}$ commitments to its coefficients: + \begin{equation*} + \begin{aligned} + T_1 &= [t_1]G + [\tau_1]B\\ + T_3 &= [t_3]G + [\tau_3]B\\ + T_4 &= [t_4]G + [\tau_4]B\\ + T_5 &= [t_5]G + [\tau_5]B\\ + T_6 &= [t_6]G + [\tau_6]B + \end{aligned} + \end{equation*} + \textbf{Note:} Prover does not send separate commitment to $t_2$ as the verifier could derive it from $\mathbf{V}$ and the circuit public parameters: + \begin{align*} + t_2 &= w_c + \langle \mathbf{w}_V, \mathbf{v} \rangle + \delta(y, z) \\ + T_2 &= \langle \mathbf{w}_V, \mathbf{V} \rangle + [\delta(y, z) + w_c]G + \end{align*} + \item Verifier samples and sends to $\mathcal{P}$ random evaluation point $u \xleftarrow{R} \mathbb{F}_p$. + \item Prover evaluates polynomials at $u$: + \begin{equation*} + \begin{aligned} + \mathbf{l}_u &= \mathbf{l}(u) \\ + \mathbf{r}_u &= \mathbf{r}(u) \\ + t_u &= \langle \mathbf{l}_u, \mathbf{r}_u \rangle = t(u) \\ + \tau_u &= \tau_1 \cdot u + \langle \mathbf{w}_V, \mathbf{r} \rangle u^2 + \tau_3 \cdot u^3 + \tau_4 \cdot u^4 + \tau_5 \cdot u^5 + \tau_6 \cdot u^6\\ + \alpha_u &= \alpha u + \gamma u^2 + \beta u^3 + \end{aligned} + \end{equation*} + and sends $(\mathbf{l}_u, \mathbf{r}_u, t_u, \alpha_u, \tau_u)$ to $\mathcal{V}$. + \item Verifier performs checks: + \begin{equation*} + \begin{aligned} + &[u]A_I + [u^2]A_O + [u^3]S - \langle \mathbf{1}, \mathbf{H} \rangle + \\ + &u \cdot (\langle \mathbf{y}^{-n} \circ \mathbf{w}_L, \mathbf{G} \rangle + \langle \mathbf{y}^{-n} \circ \mathbf{w}_R, \mathbf{H} \rangle) + \langle \mathbf{y}^{-n} \circ \mathbf{w}_O, \mathbf{H} \rangle \\ + &\stackrel{\text{?}}{=} \langle \mathbf{l}_u, \mathbf{G} \rangle + \langle \mathbf{r}_u, \mathbf{y}^{-n} \circ \mathbf{H} \rangle + [\alpha_u]B\\ + [t_u]G + [\tau_u]B &\stackrel{\text{?}}{=} [u]T_1 + u^2 \cdot (\langle \mathbf{w}_V, \mathbf{V} \rangle + [\delta(y,z) + w_c]G) +\\ + &[u^3]T_3 + [u^4]T_4 + [u^5]T_5 + [u^6]T_6\\ + t_u &\stackrel{\text{?}}{=} \langle \mathbf{l}_u, \mathbf{r}_u \rangle + \end{aligned} + \end{equation*} + \end{itemize} +\end{definition} + +\begin{remark} + As in the \textit{range proof} protocol case the last two steps of $\Pi_{sat}$ could be substituted with an inner-product argument $\Pi_{ip}$ to provide logarithmic size-proof with the following steps: + \begin{itemize} + \item After $\mathcal{P}$ evaluates polynomials at $u$ he sends $(t_u, \alpha_u, \tau_u$ to $\mathcal{V})$ and computes commitment: + $$P = \langle \mathbf{l}_u, \mathbf{G} \rangle + \langle \mathbf{r}_u, \mathbf{y}^{-n} \circ \mathbf{H} \rangle$$ + \item $\mathcal{V}$ performs check: + $$ + [t_u]G + [\tau_u]B \stackrel{\text{?}}{=} \sum_{k=1, k\neq 2}^6 [u^k]T_k + u^2 \cdot (\langle \mathbf{w}_V, \mathbf{V} \rangle + [\delta(y,z) + w_c]G) + $$ + halts if it fails and reconstructs commitment $P$ otherwise: + \begin{align*} + P =& [u]A_I + [u^2]A_O + [u^3]S - \langle \mathbf{1}, \mathbf{H} \rangle +\\ + &u \cdot (\langle \mathbf{y}^{-n} \circ \mathbf{w}_L, \mathbf{G} \rangle + \langle \mathbf{y}^{-n} \circ \mathbf{w}_R, \mathbf{H} \rangle) + \langle \mathbf{y}^{-n} \circ \mathbf{w}_O, \mathbf{H} \rangle - [\alpha_u]B + \end{align*} + \item Parties run inner-product argument $\Pi_{ip}$ on $(\mathbf{G},\mathbf{y}^{-n} \circ \mathbf{H}, P, t_u; \mathbf{l}_u, \mathbf{r}_u)$ + \end{itemize} +\end{remark} + +\begin{theorem} + The \textbf{arithmetic circuit satisfiability protocol} $\Pi_{sat}$ has \textit{perfect completeness, computational extended witness emulation, perfect honest-verifier zero-knowledge} + \label{th:arithmetic_circuit_satisfiability} +\end{theorem} +\textbf{Proof idea}. \textit{Perfect completeness} +\begin{itemize} + \item \textbf{Polynomial correctness check}: + \begin{equation*} + \begin{aligned} + LHS &= [u]A_I + [u^2]A_O + [u^3]S - \langle \mathbf{1}, \mathbf{H} \rangle +\\ + &u \cdot (\langle \mathbf{y}^{-n} \circ \mathbf{w}_L, \mathbf{G} \rangle + \langle \mathbf{y}^{-n} \circ \mathbf{w}_R, \mathbf{H} \rangle) + \langle \mathbf{y}^{-n} \circ \mathbf{w}_O, \mathbf{H} \rangle \\ + &= \textcolor{Sepia}{u \cdot \langle \mathbf{a}_L, \mathbf{G} \rangle + u \cdot \langle \mathbf{a}_R, \mathbf{H} \rangle} + \textcolor{RubineRed}{[\alpha u]B} + \textcolor{teal}{u^2 \cdot \langle \mathbf{a}_O, \mathbf{G} \rangle} + \textcolor{RubineRed}{[\gamma u^2]B} + \\ + &+ \textcolor{Violet}{u^3 \cdot \langle \mathbf{s}_L, \mathbf{G} \rangle + u^3 \cdot \langle \mathbf{s}_R, \mathbf{H} \rangle} + \textcolor{RubineRed}{[\beta u^3]B} \textcolor{gray}{- \langle \mathbf{1}, \mathbf{H} \rangle} + \\ + &+ \textcolor{Sepia}{u \cdot \langle \mathbf{y}^{-n} \circ \mathbf{w}_L, \mathbf{G} \rangle + u\cdot \langle \mathbf{y}^{-n} \circ \mathbf{w}_R, \mathbf{H} \rangle} + \textcolor{gray}{\langle \mathbf{y}^{-n} \circ \mathbf{w}_O, \mathbf{H} \rangle} \\ + RHS &= \langle \mathbf{l}_u, \mathbf{G} \rangle + \langle \mathbf{r}_u, \mathbf{y}^{-n} \circ \mathbf{H} \rangle + [\alpha_u]B \\ + &= u^3 \cdot \langle \mathbf{s}_L, \mathbf{G} \rangle + u^2 \cdot \langle \mathbf{a}_O, \mathbf{G} \rangle + u \cdot \langle \mathbf{a}_L + \mathbf{y}^{-n} \circ \mathbf{w}_R, \mathbf{G} \rangle + \\ + &+ u^3 \cdot \langle \mathbf{y}^n \circ \mathbf{s}_R, \mathbf{y}^{-n} \circ \mathbf{H} \rangle + u \cdot \langle \mathbf{y}^n \circ \mathbf{a}_R, \mathbf{y}^{-n} \circ \mathbf{H} \rangle + u \cdot \langle \mathbf{w}_L, \mathbf{y}^n \circ \mathbf{H} \rangle \\ + &- \langle \mathbf{y}^n, \mathbf{y}^{-n} \circ \mathbf{H} \rangle + \langle \mathbf{w}_O, \mathbf{y}^{-n} \circ \mathbf{H} \rangle + [\alpha u + \gamma u^2 + \beta u^3]B \\ + &= \textcolor{Violet}{u^3 \cdot \langle \mathbf{s}_L, \mathbf{G} \rangle} + \textcolor{teal}{u^2 \cdot \langle \mathbf{a}_O, \mathbf{G} \rangle} + \textcolor{Sepia}{u \cdot \langle \mathbf{a}_L, \mathbf{G}\rangle + u \cdot \langle \mathbf{y}^{-n} \circ \mathbf{w}_R, \mathbf{G} \rangle} + \\ + &+\textcolor{Violet}{u^3 \cdot \langle \mathbf{s}_R, \mathbf{H} \rangle} + \textcolor{Sepia}{u \cdot \langle \mathbf{a}_R, \mathbf{H} \rangle + u \cdot \langle \mathbf{w}_L, \mathbf{y}^n \circ \mathbf{H} \rangle} \\ + &\textcolor{gray}{- \langle \mathbf{1}^n, \mathbf{H} \rangle + \langle \mathbf{w}_O, \mathbf{y}^{-n} \circ \mathbf{H} \rangle} + \textcolor{RubineRed}{[\alpha u]B + [\gamma u^2]B + [\beta u^3]B} + \end{aligned} + \end{equation*} + As we see $LHS=RHS$ so the check holds. + \item \textbf{$t_2$ correctness check}: + \begin{equation*} + \begin{aligned} + [t_u]G + [\tau_u]B &\stackrel{\text{?}}{=} [u]T_1 + u^2 \cdot (\langle \mathbf{w}_V, \mathbf{V} \rangle + [\delta(y,z) + w_c]G) +\\ + &+ [u^3]T_3 + [u^4]T_4 + [u^5]T_5 + [u^6]T_6 + \end{aligned} + \end{equation*} + Note that + \begin{align*} + \langle \mathbf{w}_V, \mathbf{V} \rangle &= \sum_{i=1}^m [w_{V,i}]V_i = \sum_{i=1}^n ([w_{i, k} \cdot v_i]G + [w_{V, i} \cdot r_i]B) \\ + &= [\sum_{i=1}^m w_{V, i} \cdot v_i]G + [\sum_{i=1}^m w_{V, i} \cdot r_i]B \\ + &= [\langle \mathbf{w}_V, \mathbf{v} \rangle] G + [\langle \mathbf{w}_V, \mathbf{r} \rangle] B + \end{align*} + so simplyfying the LHS we get: + \begin{equation*} + \begin{aligned} + &[t_u]G + [\tau_u]B = \\ + & [t_1 u + (w_c + \langle \mathbf{w}_V, \mathbf{v} \rangle + \delta(y, z)) u^2 + t_3 u^3 + t_4 u^4 + t_5 u^5 + t_6 u^6]G + \\ + & [\tau_1 u + \langle \mathbf{w}_V, \mathbf{r} \rangle u^2 + \tau_3 u^3 + \tau_4 u^4 + \tau_5 u^5 + \tau_6 u^6]B = \\ + & [u]T_1 + u^2 \cdot (\langle \mathbf{w}_V, \mathbf{V} \rangle + [\delta(y,z) + w_c]G) +\\ + &+ [u^3]T_3 + [u^4]T_4 + [u^5]T_5 + [u^6]T_6 = RHS \quad \square + \end{aligned} + \end{equation*} +\end{itemize} +\textit{Perfect honest-verifier zero-knowledge} follows from the zero-knowledge construction of $\Pi_{zkip}$ protocol as Vertifier learns no information about $\mathbf{a}_L, \mathbf{a}_R, \mathbf{v}$. +\textit{Computational extended witness emulation} implies building extractor that combines extractors for two subprotocols: $\mathcal{E}_{ip}$ extracts witness ($\mathbf{l}_u, \mathbf{r}_u$) from $\Pi_{ip}$ than the extractor built on top of $\mathcal{E}_{zkip}$ extracts high-level witness ($\mathbf{a}_L, \mathbf{a}_R, \mathbf{a}_O, \mathbf{v}$) from $\Pi_{zkip} \quad \square$ + +The proof size of \textbf{arithmetic circuit satisfiability} protocol is $2 \log_2n +8$ group $\mathbb{G}$ elements and $5$ field $\mathbb{F}_p$ elements. + +Usually the number of miltipliers $n$ is not a power of $2$ so that the intermediate witness vectors $\mathbf{a}_L, \mathbf{a}_R, \mathbf{a}_O$ must be padded with zeroes to the next power of $2$ in order to apply the \textit{inner-product argument} efficiently. +\begin{remark} + The $\Pi_{sat}$ protocol could be slightly modified to provide intermediate random challenges inside the circuit. For example it would allow proving \textit{permutation check}: $\{a,b\} = \{c,d\} \iff (a-x)\cdot(b-x) = (c-x)\cdot(d-x)$ for some random challenge $x$. +\end{remark} \subsection*{Acknowledgements} This section was heavily inspired by: \begin{itemize} From 869ba595179c65991cf739f25e807660777ba3ed Mon Sep 17 00:00:00 2001 From: Yevhen Hrubiian Date: Wed, 23 Jul 2025 18:26:06 +0300 Subject: [PATCH 15/25] work on slides --- lectures/2-9-bulletproofs.tex | 26 +- presentations/15-bulletproofs.tex | 318 +++++++++++++++++++ presentations/zkdl-presentation-template.cls | 2 +- 3 files changed, 334 insertions(+), 12 deletions(-) create mode 100644 presentations/15-bulletproofs.tex diff --git a/lectures/2-9-bulletproofs.tex b/lectures/2-9-bulletproofs.tex index fb65403..1c34a86 100644 --- a/lectures/2-9-bulletproofs.tex +++ b/lectures/2-9-bulletproofs.tex @@ -167,21 +167,24 @@ \subsubsection{Optimized polynomial multiplication protocol} \end{theorem} \textbf{Proof}. We left to a reader proof of the theorem in the sake of brevity because it's very similar to the proof of \Cref{th:poly_mul_naive} $\quad \square$ +\subsubsection{Zero-knowledge multiplication protocol} -Finally, We could easily build the protocol for the zk-multiplication relation: $$\mathcal{R}_{abc} = \{ ;c,a,b \vert c=ab \}$$ +Finally, we could easily build the protocol for the zk-multiplication relation where each witness element presented in statement as a Pedersen commitment: +$$\mathcal{R}_{abc} = \left\{\begin{array}{l} + (G,H,B,A,T_0;a,b, \alpha, \tau_0) \vert \\ + A = [a]G + [b]H + [\alpha]B \wedge \\ + T_0 = [ab]G + [\tau_0]B +\end{array}\right\}$$ +Where $G,H,B \in \mathbb{G}$ are generators with unknown discrete log relations, $A \in \mathbb{G}$ is a Pedersen commitment to $a,b$ and $T_0 \in \mathbb{G}$ is a Pedersen commitment to their product $ab$. \begin{definition} - The \textbf{multiplication protocol} $\Pi_{abc} = (\mathsf{Setup}, \mathcal{P,V})$ for the relation $\mathcal{R}_{abc}$ with prover $\mathcal{P}$ and verifier $\mathcal{V}$ is defined as follows: + The \textbf{multiplication protocol} $\Pi_{abc} = (\mathcal{P,V})$ for the relation $\mathcal{R}_{abc}$ with prover $\mathcal{P}$ and verifier $\mathcal{V}$ is defined as follows: \begin{itemize} - \item $\mathsf{Setup}$ returns triple of group generators with unknown discrete log relations $G, H, B \in \mathbb{G}$ - \item Parties $\mathcal{P, V}$ run the following protocol: - \begin{itemize} - \item Prover $\mathcal{P}$ draws random $s_L, s_R \xleftarrow{R} \mathbb{F}_p$ and defines polynomials: $$l(x)=a+s_L x,\quad r(x)=b+s_R x,\quad t(x)=l(x)r(x)$$ - \item Parties run $\Pi_{mul}$ on inputs $(l(x),r(x),t(x))$ - \end{itemize} + \item Prover $\mathcal{P}$ draws random $s_L, s_R \xleftarrow{R} \mathbb{F}_p$ and defines polynomials: $$l(x)=a+s_L x,\quad r(x)=b+s_R x,\quad t(x)=l(x)r(x)$$ + \item Parties run $\Pi_{mul}$ on inputs $(l(x),r(x),t(x))$ along with provided a-priori setup $G,H,B \in \mathbb{G}$ and commitments $A, T_0$ \end{itemize} \end{definition} -Here we refer to $A$ as a Pedersen commitment to $a,b$ and $T_0$ as a Pedersen commitment to their product which could be given to verifier before start of the protocol. The protocol obviously has \textit{perfect completeness, special soundness, perfect honest-verifier zero-knowledge} due to the \Cref{th:poly_mul}. +The protocol obviously has \textit{perfect completeness, special soundness, perfect honest-verifier zero-knowledge} due to the \Cref{th:poly_mul}. \begin{remark} %todo: add reference for Chaum-Pedersen Now curious reader may wonder why build so overwhelmingly complicated protocol for simple multiplication and not just use classic \textit{Chaum-Pedersen protocol} for DH-triplets? Indeed, it definitely could establish that for given group elements $[a]G, [b]G, [c]G$ equality $c=ab$ holds, but unfortunatelly commitments $[a]G, [b]G, [c]G$ do not have a \textit{binding} property (though preserving hiding property) so they're almost useless for building more complex protocols where the prover usually needs to bind to specific values without having an ability to silently modify them. @@ -215,8 +218,9 @@ \subsubsection{Zero-knowledge inner-product protocol} \begin{definition} The \textbf{zero-knowledge inner-product protocol} $\Pi_{zkip} = (\mathcal{P,V})$ for the relation \begin{align*} - \mathcal{R}_{zkip} = \{ (\mathbf{G,H},G,B,A,V;\mathbf{a,b},\alpha, \gamma) \vert & A = \langle \mathbf{a,G} \rangle + \langle \mathbf{b,H} \rangle + [\alpha] B, \\ &V = [\langle \mathbf{a,b} \rangle]G + [\gamma]B \} - \end{align*} where $\mathbf{G,H} \in \mathbb{G}^n, G,B \in \mathbb{G}$ -- independent group generators with prover $\mathcal{P}$ and verifier $\mathcal{V}$ is defined as follows: + \mathcal{R}_{zkip} = \left\{ \begin{aligned} (\mathbf{G,H},G,B,A,V;\mathbf{a,b},\alpha, \gamma) \vert & A = \langle \mathbf{a,G} \rangle + \langle \mathbf{b,H} \rangle + [\alpha] B, \\ & V = [\langle \mathbf{a,b} \rangle]G + [\gamma]B \end{aligned} \right\} + \end{align*} + where $\mathbf{G,H} \in \mathbb{G}^n, G,B \in \mathbb{G}$ -- independent group generators with prover $\mathcal{P}$ and verifier $\mathcal{V}$ is defined as follows: \begin{itemize} \item Prover $\mathcal{P}$ choses blinding vectors $\mathbf{s}_L, \mathbf{s}_R \in \mathbb{F}_p^n$ and computes polynomials: diff --git a/presentations/15-bulletproofs.tex b/presentations/15-bulletproofs.tex new file mode 100644 index 0000000..9895856 --- /dev/null +++ b/presentations/15-bulletproofs.tex @@ -0,0 +1,318 @@ +\documentclass{zkdl-presentation-template} + +\title[Bulletproofs: inner-product argument]{\textbf{Bulletproofs: inner-product argument}} +\author{Distributed Lab} +\date{July 24, 2025} +\homepage{zkdl-camp.github.io} +\github{ZKDL-Camp} + +\begin{document} +\frame { + \tikz [remember picture,overlay] + \node at + ([yshift=1.5cm,xshift=-1.5cm]current page.south east) +%or: (current page.center) + {\includegraphics[width=60pt]{images/logo.png}}; + + \titlepage +} + +\begin{frame}{Plan} + \tableofcontents +\end{frame} + +\section{Introduction} + +\begin{frame}{Bulletproofs: Motivation} + \begin{itemize} + \item \textbf{Bulletproofs:} zero-knowledge proofs with logarithmic proof size + \item No trusted setup + \item Straightforward cryptographic assumption: discrete logarithm without bilinear pairings or other advanced assumptions + \item Originally developed for efficient range proofs in confidential transactions + \item Applicable for proving arithmetic circuits satisfiability (e.g., R1CS) + \item Efficient polynomial commitment scheme could be derived (e.g., \textit{IPA} polynomial commitment) + \item Heart of \textit{bulletproofs} -- \textbf{inner-product argument} + \end{itemize} +\end{frame} + +\begin{frame}{Bulletproofs: Building blocks} + \begin{itemize} + \item Zero-knowledge multiplication protocol + \item Inner-product argument + \item Application: \textit{IPA} polynomial commitment scheme + \item Application: range proofs + \item Application: arithmetic circuits satisfiability + \end{itemize} +\end{frame} + +\begin{frame}{Notation} + \begin{itemize} + \item $\mathbb{G}$: cyclic group of prime order $p$ + \item $G, B \in \mathbb{G}$ -- independent generators + \item $\mathbf{G}, \mathbf{H} \in \mathbb{G}^n$ -- vectors of generators with mutually unknown discrete log relations + \item $\langle \mathbf{a}, \mathbf{b} \rangle$ -- inner product of vectors $\mathbf{a}, \mathbf{b}$ + \item $\langle \mathbf{a}, \mathbf{G} \rangle = \sum_{i=1}^n [a_i] G_i$ + \item $\mathbf{k}^n = (1, k, k^2, \dots, k^{n-1})$ + \end{itemize} +\end{frame} + +% --- Next Sections --- + +\section{Zero-knowledge multiplication} + +\begin{frame}{Zero-knowledge multiplication} + \textbf{Goal:} Prove knowledge of $a, b$ such that $c = ab$ without revealing $a, b$, i.e. consider the relation: + $$\mathcal{R}_{abc} = \{ (;c,a,b) \in \mathbb{F}_p \mid c=ab \}$$ + Curious reader could argue that this is relatively simple problem as we have $\Sigma$-protocols framework, especially Chaum-Pedersen protocol for DH-triplets, e.g. we could prove slightly modified relation: + $$\mathcal{R}'_{abc} = \{ (P, Q_a, Q_b, Q_c \in \mathbb{G};a,b) \mid Q_c = [a]Q_b, Q_a = [a]P, Q_b = [b]P \}$$ + + \begin{alertblock}{Problem} + Prover does not bind to $a,b,c$ values so that verifier could not be sure prover silently modified $a,b,c$ values after committing to them. + \end{alertblock} +\end{frame} + +\begin{frame}{Multiplication of committed values} + \begin{block}{Solution} + Use Pedersen commitments to bind to $a,b,c$ values. + \end{block} + $$ + \mathcal{R}'_{abc} = \left\{\begin{array}{l} + (G,H,B,A,V; a,b, \alpha, \beta \in \mathbb{F}_p) \mid \\ + V = [ab]G + [\beta]B, \\ + A = [a]G + [b]H + [\alpha]B + \end{array}\right\} + $$ + Here $A$ is a binding Pedersen commitment to both $a$ and $b$ while $V$ is a binding Pedersen commitment to their product $ab$. + \begin{alertblock}{Problem} + How to prove that in zero-knowledge? + \end{alertblock} +\end{frame} + +\begin{frame}{Zero-knowledge polynomial multiplication} + In order to provide zero-knowledge proof of $\mathcal{R}'_{abc}$ we bring out polynomials! + \begin{align*} + l(x) &= a + s_L x\\ + r(x) &= b + s_R x\\ + t(x) &= l(x)r(x) = ab + (s_L + s_R)bx + s_Ls_Rx^2 + \end{align*} + What we have now: + \begin{itemize} + \item $s_L, s_R \in \mathbb{F}_p$ are blinding factors + \item $l(x)$ is a linear polynomial hiding value $a$ + \item $r(x)$ is a linear polynomial hiding value $b$ + \item $t(x)$ is a quadratic polynomial, constant term is the product $ab$ + \end{itemize} +\end{frame} + +\begin{frame}{Zero-knowledge polynomial multiplication} + \begin{block}{Idea} + If $l(x)r(x) = t(x)$ than with high probability for random $u \in \mathbb{F}_p$ we have $l(u)r(u) = t(u)$ due to \textit{Schwartz-Zippel lemma} + \end{block} + Now we can build a zero-knowledge protocol for proving $l(x)r(x) = t(x)$: + \begin{itemize} + \item Prover commits to coefficients of $l(x), r(x), t(x)$: + \begin{align*} + A &= [a]G + [b]H + [\alpha]B\\ + S &= [s_L]G + [s_R]H + [\beta]B\\ + T_0 &= [ab]G + [\tau_0]B \\ + T_1 &= [s_L + s_R]G + [\tau_1]B \\ + T_2 &= [s_Ls_R]G + [\tau_2]B + \end{align*} + \item Verifier draws random challenge $u \in \mathbb{F}_p$ and sends it to prover + \end{itemize} +\end{frame} + +\begin{frame}{Zero-knowledge polynomial multiplication} + \begin{itemize} + \item Prover evaluates and sends to Verifier: + \begin{align*} + l_u &=l(u), r_u = r(u), t_u = t(u) = l_u \cdot r_u, \\ + \alpha_u &= \alpha + \beta u, \tau_u = \tau_0 + \tau_1 u + \tau_2 u^2 + \end{align*} + \item Verifier checks: + \begin{align*} + A + [u]S &\stackrel{?}{=} [l_u]G + [r_u]H + [\alpha_u]B \\ + [t_u]G + [\tau_u]B &\stackrel{?}{=} T_0 + [u]T_1 + [u^2]T_2 \\ + t_u &\stackrel{?}{=} l_u r_u + \end{align*} + \end{itemize} +\end{frame} + +\begin{frame}{zk-mul protocol: Security} + \begin{theorem} + Zero-knowledge polynomial multiplication protocol is perfect complete, special sound and perfect honest-verifier zero-knowledge. + \end{theorem} + Here we briefly show the completeness: + \begin{itemize} + \item First check: \begin{align*} + A + [u]S &\stackrel{?}{=} [l_u]G + [r_u]H + [\alpha_u]B \\ + LHS &= [a]G + [b]H + [\alpha]B + [us_L]G + [us_R]H + [u\beta]B \\ + RHS &= [a+ u s_L]G + [b + u s_R]H + [\alpha + u \beta]B = \\ + &=[a]G + [b]H + [\alpha]B + [us_L]G + [us_R]H + [u\beta]B + \end{align*} + As $LHS = RHS$ the check is satisfied. + \end{itemize} +\end{frame} + +\begin{frame}{zk-mul protocol: Security} + \begin{itemize} + \item Second check: \begin{align*} + [t_u]G + [\tau_u]B &\stackrel{?}{=} T_0 + [u]T_1 + [u^2]T_2 \\ + LHS &= [ab + t_1 u + t_2 u^2]G + [\tau_0 + \tau_1 u + \tau_2 u^2]B\\ + RHS &=[ab]G + [\tau_0]B + [u]([t_1]G + [\tau_1]B) + \\ + &+ [u^2]([t_2]G + [\tau_2]B) + \end{align*} + As $LHS = RHS$ the check is satisfied. + \item The third check is $t_u = l_u r_u$ holds by definition + \end{itemize} + Intuitively \textbf{zk-mul} protocol is also \textit{zero-knowledge} as it is easy to simulate every step of the honest prover. \textit{Special soundness} holds as it's easy to build an extractor similar to Okamoto's protocol extractor for extracting openings of Pedersen commitments. +\end{frame} + +\begin{frame}{zk-mul: inner-product version} + We could easily generalize \textbf{zk-mul} protocol to prove $t(x) = \langle \mathbf{l}(x), \mathbf{r}(x) \rangle $ for polynomials with vector coefficients $\mathbf{l}(x), \mathbf{r}(x), \mathbf{t}(x) \in \mathbb{F}_p^n[x]$. Specifically using generalized \textbf{zk-mul} protocol we could also prove in zero-knowledge that inner-product relation holds for vectors $\mathbf{a}, \mathbf{b} \in \mathbb{F}_p^n$: + \begin{align*} + \mathcal{R}_{zkip} = \left\{ \begin{aligned} (\mathbf{G,H},G,B,A,V;\mathbf{a,b},\alpha, \gamma) \vert & A = \langle \mathbf{a,G} \rangle + \langle \mathbf{b,H} \rangle + [\alpha] B, \\ & V = [\langle \mathbf{a,b} \rangle]G + [\gamma]B \end{aligned} \right\} + \end{align*} + \begin{alertblock}{Problem} + Proof size is linear in $n$ as the prover should send evaluated vectors $\mathbf{l_u}, \mathbf{r_u}$. + \end{alertblock} +\end{frame} + +\begin{frame}{Zero-Knowledge Multiplication Protocol} + \begin{itemize} + \item $\mathsf{Setup}$: $G, H, B \in \mathbb{G}$ + \item Prover samples random $s_L, s_R$ + \item Defines $l(x) = a + s_L x$, $r(x) = b + s_R x$, $t(x) = l(x)r(x)$ + \item Parties run the multiplication protocol on $(l(x), r(x), t(x))$ + \end{itemize} +\end{frame} + +\begin{frame}{Summary: ZK Multiplication} + \begin{itemize} + \item Efficient protocol for proving $c = ab$ in zero-knowledge + \item Foundation for more complex protocols (range proofs, circuits) + \item Next: Inner-product argument + \end{itemize} +\end{frame} + +\section{Inner-product argument} + +\begin{frame}{Motivation: Inner-product Argument} + \begin{itemize} + \item Core of Bulletproofs: enables short proofs for range and circuit satisfiability + \item Compresses multiple constraints into a single inner-product relation + \item Achieves logarithmic proof size + \end{itemize} +\end{frame} + +\begin{frame}{Relation} + \begin{itemize} + \item Prover knows $\mathbf{a}, \mathbf{b} \in \mathbb{F}_p^n$ such that $P' = \langle \mathbf{a}, \mathbf{G} \rangle + \langle \mathbf{b}, \mathbf{H} \rangle + [\langle \mathbf{a}, \mathbf{b} \rangle]Q$ + \item $\mathbf{G}, \mathbf{H} \in \mathbb{G}^n$, $Q \in \mathbb{G}$ + \item All vectors have length $n = 2^d$ + \end{itemize} +\end{frame} + +\begin{frame}{Protocol Overview} + \begin{itemize} + \item Recursive protocol: reduces $n$-dimensional inner product to $1$-dimensional + \item At each round, splits vectors in half and sends commitments $L_k, R_k$ + \item Verifier sends challenge $u_k$ + \item Prover and verifier update vectors and generators + \item Repeat until $n = 1$ + \end{itemize} +\end{frame} + +\begin{frame}{Protocol: Step-by-Step} + \begin{enumerate} + \item For $k = d$ down to $1$: + \begin{itemize} + \item Prover splits $\mathbf{a}^{(k)}, \mathbf{b}^{(k)}, \mathbf{G}^{(k)}, \mathbf{H}^{(k)}$ into halves + \item Computes $L_k, R_k$ as commitments to cross terms + \item Sends $L_k, R_k$ to verifier + \item Verifier samples random $u_k$ + \item Prover and verifier update vectors and generators + \end{itemize} + \item When $n=1$, prover sends $a, b$ to verifier + \end{enumerate} +\end{frame} + +\begin{frame}{Commitment Equations} + \begin{itemize} + \item $L_k = \langle \mathbf{a}_{lo}^{(k)}, \mathbf{G}_{hi}^{(k)} \rangle + \langle \mathbf{b}_{hi}^{(k)}, \mathbf{H}_{lo}^{(k)} \rangle + [\langle \mathbf{a}_{lo}^{(k)}, \mathbf{b}_{hi}^{(k)} \rangle]Q$ + \item $R_k = \langle \mathbf{a}_{hi}^{(k)}, \mathbf{G}_{lo}^{(k)} \rangle + \langle \mathbf{b}_{lo}^{(k)}, \mathbf{H}_{hi}^{(k)} \rangle + [\langle \mathbf{a}_{hi}^{(k)}, \mathbf{b}_{lo}^{(k)} \rangle]Q$ + \end{itemize} +\end{frame} + +\begin{frame}{Recursive Update} + \begin{itemize} + \item After challenge $u_k$, update: + \begin{align*} + \mathbf{a}^{(k-1)} &= u_k \mathbf{a}_{lo}^{(k)} + u_k^{-1} \mathbf{a}_{hi}^{(k)} \\ + \mathbf{b}^{(k-1)} &= u_k^{-1} \mathbf{b}_{lo}^{(k)} + u_k \mathbf{b}_{hi}^{(k)} \\ + \mathbf{G}^{(k-1)} &= u_k^{-1} \mathbf{G}_{lo}^{(k)} + u_k \mathbf{G}_{hi}^{(k)} \\ + \mathbf{H}^{(k-1)} &= u_k \mathbf{H}_{lo}^{(k)} + u_k^{-1} \mathbf{H}_{hi}^{(k)} + \end{align*} + \item Repeat until $n=1$ + \end{itemize} +\end{frame} + +\begin{frame}{Final Check} + \begin{itemize} + \item Prover sends $a, b$ (scalars) to verifier + \item Verifier checks: + \begin{equation*} + P' + \sum_{i=1}^d ([u_i^2]L_i + [u_i^{-2}]R_i) = [a]G_1^{(0)} + [b]H_1^{(0)} + [ab]Q + \end{equation*} + \item Accept if equality holds + \end{itemize} +\end{frame} + +\begin{frame}{Communication Complexity} + \begin{itemize} + \item Each round: send $L_k, R_k$ ($2 \log_2 n$ group elements in total) + \item Final: send $a, b$ (2 field elements) + \item Logarithmic proof size + \end{itemize} +\end{frame} + +\begin{frame}{Security Properties} + \begin{itemize} + \item \textbf{Perfect completeness}: Honest prover always convinces verifier + \item \textbf{Statistical soundness}: Extractor can recover witness or find discrete log relation + \item \textbf{Not zero-knowledge} (unless $n>1$ and combined with ZK techniques) + \end{itemize} +\end{frame} + +\begin{frame}{Making it Zero-Knowledge} + \begin{itemize} + \item Combine with zero-knowledge inner-product protocol $\Pi_{zkip}$ + \item Add blinding to commitments + \item Ensures verifier learns nothing about witness + \end{itemize} +\end{frame} + +\begin{frame}{Diagram: Inner-product Protocol} + % TODO: Insert or reference TikZ diagram from lecture notes + \begin{center} + \textit{[Protocol sequence diagram here]} + \end{center} +\end{frame} + +\begin{frame}{Use in Bulletproofs} + \begin{itemize} + \item Inner-product argument enables short range proofs and circuit proofs + \item Used recursively to compress constraints + \item Key to logarithmic proof size in Bulletproofs + \end{itemize} +\end{frame} + +\begin{frame}{Summary: Inner-product Argument} + \begin{itemize} + \item Efficient, recursive protocol for proving inner-product relations + \item Logarithmic proof size + \item Foundation for Bulletproofs and other modern ZK systems + \end{itemize} +\end{frame} + +\end{document} \ No newline at end of file diff --git a/presentations/zkdl-presentation-template.cls b/presentations/zkdl-presentation-template.cls index 5d74fa2..ab64c00 100644 --- a/presentations/zkdl-presentation-template.cls +++ b/presentations/zkdl-presentation-template.cls @@ -10,7 +10,7 @@ \usepackage[english]{babel} % --- Listing settings --- -\RequirePackage{listings, ../listings-circom} +%^\RequirePackage{listings, ../listings-circom} % --- Algorithm packages --- \RequirePackage[ From c9b728356a74e7c83702fde320fe83e5c1e468ba Mon Sep 17 00:00:00 2001 From: Yevhen Hrubiian Date: Fri, 25 Jul 2025 09:43:36 +0300 Subject: [PATCH 16/25] add slides, fix typos, fix binding/hiding bug --- .vscode/settings.json | 8 + lectures/2-9-bulletproofs.tex | 72 ++-- presentations/15-bulletproofs.tex | 318 ---------------- presentations/17-bulletproofs.pdf | Bin 0 -> 633824 bytes presentations/17-bulletproofs.tex | 346 ++++++++++++++++++ .../images/lecture_17/compressed.png | Bin 0 -> 49844 bytes presentations/images/lecture_17/ipa.png | Bin 0 -> 78716 bytes presentations/images/lecture_17/ipa_meme.jpg | Bin 0 -> 101350 bytes presentations/images/lecture_17/meme.jpg | Bin 0 -> 74857 bytes 9 files changed, 392 insertions(+), 352 deletions(-) create mode 100644 .vscode/settings.json delete mode 100644 presentations/15-bulletproofs.tex create mode 100644 presentations/17-bulletproofs.pdf create mode 100644 presentations/17-bulletproofs.tex create mode 100644 presentations/images/lecture_17/compressed.png create mode 100644 presentations/images/lecture_17/ipa.png create mode 100644 presentations/images/lecture_17/ipa_meme.jpg create mode 100644 presentations/images/lecture_17/meme.jpg diff --git a/.vscode/settings.json b/.vscode/settings.json new file mode 100644 index 0000000..73370dd --- /dev/null +++ b/.vscode/settings.json @@ -0,0 +1,8 @@ +{ + "cSpell.words": [ + "arithmetization", + "indistinguishability", + "Naїve", + "satisfiability" + ] +} \ No newline at end of file diff --git a/lectures/2-9-bulletproofs.tex b/lectures/2-9-bulletproofs.tex index 1c34a86..7e19a30 100644 --- a/lectures/2-9-bulletproofs.tex +++ b/lectures/2-9-bulletproofs.tex @@ -20,19 +20,19 @@ \subsection{Introduction} Also, \textbf{bulletproofs}' \textbf{inner-product argument} could be used to build various polynomial commitment schemes -- crucial building block of proving systems built with \textit{IOP} framework (\textit{Halo, Nova, etc}). -The main advantages of \textbf{bulletproofs} are an absence of a trusted setup and security against eavesdropping that relies on the \textit{discrete-logarithm} assumption without any other auxiliary structures like bilinear pairings. Also it has quite fast prover for small circuits making it practically usefull for client-side proving. However, the main disadvantage of \textbf{bulletproofs} is that it isn't a classic \textit{SNARK} due to linear in circuit size verification time, however still very efficient for small circuits. +The main advantages of \textbf{bulletproofs} are an absence of a trusted setup and security against eavesdropping that relies on the \textit{discrete-logarithm} assumption without any other auxiliary structures like bilinear pairings. Also it has quite fast prover for small circuits making it practically useful for client-side proving. However, the main disadvantage of \textbf{bulletproofs} is that it isn't a classic \textit{SNARK} due to linear in circuit size verification time, however still very efficient for small circuits. \subsection{Notation} Let $\mathbb{G}$ - cyclic group of prime order $p$ written additively, $\mathbf{G} = (G_1, \dots, G_n), \mathbf{H} = (H_1, \dots, H_n) \in \mathbb{G}^n$ - vectors of independent generators. We denote by $\langle \mathbf{a,b} \rangle$ - inner product of vectors $\mathbf{a} = (a_1, \dots, a_n), \mathbf{b} = (b_1, \dots, b_n) \in \mathbb{F}_p^n$ and $\langle \mathbf{a,G} \rangle = \sum_{i=1}^n [a_i] G_i \in \mathbb{G}$ - inner product of vector $\mathbf{a}$ with vector of generators $\mathbf{G}$. Denote by $\mathbf{k}^n$ vector of $k$'s first $n$ powers: $\mathbf{k}^n = (1, k, k^2, \dots, k^{n-1})$, for example $\mathbf{0}^n, \mathbf{1}^n$ represents vectors of zeros and ones respectively, while $\mathbf{2}^{n} = (1, 2, 4, \dots, 2^{n-1})$ \subsection{Zero-knowledge multiplication} -Let $a,b,c \in \mathbb{F}_p$. Here we build a zero-knowledge protocol for relation $\mathcal{R}_{abc} = \{ ;c,a,b: c=ab \}$. We use well-known $\Sigma$-protocol framework for that, but firstly we make very useful generalization that could allow us to prove much larger class of relations. +Let $a,b,c \in \mathbb{F}_p$. Here we build a zero-knowledge protocol for relation $\mathcal{R}_{abc} = \{ (\bot;c,a,b) \vert c=ab \}$. We use well-known $\Sigma$-protocol framework for that, but firstly we make very useful generalization that could allow us to prove much larger class of relations. Consider the first-degree polynomials $l(x) = a + s_L x, r(x) = b + s_R x \in \mathbb{F}_p[x]$. Let $t(x) = l(x)r(x)$ and relation -$$\mathcal{R}_{mul} = \{ (;l(x),r(x),t(x))\in (\emptyset \times \mathbb{F}_p^3) \vert t(x) = l(x)r(x)\}$$ +$$\mathcal{R}_{mul} = \{ (\bot;l(x),r(x),t(x)) \vert t(x) = l(x)r(x)\}$$ Firstly, observe that proving $t(x) = l(x)r(x)$ may be reduced to evaluation check at some challenge point $u \in \mathbb{F}_p$: $t(u) = l(u)r(u)$, due to the \textit{Schwartz-Zippel lemma}: $$\mathsf{Pr}[l(u)r(u) = t(u) \vert l(x)r(x) \neq t(x)] \le \frac{max(\deg(l(x)r(x)), \deg(t(x)))}{p} = \frac{2}{p}$$ -is typycally negligible function from security level which makes this check \textit{sound}. +is typically negligible function from security level which makes this check \textit{sound}. \subsubsection{Naїve polynomial multiplication protocol} Let's describe naїve unoptimized version of \textbf{polynomial multiplication} protocol $\Pi'_{mul} = (\mathsf{Setup},\mathcal{P,V})$ for relation $\mathcal{R}_{mul}$. @@ -121,7 +121,7 @@ \subsubsection{Optimized polynomial multiplication protocol} We could optimize our \textbf{polynomial multiplication protocol} furthermore. Note that we could simply apply vector Pedersen commitment for constant and linear terms using one more group element $H \in \mathbb{G}$. \begin{definition} - The \textbf{polynomial multiplication protocol} $\Pi_{mul} = (\mathsf{Setup}, \mathcal{P,V})$ for the relation $\mathcal{R}_{mul} = \{ (;l(x),r(x),t(x)) \vert t(x) = l(x)r(x)\}$ with prover $\mathcal{P}$ and verifier $\mathcal{V}$ is defined as follows: + The \textbf{polynomial multiplication protocol} $\Pi_{mul} = (\mathsf{Setup}, \mathcal{P,V})$ for the relation $$\mathcal{R}_{mul} = \{ (\bot;l(x),r(x),t(x)) \vert t(x) = l(x)r(x)\}$$ with prover $\mathcal{P}$ and verifier $\mathcal{V}$ is defined as follows: \begin{itemize} \item $\mathsf{Setup}$ returns triple of group generators with unknown discrete log relations $G, H, B \in \mathbb{G}$ \item Parties $\mathcal{P, V}$ run the following protocol: @@ -187,14 +187,14 @@ \subsubsection{Zero-knowledge multiplication protocol} The protocol obviously has \textit{perfect completeness, special soundness, perfect honest-verifier zero-knowledge} due to the \Cref{th:poly_mul}. \begin{remark} %todo: add reference for Chaum-Pedersen - Now curious reader may wonder why build so overwhelmingly complicated protocol for simple multiplication and not just use classic \textit{Chaum-Pedersen protocol} for DH-triplets? Indeed, it definitely could establish that for given group elements $[a]G, [b]G, [c]G$ equality $c=ab$ holds, but unfortunatelly commitments $[a]G, [b]G, [c]G$ do not have a \textit{binding} property (though preserving hiding property) so they're almost useless for building more complex protocols where the prover usually needs to bind to specific values without having an ability to silently modify them. + Now curious reader may wonder why build so overwhelmingly complicated protocol for simple multiplication and not just use classic \textit{Chaum-Pedersen protocol} for DH-triplets? Indeed, it definitely could establish that for given group elements $[a]G, [b]G, [c]G$ equality $c=ab$ holds, but unfortunately commitments $[a]G, [b]G, [c]G$ do not have a \textit{perfect hiding} property (though preserving \textit{computational binding} property) so an adversary could potentially learn $a,b,c$ values if they are not uniformly distributed. \end{remark} Also, there's a folklore version of very similar protocol for establishing product relationship between Pedersen committed values described in \href{https://people.cs.georgetown.edu/jthaler/ProofsArgsAndZK.pdf}{section 12.3 of Thaler's book} \subsubsection{Zero-knowledge inner-product protocol} -We could extend our $\Pi_{mul}$ protocol even further to provide zero-knowledge proof for the inner-product of vectors: $\langle \mathbf{a, b} \rangle = v$. The main trick is to substitude polynomials $l(x), r(x) \in \mathbb{F}_p[x]$ by vector polynomials $\mathbf{l}(x), \mathbf{r}(x) \in \mathbb{F}_p^n[x]$ where constant terms are equal to $\mathbf{a}$ and $\mathbf{b}$ respectively, taking inner-product $\langle \mathbf{l}(x), \mathbf{r}(x) \rangle$ results in polynomial with scalar coefficients where constant term is equal to $\langle \mathbf{a, b} \rangle$. +We could extend our $\Pi_{mul}$ protocol even further to provide zero-knowledge proof for the inner-product of vectors: $\langle \mathbf{a, b} \rangle = v$. The main trick is to substitute polynomials $l(x), r(x) \in \mathbb{F}_p[x]$ by vector polynomials $\mathbf{l}(x), \mathbf{r}(x) \in \mathbb{F}_p^n[x]$ where constant terms are equal to $\mathbf{a}$ and $\mathbf{b}$ respectively, taking inner-product $\langle \mathbf{l}(x), \mathbf{r}(x) \rangle$ results in polynomial with scalar coefficients where constant term is equal to $\langle \mathbf{a, b} \rangle$. \begin{example} Let $\mathbf{a} = (a_1, a_2)$ and $\mathbf{b} = (b_1, b_2)$ be vectors in $\mathbb{F}_p^2$. Consider vector polynomials with vector coefficients: @@ -258,11 +258,11 @@ \subsubsection{Zero-knowledge inner-product protocol} \end{itemize} \end{definition} -The protocol also has \textit{perfect completeness, special soundness, perfect honest-verifier zero-knowledge} due to the \Cref{th:poly_mul}, however building the extractor needs some vector equations we omit for the sake of brevity. Also note that transcript size is linear in size of vectors $\mathbf{l}_u, \mathbf{r}_u$ which is extremly inefficient when vectors are large. So in the next section we present so called \textbf{inner-product argument} which is summoned to reduce conversational complexity to logarithmic in vector length making the last check $t_u \stackrel{\text{?}}{=} \langle \mathbf{l}_u \mathbf{r}_u \rangle$ quite efficient. +The protocol also has \textit{perfect completeness, special soundness, perfect honest-verifier zero-knowledge} due to the \Cref{th:poly_mul}, however building the extractor needs some vector equations we omit for the sake of brevity. Also note that transcript size is linear in size of vectors $\mathbf{l}_u, \mathbf{r}_u$ which is extremely inefficient when vectors are large. So in the next section we present so called \textbf{inner-product argument} which is summoned to reduce conversational complexity to logarithmic in vector length making the last check $t_u \stackrel{\text{?}}{=} \langle \mathbf{l}_u \mathbf{r}_u \rangle$ quite efficient. \subsection{Inner-product argument} -Here we describe the further generalization of $\Pi_{mul}$ -- efficient protocol for the \textbf{inner-product argument} - core component of the \textbf{bulletproofs} protocol. After that we will apply it to range proofs and arithmetic circuits. We have already seen that inner-products are the main ingridients for R1CS language because any R1CS relation could be seen as a batch of inner-products though it's not the most efficient representation and we'll see how to amortize all the constraints into inner-products more efficiently. +Here we describe the further generalization of $\Pi_{mul}$ -- efficient protocol for the \textbf{inner-product argument} - core component of the \textbf{bulletproofs} protocol. After that we will apply it to range proofs and arithmetic circuits. We have already seen that inner-products are the main ingredients for R1CS language because any R1CS relation could be seen as a batch of inner-products though it's not the most efficient representation and we'll see how to amortize all the constraints into inner-products more efficiently. The \textbf{inner-product argument} allows to prove that two vectors $\mathbf{a,b} \in \mathbb{F}_p^n$ satisfy the relation: $$\mathcal{R}_{ip} = \{ (\mathbf{G,H}, P, c; \mathbf{a,b}) \vert P = \langle \mathbf{a,G} \rangle + \langle \mathbf{b,H} \rangle \wedge \langle \mathbf{a,b} \rangle = c \}$$ @@ -289,8 +289,8 @@ \subsection{Inner-product argument} \subsubsection{Inner-product compression} -Here we describe \textbf{inner-product compression} algorithm -- main building block of the interactive \textbf{inner-product} procotol. -Firstly, assuming that $n = 2^d$ define by $\mathbf{G_{lo}} = (G_1, \dots, G_{n/2}), \mathbf{G_{hi}} = (G_{n/2+1},\dots, G_n) \in \mathbb{G}^{n/2}$ -- lower and higher halves of vector $\mathbf{G}$ and $\mathbf{a_{lo}} = (a_1, \dots, a_{n/2}), \mathbf{a_{hi}} = (a_{n/2+1},\dots,a_n) \in \mathbb{F}_{n/2}$ -- lower and higher halves of $\mathbf{a} \in \mathbb{F}_p^{n}$. +Here we describe \textbf{inner-product compression} algorithm -- main building block of the interactive \textbf{inner-product} protocol. +Firstly, assuming that $n = 2^d$ define by $\mathbf{G_{lo}} = (G_1, \dots, G_{n/2}), \mathbf{G_{hi}} = (G_{n/2+1},\dots, G_n) \in \mathbb{G}^{n/2}$ -- lower and higher halves of vector $\mathbf{G}$ and $\mathbf{a_{lo}} = (a_1, \dots, a_{n/2}), \mathbf{a_{hi}} = (a_{n/2+1},\dots,a_n) \in \mathbb{F}_p^{n/2}$ -- lower and higher halves of $\mathbf{a} \in \mathbb{F}_p^{n}$. Let $u_k \in \mathbb{F}_p$ - be some scalar, define compressed vectors: \begin{align*} @@ -300,7 +300,7 @@ \subsubsection{Inner-product compression} \mathbf{H}^{(k-1)} &= \mathbf{H_{lo}} \cdot u_k + u_k^{-1} \cdot \mathbf{H_{hi}} \end{align*} -Define $P_k \gets P' = \langle \mathbf{a,G} \rangle + \langle \mathbf{b,H} \rangle + [\langle \mathbf{a,b} \rangle]Q$ and define $P_{k-1}$ using compressed vectors to have the same form as $P_k$, but in new basis $(\mathbf{G}^{(k-1)}, \mathbf{H}^{(k-1)})$: +Define $P_k \gets P' = \langle \mathbf{a,G} \rangle + \langle \mathbf{b,H} \rangle + [\langle \mathbf{a,b} \rangle]Q$ -- current commitment to vectors $\mathbf{a,b}$ and define $P_{k-1}$ using compressed vectors to have the same form as $P_k$, but in new basis $(\mathbf{G}^{(k-1)}, \mathbf{H}^{(k-1)})$: \begin{equation} P_{k-1} = \langle \mathbf{a}^{(k-1)}, \mathbf{G}^{(k-1)} \rangle + \langle \mathbf{b}^{(k-1)}, \mathbf{H}^{(k-1)} \rangle + [\langle \mathbf{a}^{(k-1)}, \mathbf{b}^{(k-1)} \rangle]Q @@ -308,13 +308,15 @@ \subsubsection{Inner-product compression} \end{equation} Or alternatively, expressing $P_{k-1}$ in old basis $(\mathbf{G}^{(k)}, \mathbf{H}^{(k)})$ we get: -\begin{align} +\begin{equation} + \begin{aligned} P_{k-1} &= \langle u_k^{-1} \cdot \mathbf{a}^{(k-1)}, \mathbf{G_{lo}}^{(k)} \rangle + \langle u_k \cdot \mathbf{a}^{(k-1)}, \mathbf{G_{hi}}^{(k)} \rangle + \langle u_k \cdot \mathbf{b}^{(k-1)}, \mathbf{H_{lo}}^{(k)} \rangle \\ &+ \langle u_k^{-1} \cdot \mathbf{b}^{(k-1)}, \mathbf{H_{hi}}^{(k)} \rangle + [\langle \mathbf{a}^{(k-1)}, \mathbf{b}^{(k-1)} \rangle]Q \label{eq:p_k_1_from_old_basis} -\end{align} + \end{aligned} +\end{equation} -Subsituting compressed vectors and applying bilinearity property of inner product we get: +Substituting compressed vectors and applying bilinearity property of inner product we get: \begin{align*} P_{k-1} = & \langle \mathbf{a_{lo}}, \mathbf{G_{lo}}\rangle + \langle \mathbf{a_{hi}}, \mathbf{G_{hi}}\rangle &+ u_k^2\langle \mathbf{a_{lo}}, \mathbf{G_{hi}}\rangle + u_k^{-2}\langle \mathbf{a_{hi}}, \mathbf{G_{lo}}\rangle + \\ & \langle \mathbf{b_{lo}}, \mathbf{H_{lo}}\rangle + \langle \mathbf{b_{hi}}, \mathbf{H_{hi}}\rangle &+ u_k^2\langle \mathbf{b_{hi}}, \mathbf{H_{lo}}\rangle + u_k^{-2}\langle \mathbf{b_{lo}}, \mathbf{H_{hi}}\rangle + \\ @@ -331,15 +333,17 @@ \subsubsection{Inner-product compression} R_{k} &= \langle \mathbf{a_{hi}}, \mathbf{G_{lo}}\rangle + \langle \mathbf{b_{lo}}, \mathbf{H_{hi}}\rangle + [\langle \mathbf{a_{hi}}, \mathbf{b_{lo}}\rangle]Q \end{align*} -Here we come up with some kind of statement compression algorithm reducing size of all vectors in half per compression step. Repeating comression algorithm $k$ times we end up with vectors $\mathbf{a}^{(0)}, \mathbf{b}^{(0)}, \mathbf{G}^{(0)}, \mathbf{H}^{(0)}$ each of length one and $P_0$ containing all accumulated cross-terms: +The first equation $P_{k-1} = P_k + [u_k^2] L_k + [u_k^{-2}] R_k$ could be used as a check for asserting correctness of next commitment $P_{k-1}$ given cross-terms $L_k, R_k$, half-sized vectors $\mathbf{a}^{(k-1)}, \mathbf{b}^{(k-1)}$ from which $P_{k-1}$ was computed (\ref{eq:p_k_1_from_new_basis}) using updated basis $\mathbf{G}^{(k-1)}, \mathbf{H}^{(k-1)}$ and current commitment value $P_k$. + +But we wish not send $\mathbf{a}^{(k-1)}, \mathbf{b}^{(k-1)}$ directly as this's inefficient due to still linear sizes, instead we apply recursion to compress this vectors to just one element. Here we come up with some kind of statement compression algorithm reducing size of all vectors in half per compression step. Repeating compression algorithm $k$ times we end up with sending vectors $\mathbf{a}^{(0)}, \mathbf{b}^{(0)}$ each of length one and $P_0$ containing all accumulated cross-terms: \begin{align*} P_0 &= [a_1^{(0)}]G_1^{(0)} + [b_1^{(0)}]H_1^{(0)} + [a_1^{(0)}b_1^{(0)}]Q \\ - P_0 &= P_k + \sum_{i=1}^d ([u_i^2]L_i + [u_i^{-2}]R_i) + P_0 &= P_k + \sum_{i=1}^k ([u_i^2]L_i + [u_i^{-2}]R_i) \end{align*} Recalling that $P_k = P'$, the final compressed statement asserting inner-product value will have the following form: \begin{equation} - P' + \sum_{i=1}^d ([u_i^2]L_i + [u_i^{-2}]R_i) = [a_1^{(0)}]G_1^{(0)} + [b_1^{(0)}]H_1^{(0)} + [a_1^{(0)}b_1^{(0)}]Q + P' + \sum_{i=1}^k ([u_i^2]L_i + [u_i^{-2}]R_i) = [a_1^{(0)}]G_1^{(0)} + [b_1^{(0)}]H_1^{(0)} + [a_1^{(0)}b_1^{(0)}]Q \label{eq:ip-final-compressed} \end{equation} @@ -458,13 +462,13 @@ \subsubsection{Proving $\mathcal{R}'_{ip}$} \label{fig:interactive_ip} \end{figure} -As we can see, overall communication complexity of $\Pi_{ip}$ is $2\log_2 n$ group elements plus $2$ field elements so we come up with logarithmic proof size for our inner-product relation $\mathcal{R}_{ip}$. Now one might ask whether $\Pi_{ip}$ has any desired properties such as completeness, soundness and zero-knowledge. It appears that the first two holds under some generalizations needed for security proofs but not \textit{zero-knowledge} (indeed, if $n=1$ then $\mathcal{P}$ sends witness pair $a,b$ directly). We'll combile efficient \textbf{inner-product argument} $\Pi_{ip}$ with zero-knowledge $\Pi_{zkip}$ to achieve efficient zero-knowledge proofs for range proofs and arithmetic circuits. +As we can see, overall communication complexity of $\Pi_{ip}$ is $2\log_2 n$ group elements plus $2$ field elements so we come up with logarithmic proof size for our inner-product relation $\mathcal{R}_{ip}$. Now one might ask whether $\Pi_{ip}$ has any desired properties such as completeness, soundness and zero-knowledge. It appears that the first two holds under some generalizations needed for security proofs but not \textit{zero-knowledge} (indeed, if $n=1$ then $\mathcal{P}$ sends witness pair $a,b$ directly). We'll compile efficient \textbf{inner-product argument} $\Pi_{ip}$ with zero-knowledge $\Pi_{zkip}$ to achieve efficient zero-knowledge proofs for range proofs and arithmetic circuits. \begin{theorem}[Inner-Product Argument] The argument system $\Pi_{ip}$ for relation $\mathcal{R}_{ip}$ has \textit{perfect completeness and statistical witness-extended emulation} for either extracting a non-trivial discrete logarithm relation between $\mathbf{G,H}, Q$ or extracting valid witness $\mathbf{a,b}$. \end{theorem} -\textbf{Proof idea}. \textit{Perfect completeness} of $Pi_{ip}$ follows because $Pi_{ip}$ converts instance of $\mathcal{R}_{ip}$ to instance of $\mathcal{P}'_{ip}$ and $\Pi'_{ip}$ is trivially complete by construction due to \Cref{eq:ip-final-compressed}. Notation \textit{statistical witness-extended emulation} generalizes \textit{special soundness} in the way applicable for multi-stage complex argument systems where each step of the protocol could be rewinded to extract part of the witness so that more accurate definition of protocol security is achieved despite \textit{special soundness} implies building the whole knowledge extractor which might has non-polynomial running time for multi-stage protocols. +\textbf{Proof idea}. \textit{Perfect completeness} of $Pi_{ip}$ follows because $Pi_{ip}$ converts instance of $\mathcal{R}_{ip}$ to instance of $\mathcal{P}'_{ip}$ and $\Pi'_{ip}$ is trivially complete by construction due to \Cref{eq:ip-final-compressed}. Notation \textit{statistical witness-extended emulation} generalizes \textit{special soundness} in the way applicable for multi-stage complex argument systems where each step of the protocol could be rewound to extract part of the witness so that more accurate definition of protocol security is achieved despite \textit{special soundness} implies building the whole knowledge extractor which might has non-polynomial running time for multi-stage protocols. Here we briefly describe a knowledge extractor $\mathcal{E}'_{ip}$ for a witness $(\mathbf{a}, \mathbf{b})$ or non-trivial discrete logarithm relation for $(\mathbf{G, H}, Q)$ for $\Pi'_{ip}$. \begin{enumerate} @@ -515,7 +519,7 @@ \subsubsection{Proving $\mathcal{R}'_{ip}$} We need the fourth rewinding to assert equality of inner product: $$\langle \mathbf{a}^{(1)}, \mathbf{b}^{(1)} \rangle = \sum_{i=1}^3 v_i \langle \mathbf{a}_i^{(0)}, \mathbf{b}_i^{(0)} \rangle$$ - We won't describe it fully since it takes some unwiedly technical details and refer a reder to the original \textit{bulletproofs} paper, where the full proof of extraction is described in Theorem 1. + We won't describe it fully since it takes some unwieldy technical details and refer a reader to the original \textit{bulletproofs} paper, where the full proof of extraction is described in Theorem 1. \item The extractor $\mathcal{E}'_{ip}$ recursively extracts $\mathbf{a}^{(k+1)}, \mathbf{b}^{(k+1)}$ from $ \mathbf{a}^{(k)}, \mathbf{b}^{(k)}$ using the method described in steps 2-5 until it reaches the final witness $\mathbf{a}^{(d)}, \mathbf{b}^{(d)} = \mathbf{a}, \mathbf{b}$ for which the relation $\mathcal{R}'_{ip}$ holds: $$P' = \langle \mathbf{a,G} \rangle + \langle \mathbf{b,H} \rangle + [\langle \mathbf{a,b} \rangle]Q$$ \end{enumerate} @@ -572,7 +576,7 @@ \subsection{Range proofs} For the first view it seems very inconspicuous why \textbf{inner-product argument} is useful for proving the range proof relation, but we'll show it ab initio. -Firstly, write $v$ in base-2 representation: $v = \sum_{i=0}^{\lfloor \log_2 v \rfloor} 2^i v_i$ and $\mathbf{a}_L = (v_0, v_1, \dots, v_{n-1})$ be the vector of bits padded with zeroes to length $n$, so the range validation that $v$ lays in $[0, 2^n)$ imlpies two checks: +Firstly, write $v$ in base-2 representation: $v = \sum_{i=0}^{\lfloor \log_2 v \rfloor} 2^i v_i$ and $\mathbf{a}_L = (v_0, v_1, \dots, v_{n-1})$ be the vector of bits padded with zeroes to length $n$, so the range validation that $v$ lays in $[0, 2^n)$ implies two checks: \begin{itemize} \item Each bit $v_i$ must be either $0$ or $1$ \item The following inner-product equality holds: $\langle \mathbf{a}_L, \mathbf{2}^n \rangle = v$ @@ -603,7 +607,7 @@ \subsection{Range proofs} \item $\langle \mathbf{a}_L - \mathbf{a}_R - \mathbf{1}^n, \mathbf{y}^n \rangle = 0$ \end{enumerate} -Here we could soundly combine all three checks into one using random linear combintation with some verifier-provided challenge $z \in \mathbb{F}_p$: +Here we could soundly combine all three checks into one using random linear combination with some verifier-provided challenge $z \in \mathbb{F}_p$: $$z^2 \cdot \langle \mathbf{a}_L, \mathbf{2}^n \rangle + z \cdot \langle \mathbf{a}_L - \mathbf{a}_R - \mathbf{1}^n, \mathbf{y}^n \rangle + \langle \mathbf{a}_L, \mathbf{a}_R \circ \mathbf{y}^n \rangle = z^2v$$ \begin{remark} @@ -760,13 +764,13 @@ \subsection{Range proofs} \end{equation*} Which holds since $t_u = t(u) = z^2v + \delta(y,z) + t_1u + t_2u^2$. The third inner-product also holds as $t_u = \langle \mathbf{l}_u \mathbf{r}_u \rangle$. -\textit{Perfect honest-verifier zero-knowledge} follows from the zero-knowledge construction of $\Pi_{zkip}$ protocol as Vertifier learns no information about $\mathbf{a}_L, \mathbf{a}_R$. +\textit{Perfect honest-verifier zero-knowledge} follows from the zero-knowledge construction of $\Pi_{zkip}$ protocol as Verifier learns no information about $\mathbf{a}_L, \mathbf{a}_R$. \textit{Computational extended witness emulation} implies building extractor that combines extractors for two subprotocols: $\mathcal{E}_{ip}$ extracts witness ($\mathbf{l}_u, \mathbf{r}_u$) from $\Pi_{ip}$ than the extractor $\mathcal{E}_{zkip}$ extracts high-level witness ($\mathbf{a}_L, \mathbf{a}_R$) from $\Pi_{zkip} \quad \square$ The proof size of \textbf{range-proof} protocol is $2 \log_2n +4$ group $\mathbb{G}$ elements and $5$ field $\mathbb{F}_p$ elements. \begin{remark} - Range proofs could be efficiently aggregated: e.g. one could prove the relation using slighly modified range proof protocol $\Pi_{rp}$ + Range proofs could be efficiently aggregated: e.g. one could prove the relation using slightly modified range proof protocol $\Pi_{rp}$ $$\mathcal{R}_{rpm} = \{ (G, B, \vec{V}, n; \vec{v}, \vec{\gamma}) \vert \forall i \in 1..m: V_i = [v_i]G + [\gamma_i]B, v_i \in [0, 2^n) \}$$ Where $\vec{v} = (v_1, v_2, \dots, v_m)$ and $\vec{\gamma} = (\gamma_1, \gamma_2, \dots, \gamma_m)$ -- respectively secrets and blinding factors. Detail explanation of aggregation protocol could be found in \href{https://eprint.iacr.org/2017/1066.pdf}{original bulletroofs paper} \end{remark} @@ -780,13 +784,13 @@ \subsection{Range proofs} \end{example} \subsection{Arithmetic circuits proofs} -\textbf{Bulletproofs} presents not only range proofs, but also efficient proofs for arithmetic circuits satisfiability. As we could see before, inner-product relation is quite powerful tool and could be used to prove a knowledge of witness to any $NP$-problem. But here we present a more convinient way to compile arithmetic circuits into inner-product relation. +\textbf{Bulletproofs} presents not only range proofs, but also efficient proofs for arithmetic circuits satisfiability. As we could see before, inner-product relation is quite powerful tool and could be used to prove a knowledge of witness to any $NP$-problem. But here we present a more convenient way to compile arithmetic circuits into inner-product relation. \subsubsection{Arithmetization} -\textbf{Bulletproofs} arithmetization slightly differs from the classic R1CS we studied before, however it could be transformed vice-versa easily. Also \textbf{bulletproofs} arithmetization is more convinient and human-friendly for encoding most of the arithmetic circuits than the R1CS. +\textbf{Bulletproofs} arithmetization slightly differs from the classic R1CS we studied before, however it could be transformed vice-versa easily. Also \textbf{bulletproofs} arithmetization is more convenient and human-friendly for encoding most of the arithmetic circuits than the R1CS. -There are two types of variables in \textit{bulletproofs} constraint system: \textit{low-level} and \textit{high-level}. Typycally \textit{high-level} variables are provided with Pedersen commitments $\mathbf{V} \in \mathbb{G}^m$ as the private witness inputs $\mathbf{v} \in \mathbb{F}_p^m$ to the circuit, while \textit{low-level} variables $\mathbf{a}_L, \mathbf{a}_R, \mathbf{a}_O \in \mathbb{F}_p^n$ are the intermediate witness values of computation. We will define circuit as a set of multiplication constraints operating with \textit{low-level} variables and set of linear constraints which links \textit{low-level} variables between each other and \textit{high-level} variables as well. +There are two types of variables in \textit{bulletproofs} constraint system: \textit{low-level} and \textit{high-level}. Typically \textit{high-level} variables are provided with Pedersen commitments $\mathbf{V} \in \mathbb{G}^m$ as the private witness inputs $\mathbf{v} \in \mathbb{F}_p^m$ to the circuit, while \textit{low-level} variables $\mathbf{a}_L, \mathbf{a}_R, \mathbf{a}_O \in \mathbb{F}_p^n$ are the intermediate witness values of computation. We will define circuit as a set of multiplication constraints operating with \textit{low-level} variables and set of linear constraints which links \textit{low-level} variables between each other and \textit{high-level} variables as well. Multiplication constraints are defined with one vector equation: $$ \mathbf{a}_L \circ \mathbf{a}_R = \mathbf{a}_O $$ @@ -794,7 +798,7 @@ \subsubsection{Arithmetization} Linear constraints are defined via: $$ \mathbf{W}_L \cdot \mathbf{a}_L + \mathbf{W}_R \cdot \mathbf{a}_R + \mathbf{W}_O \cdot \mathbf{a}_O = \mathbf{W}_V \cdot \mathbf{v} + \mathbf{c} $$ -Where $\mathbf{a_L, a_R, a_O}$ -- vectors of left and right inputs for multiplication gates and output values (all of them are low-level variables). $\mathbf{W_L, W_R, W_O} \in \mathbb{F}_p^{q \times n}, \mathbf{W}_V \in \mathbb{F}_p^{q \times m}$ -- public matrices of weights for linear constraints(obviously known to verifier). $\mathbf{c} \in \mathbb{F}_p^q$ -- public vector of constants. Typycally they encode wiring of the circuit and other linear relations between variables. +Where $\mathbf{a_L, a_R, a_O}$ -- vectors of left and right inputs for multiplication gates and output values (all of them are low-level variables). $\mathbf{W_L, W_R, W_O} \in \mathbb{F}_p^{q \times n}, \mathbf{W}_V \in \mathbb{F}_p^{q \times m}$ -- public matrices of weights for linear constraints(obviously known to verifier). $\mathbf{c} \in \mathbb{F}_p^q$ -- public vector of constants. Typically they encode wiring of the circuit and other linear relations between variables. \begin{example} Consider the following elliptic curve membership circuit. Here witness $(v_1, v_2)$ should satisfy elliptic curve equation: @@ -1044,7 +1048,7 @@ \subsubsection{Proving a circuit satisfiability} \end{equation} So than we could get desired sum of inner products as the second-degree coefficient $s_2$: $$ w_c + \langle \mathbf{w}_V, \mathbf{v} \rangle + \delta(y, z) = s_2$$ -In order to obtain final polynimials $\mathbf{l}(x), \mathbf{r}(x)$ we must firstly blind $\mathbf{a}_L, \mathbf{a}_R$: +In order to obtain final polynomials $\mathbf{l}(x), \mathbf{r}(x)$ we must firstly blind $\mathbf{a}_L, \mathbf{a}_R$: \begin{equation} \begin{aligned} \mathbf{a}_L \gets \mathbf{a}_L + \mathbf{s}_Lx^2 && \mathbf{a}_R \gets \mathbf{a}_R + \mathbf{s}_Rx^2 @@ -1057,7 +1061,7 @@ \subsubsection{Proving a circuit satisfiability} $\mathbf{a}_O$ does not need separate blinding as it's located on the left side of the inner-product (\ref{eq:inner-product-circuit-1}) along with $\mathbf{a}_L$, which is already blinded by $\mathbf{s}_L$. \end{remark} -Now we could compute polynomials $\mathbf{l}(x), \mathbf{r}(x)$ from (\ref{eq:inner-product-circuit-1}) using assignments from (\ref{eq:inner-product-circuit-assignment}) and blindings from (\ref{eq:inner-product-circuit-blinding}): +Now we could compute polynomials $\mathbf{l}(x), \mathbf{r}(x)$ from (\ref{eq:inner-product-circuit-1}) using assignments from (\ref{eq:inner-product-circuit-assignment}) and blinders from (\ref{eq:inner-product-circuit-blinding}): \begin{equation} \begin{aligned} {\mathbf{l}}(x) &= (\mathbf{a}_L + \mathbf{s}_L \cdot x^2) \cdot x + \mathbf{y}^{-n} \circ \mathbf{w}_R \cdot x + \mathbf{a}_O \cdot x^2 \\ @@ -1209,7 +1213,7 @@ \subsubsection{Proving a circuit satisfiability} &= [\sum_{i=1}^m w_{V, i} \cdot v_i]G + [\sum_{i=1}^m w_{V, i} \cdot r_i]B \\ &= [\langle \mathbf{w}_V, \mathbf{v} \rangle] G + [\langle \mathbf{w}_V, \mathbf{r} \rangle] B \end{align*} - so simplyfying the LHS we get: + so simplifying the LHS we get: \begin{equation*} \begin{aligned} &[t_u]G + [\tau_u]B = \\ @@ -1221,13 +1225,13 @@ \subsubsection{Proving a circuit satisfiability} \end{equation*} \end{itemize} -\textit{Perfect honest-verifier zero-knowledge} follows from the zero-knowledge construction of $\Pi_{zkip}$ protocol as Vertifier learns no information about $\mathbf{a}_L, \mathbf{a}_R, \mathbf{v}$. +\textit{Perfect honest-verifier zero-knowledge} follows from the zero-knowledge construction of $\Pi_{zkip}$ protocol as Verifier learns no information about $\mathbf{a}_L, \mathbf{a}_R, \mathbf{v}$. \textit{Computational extended witness emulation} implies building extractor that combines extractors for two subprotocols: $\mathcal{E}_{ip}$ extracts witness ($\mathbf{l}_u, \mathbf{r}_u$) from $\Pi_{ip}$ than the extractor built on top of $\mathcal{E}_{zkip}$ extracts high-level witness ($\mathbf{a}_L, \mathbf{a}_R, \mathbf{a}_O, \mathbf{v}$) from $\Pi_{zkip} \quad \square$ The proof size of \textbf{arithmetic circuit satisfiability} protocol is $2 \log_2n +8$ group $\mathbb{G}$ elements and $5$ field $\mathbb{F}_p$ elements. -Usually the number of miltipliers $n$ is not a power of $2$ so that the intermediate witness vectors $\mathbf{a}_L, \mathbf{a}_R, \mathbf{a}_O$ must be padded with zeroes to the next power of $2$ in order to apply the \textit{inner-product argument} efficiently. +Usually the number of multipliers $n$ is not a power of $2$ so that the intermediate witness vectors $\mathbf{a}_L, \mathbf{a}_R, \mathbf{a}_O$ must be padded with zeroes to the next power of $2$ in order to apply the \textit{inner-product argument} efficiently. \begin{remark} The $\Pi_{sat}$ protocol could be slightly modified to provide intermediate random challenges inside the circuit. For example it would allow proving \textit{permutation check}: $\{a,b\} = \{c,d\} \iff (a-x)\cdot(b-x) = (c-x)\cdot(d-x)$ for some random challenge $x$. \end{remark} diff --git a/presentations/15-bulletproofs.tex b/presentations/15-bulletproofs.tex deleted file mode 100644 index 9895856..0000000 --- a/presentations/15-bulletproofs.tex +++ /dev/null @@ -1,318 +0,0 @@ -\documentclass{zkdl-presentation-template} - -\title[Bulletproofs: inner-product argument]{\textbf{Bulletproofs: inner-product argument}} -\author{Distributed Lab} -\date{July 24, 2025} -\homepage{zkdl-camp.github.io} -\github{ZKDL-Camp} - -\begin{document} -\frame { - \tikz [remember picture,overlay] - \node at - ([yshift=1.5cm,xshift=-1.5cm]current page.south east) -%or: (current page.center) - {\includegraphics[width=60pt]{images/logo.png}}; - - \titlepage -} - -\begin{frame}{Plan} - \tableofcontents -\end{frame} - -\section{Introduction} - -\begin{frame}{Bulletproofs: Motivation} - \begin{itemize} - \item \textbf{Bulletproofs:} zero-knowledge proofs with logarithmic proof size - \item No trusted setup - \item Straightforward cryptographic assumption: discrete logarithm without bilinear pairings or other advanced assumptions - \item Originally developed for efficient range proofs in confidential transactions - \item Applicable for proving arithmetic circuits satisfiability (e.g., R1CS) - \item Efficient polynomial commitment scheme could be derived (e.g., \textit{IPA} polynomial commitment) - \item Heart of \textit{bulletproofs} -- \textbf{inner-product argument} - \end{itemize} -\end{frame} - -\begin{frame}{Bulletproofs: Building blocks} - \begin{itemize} - \item Zero-knowledge multiplication protocol - \item Inner-product argument - \item Application: \textit{IPA} polynomial commitment scheme - \item Application: range proofs - \item Application: arithmetic circuits satisfiability - \end{itemize} -\end{frame} - -\begin{frame}{Notation} - \begin{itemize} - \item $\mathbb{G}$: cyclic group of prime order $p$ - \item $G, B \in \mathbb{G}$ -- independent generators - \item $\mathbf{G}, \mathbf{H} \in \mathbb{G}^n$ -- vectors of generators with mutually unknown discrete log relations - \item $\langle \mathbf{a}, \mathbf{b} \rangle$ -- inner product of vectors $\mathbf{a}, \mathbf{b}$ - \item $\langle \mathbf{a}, \mathbf{G} \rangle = \sum_{i=1}^n [a_i] G_i$ - \item $\mathbf{k}^n = (1, k, k^2, \dots, k^{n-1})$ - \end{itemize} -\end{frame} - -% --- Next Sections --- - -\section{Zero-knowledge multiplication} - -\begin{frame}{Zero-knowledge multiplication} - \textbf{Goal:} Prove knowledge of $a, b$ such that $c = ab$ without revealing $a, b$, i.e. consider the relation: - $$\mathcal{R}_{abc} = \{ (;c,a,b) \in \mathbb{F}_p \mid c=ab \}$$ - Curious reader could argue that this is relatively simple problem as we have $\Sigma$-protocols framework, especially Chaum-Pedersen protocol for DH-triplets, e.g. we could prove slightly modified relation: - $$\mathcal{R}'_{abc} = \{ (P, Q_a, Q_b, Q_c \in \mathbb{G};a,b) \mid Q_c = [a]Q_b, Q_a = [a]P, Q_b = [b]P \}$$ - - \begin{alertblock}{Problem} - Prover does not bind to $a,b,c$ values so that verifier could not be sure prover silently modified $a,b,c$ values after committing to them. - \end{alertblock} -\end{frame} - -\begin{frame}{Multiplication of committed values} - \begin{block}{Solution} - Use Pedersen commitments to bind to $a,b,c$ values. - \end{block} - $$ - \mathcal{R}'_{abc} = \left\{\begin{array}{l} - (G,H,B,A,V; a,b, \alpha, \beta \in \mathbb{F}_p) \mid \\ - V = [ab]G + [\beta]B, \\ - A = [a]G + [b]H + [\alpha]B - \end{array}\right\} - $$ - Here $A$ is a binding Pedersen commitment to both $a$ and $b$ while $V$ is a binding Pedersen commitment to their product $ab$. - \begin{alertblock}{Problem} - How to prove that in zero-knowledge? - \end{alertblock} -\end{frame} - -\begin{frame}{Zero-knowledge polynomial multiplication} - In order to provide zero-knowledge proof of $\mathcal{R}'_{abc}$ we bring out polynomials! - \begin{align*} - l(x) &= a + s_L x\\ - r(x) &= b + s_R x\\ - t(x) &= l(x)r(x) = ab + (s_L + s_R)bx + s_Ls_Rx^2 - \end{align*} - What we have now: - \begin{itemize} - \item $s_L, s_R \in \mathbb{F}_p$ are blinding factors - \item $l(x)$ is a linear polynomial hiding value $a$ - \item $r(x)$ is a linear polynomial hiding value $b$ - \item $t(x)$ is a quadratic polynomial, constant term is the product $ab$ - \end{itemize} -\end{frame} - -\begin{frame}{Zero-knowledge polynomial multiplication} - \begin{block}{Idea} - If $l(x)r(x) = t(x)$ than with high probability for random $u \in \mathbb{F}_p$ we have $l(u)r(u) = t(u)$ due to \textit{Schwartz-Zippel lemma} - \end{block} - Now we can build a zero-knowledge protocol for proving $l(x)r(x) = t(x)$: - \begin{itemize} - \item Prover commits to coefficients of $l(x), r(x), t(x)$: - \begin{align*} - A &= [a]G + [b]H + [\alpha]B\\ - S &= [s_L]G + [s_R]H + [\beta]B\\ - T_0 &= [ab]G + [\tau_0]B \\ - T_1 &= [s_L + s_R]G + [\tau_1]B \\ - T_2 &= [s_Ls_R]G + [\tau_2]B - \end{align*} - \item Verifier draws random challenge $u \in \mathbb{F}_p$ and sends it to prover - \end{itemize} -\end{frame} - -\begin{frame}{Zero-knowledge polynomial multiplication} - \begin{itemize} - \item Prover evaluates and sends to Verifier: - \begin{align*} - l_u &=l(u), r_u = r(u), t_u = t(u) = l_u \cdot r_u, \\ - \alpha_u &= \alpha + \beta u, \tau_u = \tau_0 + \tau_1 u + \tau_2 u^2 - \end{align*} - \item Verifier checks: - \begin{align*} - A + [u]S &\stackrel{?}{=} [l_u]G + [r_u]H + [\alpha_u]B \\ - [t_u]G + [\tau_u]B &\stackrel{?}{=} T_0 + [u]T_1 + [u^2]T_2 \\ - t_u &\stackrel{?}{=} l_u r_u - \end{align*} - \end{itemize} -\end{frame} - -\begin{frame}{zk-mul protocol: Security} - \begin{theorem} - Zero-knowledge polynomial multiplication protocol is perfect complete, special sound and perfect honest-verifier zero-knowledge. - \end{theorem} - Here we briefly show the completeness: - \begin{itemize} - \item First check: \begin{align*} - A + [u]S &\stackrel{?}{=} [l_u]G + [r_u]H + [\alpha_u]B \\ - LHS &= [a]G + [b]H + [\alpha]B + [us_L]G + [us_R]H + [u\beta]B \\ - RHS &= [a+ u s_L]G + [b + u s_R]H + [\alpha + u \beta]B = \\ - &=[a]G + [b]H + [\alpha]B + [us_L]G + [us_R]H + [u\beta]B - \end{align*} - As $LHS = RHS$ the check is satisfied. - \end{itemize} -\end{frame} - -\begin{frame}{zk-mul protocol: Security} - \begin{itemize} - \item Second check: \begin{align*} - [t_u]G + [\tau_u]B &\stackrel{?}{=} T_0 + [u]T_1 + [u^2]T_2 \\ - LHS &= [ab + t_1 u + t_2 u^2]G + [\tau_0 + \tau_1 u + \tau_2 u^2]B\\ - RHS &=[ab]G + [\tau_0]B + [u]([t_1]G + [\tau_1]B) + \\ - &+ [u^2]([t_2]G + [\tau_2]B) - \end{align*} - As $LHS = RHS$ the check is satisfied. - \item The third check is $t_u = l_u r_u$ holds by definition - \end{itemize} - Intuitively \textbf{zk-mul} protocol is also \textit{zero-knowledge} as it is easy to simulate every step of the honest prover. \textit{Special soundness} holds as it's easy to build an extractor similar to Okamoto's protocol extractor for extracting openings of Pedersen commitments. -\end{frame} - -\begin{frame}{zk-mul: inner-product version} - We could easily generalize \textbf{zk-mul} protocol to prove $t(x) = \langle \mathbf{l}(x), \mathbf{r}(x) \rangle $ for polynomials with vector coefficients $\mathbf{l}(x), \mathbf{r}(x), \mathbf{t}(x) \in \mathbb{F}_p^n[x]$. Specifically using generalized \textbf{zk-mul} protocol we could also prove in zero-knowledge that inner-product relation holds for vectors $\mathbf{a}, \mathbf{b} \in \mathbb{F}_p^n$: - \begin{align*} - \mathcal{R}_{zkip} = \left\{ \begin{aligned} (\mathbf{G,H},G,B,A,V;\mathbf{a,b},\alpha, \gamma) \vert & A = \langle \mathbf{a,G} \rangle + \langle \mathbf{b,H} \rangle + [\alpha] B, \\ & V = [\langle \mathbf{a,b} \rangle]G + [\gamma]B \end{aligned} \right\} - \end{align*} - \begin{alertblock}{Problem} - Proof size is linear in $n$ as the prover should send evaluated vectors $\mathbf{l_u}, \mathbf{r_u}$. - \end{alertblock} -\end{frame} - -\begin{frame}{Zero-Knowledge Multiplication Protocol} - \begin{itemize} - \item $\mathsf{Setup}$: $G, H, B \in \mathbb{G}$ - \item Prover samples random $s_L, s_R$ - \item Defines $l(x) = a + s_L x$, $r(x) = b + s_R x$, $t(x) = l(x)r(x)$ - \item Parties run the multiplication protocol on $(l(x), r(x), t(x))$ - \end{itemize} -\end{frame} - -\begin{frame}{Summary: ZK Multiplication} - \begin{itemize} - \item Efficient protocol for proving $c = ab$ in zero-knowledge - \item Foundation for more complex protocols (range proofs, circuits) - \item Next: Inner-product argument - \end{itemize} -\end{frame} - -\section{Inner-product argument} - -\begin{frame}{Motivation: Inner-product Argument} - \begin{itemize} - \item Core of Bulletproofs: enables short proofs for range and circuit satisfiability - \item Compresses multiple constraints into a single inner-product relation - \item Achieves logarithmic proof size - \end{itemize} -\end{frame} - -\begin{frame}{Relation} - \begin{itemize} - \item Prover knows $\mathbf{a}, \mathbf{b} \in \mathbb{F}_p^n$ such that $P' = \langle \mathbf{a}, \mathbf{G} \rangle + \langle \mathbf{b}, \mathbf{H} \rangle + [\langle \mathbf{a}, \mathbf{b} \rangle]Q$ - \item $\mathbf{G}, \mathbf{H} \in \mathbb{G}^n$, $Q \in \mathbb{G}$ - \item All vectors have length $n = 2^d$ - \end{itemize} -\end{frame} - -\begin{frame}{Protocol Overview} - \begin{itemize} - \item Recursive protocol: reduces $n$-dimensional inner product to $1$-dimensional - \item At each round, splits vectors in half and sends commitments $L_k, R_k$ - \item Verifier sends challenge $u_k$ - \item Prover and verifier update vectors and generators - \item Repeat until $n = 1$ - \end{itemize} -\end{frame} - -\begin{frame}{Protocol: Step-by-Step} - \begin{enumerate} - \item For $k = d$ down to $1$: - \begin{itemize} - \item Prover splits $\mathbf{a}^{(k)}, \mathbf{b}^{(k)}, \mathbf{G}^{(k)}, \mathbf{H}^{(k)}$ into halves - \item Computes $L_k, R_k$ as commitments to cross terms - \item Sends $L_k, R_k$ to verifier - \item Verifier samples random $u_k$ - \item Prover and verifier update vectors and generators - \end{itemize} - \item When $n=1$, prover sends $a, b$ to verifier - \end{enumerate} -\end{frame} - -\begin{frame}{Commitment Equations} - \begin{itemize} - \item $L_k = \langle \mathbf{a}_{lo}^{(k)}, \mathbf{G}_{hi}^{(k)} \rangle + \langle \mathbf{b}_{hi}^{(k)}, \mathbf{H}_{lo}^{(k)} \rangle + [\langle \mathbf{a}_{lo}^{(k)}, \mathbf{b}_{hi}^{(k)} \rangle]Q$ - \item $R_k = \langle \mathbf{a}_{hi}^{(k)}, \mathbf{G}_{lo}^{(k)} \rangle + \langle \mathbf{b}_{lo}^{(k)}, \mathbf{H}_{hi}^{(k)} \rangle + [\langle \mathbf{a}_{hi}^{(k)}, \mathbf{b}_{lo}^{(k)} \rangle]Q$ - \end{itemize} -\end{frame} - -\begin{frame}{Recursive Update} - \begin{itemize} - \item After challenge $u_k$, update: - \begin{align*} - \mathbf{a}^{(k-1)} &= u_k \mathbf{a}_{lo}^{(k)} + u_k^{-1} \mathbf{a}_{hi}^{(k)} \\ - \mathbf{b}^{(k-1)} &= u_k^{-1} \mathbf{b}_{lo}^{(k)} + u_k \mathbf{b}_{hi}^{(k)} \\ - \mathbf{G}^{(k-1)} &= u_k^{-1} \mathbf{G}_{lo}^{(k)} + u_k \mathbf{G}_{hi}^{(k)} \\ - \mathbf{H}^{(k-1)} &= u_k \mathbf{H}_{lo}^{(k)} + u_k^{-1} \mathbf{H}_{hi}^{(k)} - \end{align*} - \item Repeat until $n=1$ - \end{itemize} -\end{frame} - -\begin{frame}{Final Check} - \begin{itemize} - \item Prover sends $a, b$ (scalars) to verifier - \item Verifier checks: - \begin{equation*} - P' + \sum_{i=1}^d ([u_i^2]L_i + [u_i^{-2}]R_i) = [a]G_1^{(0)} + [b]H_1^{(0)} + [ab]Q - \end{equation*} - \item Accept if equality holds - \end{itemize} -\end{frame} - -\begin{frame}{Communication Complexity} - \begin{itemize} - \item Each round: send $L_k, R_k$ ($2 \log_2 n$ group elements in total) - \item Final: send $a, b$ (2 field elements) - \item Logarithmic proof size - \end{itemize} -\end{frame} - -\begin{frame}{Security Properties} - \begin{itemize} - \item \textbf{Perfect completeness}: Honest prover always convinces verifier - \item \textbf{Statistical soundness}: Extractor can recover witness or find discrete log relation - \item \textbf{Not zero-knowledge} (unless $n>1$ and combined with ZK techniques) - \end{itemize} -\end{frame} - -\begin{frame}{Making it Zero-Knowledge} - \begin{itemize} - \item Combine with zero-knowledge inner-product protocol $\Pi_{zkip}$ - \item Add blinding to commitments - \item Ensures verifier learns nothing about witness - \end{itemize} -\end{frame} - -\begin{frame}{Diagram: Inner-product Protocol} - % TODO: Insert or reference TikZ diagram from lecture notes - \begin{center} - \textit{[Protocol sequence diagram here]} - \end{center} -\end{frame} - -\begin{frame}{Use in Bulletproofs} - \begin{itemize} - \item Inner-product argument enables short range proofs and circuit proofs - \item Used recursively to compress constraints - \item Key to logarithmic proof size in Bulletproofs - \end{itemize} -\end{frame} - -\begin{frame}{Summary: Inner-product Argument} - \begin{itemize} - \item Efficient, recursive protocol for proving inner-product relations - \item Logarithmic proof size - \item Foundation for Bulletproofs and other modern ZK systems - \end{itemize} -\end{frame} - -\end{document} \ No newline at end of file diff --git a/presentations/17-bulletproofs.pdf b/presentations/17-bulletproofs.pdf new file mode 100644 index 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E12e!dh5!Hn literal 0 HcmV?d00001 diff --git a/presentations/17-bulletproofs.tex b/presentations/17-bulletproofs.tex new file mode 100644 index 0000000..f0e35ca --- /dev/null +++ b/presentations/17-bulletproofs.tex @@ -0,0 +1,346 @@ +\documentclass{zkdl-presentation-template} + +\title[Bulletproofs: inner-product argument]{\textbf{Bulletproofs: inner-product argument}} +\author{Distributed Lab} +\date{July 24, 2025} +\homepage{zkdl-camp.github.io} +\github{ZKDL-Camp} + +\begin{document} +\frame { + \tikz [remember picture,overlay] + \node at + ([yshift=1.5cm,xshift=-1.5cm]current page.south east) +%or: (current page.center) + {\includegraphics[width=60pt]{images/logo.png}}; + + \titlepage +} + +\begin{frame}{Plan} + \tableofcontents +\end{frame} + +\section{Introduction} + +\begin{frame}{Bulletproofs: just some basic linear algebra} + \begin{figure}[h!] + \centering + \includegraphics[width=0.7\textwidth]{images/lecture_17/meme.jpg} + \end{figure} +\end{frame} + +\begin{frame}{Bulletproofs: Motivation} + \begin{itemize} + \item \textbf{Bulletproofs:} zero-knowledge proofs with logarithmic proof size + \item No trusted setup, just some basic linear algebra + \item Straightforward cryptographic assumption: discrete logarithm without bilinear pairings or other advanced assumptions + \item Originally developed for efficient range proofs in confidential transactions + \item Applicable for proving arithmetic circuits satisfiability (e.g., R1CS) + \item Efficient polynomial commitment scheme could be derived (e.g., \textit{IPA} polynomial commitment) + \item Heart of \textit{bulletproofs} -- \textbf{inner-product argument} + \end{itemize} +\end{frame} + +\begin{frame}{Bulletproofs: Building blocks} + \begin{itemize} + \item Zero-knowledge multiplication protocol \textbf{zk-mul} + \item Inner-product argument \textbf{IPA} + \item Application: \textit{IPA} polynomial commitment scheme + \item Application: range proofs + \item Application: arithmetic circuits satisfiability + \end{itemize} +\end{frame} + +\begin{frame}{Preliminaries} + \begin{itemize} + \item $\mathbb{G}$ -- cyclic group of prime order $p$ where \textbf{DLog} assumption holds + \item $G, B \in \mathbb{G}$ -- independent generators + \item $\mathbf{G}, \mathbf{H} \in \mathbb{G}^n$ -- vectors of generators with mutually unknown discrete log relations + \item $\langle \mathbf{a}, \mathbf{b} \rangle = \sum_{i=1}^n a_i b_i$ -- inner product of scalar vectors $\mathbf{a}, \mathbf{b} \in \mathbb{F}_p^n$ + \item $\langle \mathbf{a}, \mathbf{G} \rangle = \sum_{i=1}^n [a_i] G_i$ -- inner product of a scalar vector $\mathbf{a} \in \mathbb{F}_p^n$ with a vector of generators $\mathbf{G} \in \mathbb{G}^n$ + \item $\mathbf{k}^n = (1, k, k^2, \dots, k^{n-1})$ + \end{itemize} +\end{frame} + +% --- Next Sections --- + +\section{Zero-knowledge multiplication} + +\begin{frame}{Zero-knowledge multiplication} + \textbf{Goal:} Prove knowledge of $a, b \in \mathbb{F}_p$ such that $c = ab$ without revealing $a, b, c$, i.e. consider the relation: + $$\mathcal{R}_{abc} = \{ (\bot;c,a,b) \mid c=ab \}$$ + Curious reader could argue that this is relatively simple problem as we have $\Sigma$-protocols framework, especially Chaum-Pedersen protocol for DH-triplets, e.g. we could prove slightly modified relation: + $$\mathcal{R}'_{abc} = \{ (P, Q_a, Q_b, Q_c \in \mathbb{G};a,b) \mid Q_c = [a]Q_b, Q_a = [a]P, Q_b = [b]P \}$$ + + \begin{alertblock}{Problem} + Prover does not hide $a,b,c$ values so that adversary could potentially learn them if they are not uniformly distributed. + \end{alertblock} +\end{frame} + +\begin{frame}{Multiplication of committed values} + \begin{block}{Solution} + Use Pedersen commitments to bind to $a,b,c$ values. + \end{block} + $$ + \mathcal{R}'_{abc} = \left\{\begin{array}{l} + (G,H,B,A,V; a,b, \alpha, \beta \in \mathbb{F}_p) \mid \\ + V = [ab]G + [\beta]B, \\ + A = [a]G + [b]H + [\alpha]B + \end{array}\right\} + $$ + Here $A$ is a binding Pedersen commitment to both $a$ and $b$ while $V$ is a binding Pedersen commitment to their product $ab$. + \begin{alertblock}{Problem} + How to prove that in zero-knowledge? + \end{alertblock} +\end{frame} + +\begin{frame}{Zero-knowledge polynomial multiplication} + In order to provide zero-knowledge proof of $\mathcal{R}'_{abc}$ we bring out polynomials! + \begin{align*} + l(x) &= a + s_L x\\ + r(x) &= b + s_R x\\ + t(x) &= l(x)r(x) = ab + (s_L + s_R)bx + s_Ls_Rx^2 + \end{align*} + What we have now: + \begin{itemize} + \item $s_L, s_R \in \mathbb{F}_p$ are blinding factors + \item $l(x)$ is a linear polynomial hiding value $a$ + \item $r(x)$ is a linear polynomial hiding value $b$ + \item $t(x)$ is a quadratic polynomial, constant term is the product $ab$, the product we typically want to prove knowledge of. + \end{itemize} +\end{frame} + +\begin{frame}{Zero-knowledge polynomial multiplication} + \begin{block}{Idea} + If $l(x)r(x) = t(x)$ than with high probability for random $u \in \mathbb{F}_p$ we have $l(u)r(u) = t(u)$ due to \textit{Schwartz-Zippel lemma} + \end{block} + Now we can build a zero-knowledge protocol \textbf{zk-mul} for proving a product of degree-one polynomials $l(x)r(x) = t(x)$: + \begin{itemize} + \item Prover computes and sends to $\mathcal{V}$ commitments to coefficients of $l(x), r(x), t(x)$: + \begin{align*} + A &= [a]G + [b]H + [\alpha]B & T_0 &= [ab]G + [\tau_0]B\\ + S &= [s_L]G + [s_R]H + [\beta]B &T_1 &= [s_L + s_R]G + [\tau_1]B\\ + &&T_2 &= [s_Ls_R]G + [\tau_2]B + \end{align*} + Here $A$ -- commitment to constant terms, $S$ -- commitment to degree-one coefficients of $l(x),r(x)$, $T_{i}$ for $i=0..2$ -- commitments to coefficients of $t(x)$. + \end{itemize} +\end{frame} + +\begin{frame}{Zero-knowledge polynomial multiplication} + \begin{itemize} + \item Verifier draws random challenge $u \in \mathbb{F}_p$ and sends it to prover + \item Prover evaluates and sends to Verifier $(l_u, r_u, t_u, \alpha_u, \tau_u)$: + \begin{align*} + l_u &=l(u), r_u = r(u), t_u = t(u) = l_u \cdot r_u, \\ + \alpha_u &= \alpha + \beta u, \tau_u = \tau_0 + \tau_1 u + \tau_2 u^2 + \end{align*} + \item Verifier checks: + \begin{align*} + A + [u]S &\stackrel{?}{=} [l_u]G + [r_u]H + [\alpha_u]B \\ + [t_u]G + [\tau_u]B &\stackrel{?}{=} T_0 + [u]T_1 + [u^2]T_2 \\ + t_u &\stackrel{?}{=} l_u r_u + \end{align*} + \end{itemize} +\end{frame} + +\begin{frame}{Zero-knowledge numbers multiplication} + Now we could easily tweak our \textbf{zk-mul} protocol to prove relation + $$ + \mathcal{R}'_{abc} = \left\{\begin{array}{l} + (G,H,B,A,V; a,b, \alpha, \beta \in \mathbb{F}_p) \mid \\ + V = [ab]G + [\beta]B, \\ + A = [a]G + [b]H + [\alpha]B + \end{array}\right\} + $$ + in zero-knowledge: just use prescribed commitment $A$ as a commitment to constant terms of $l(x), r(x)$ and $V$ as a commitment $T_0$ to constant coefficient of $t(x)$ +\end{frame} + +\begin{frame}{zk-mul protocol: Security} + \begin{theorem} + Zero-knowledge polynomial multiplication protocol \textbf{zk-mul} is perfect complete, special sound and perfect honest-verifier zero-knowledge. + \end{theorem} + Here we briefly show the completeness: + \begin{itemize} + \item First check: \begin{align*} + A + [u]S &\stackrel{?}{=} [l_u]G + [r_u]H + [\alpha_u]B \\ + LHS &= [a]G + [b]H + [\alpha]B + [us_L]G + [us_R]H + [u\beta]B \\ + RHS &= [a+ u s_L]G + [b + u s_R]H + [\alpha + u \beta]B = \\ + &=[a]G + [b]H + [\alpha]B + [us_L]G + [us_R]H + [u\beta]B + \end{align*} + As $LHS = RHS$ the check is satisfied. + \end{itemize} +\end{frame} + +\begin{frame}{zk-mul protocol: Security} + \begin{itemize} + \item Second check: \begin{align*} + [t_u]G + [\tau_u]B &\stackrel{?}{=} T_0 + [u]T_1 + [u^2]T_2 \\ + LHS &= [ab + t_1 u + t_2 u^2]G + [\tau_0 + \tau_1 u + \tau_2 u^2]B\\ + RHS &=[ab]G + [\tau_0]B + [u]([t_1]G + [\tau_1]B) + \\ + &+ [u^2]([t_2]G + [\tau_2]B) + \end{align*} + As $LHS = RHS$ the check is satisfied. + \item The third check is $t_u = l_u r_u$ holds by definition + \end{itemize} + Intuitively \textbf{zk-mul} protocol is also \textit{zero-knowledge} as it is easy to simulate every step of the honest prover. \textit{Special soundness} holds as it's easy to build an extractor similar to Okamoto's protocol extractor for extracting openings of Pedersen commitments. +\end{frame} + +\begin{frame}{zk-mul: inner-product version} + We could easily generalize \textbf{zk-mul} protocol to prove $t(x) = \langle \mathbf{l}(x), \mathbf{r}(x) \rangle $ for polynomials with vector coefficients $\mathbf{l}(x), \mathbf{r}(x), \mathbf{t}(x) \in \mathbb{F}_p^n[x]$. Specifically using generalized \textbf{zk-mul} protocol we could also prove in zero-knowledge that inner-product relation holds for vectors $\mathbf{a}, \mathbf{b} \in \mathbb{F}_p^n$: + \begin{align*} + \mathcal{R}_{zkip} = \left\{ \begin{aligned} (\mathbf{G,H},G,B,A,V;\mathbf{a,b},\alpha, \gamma) \vert & A = \langle \mathbf{a,G} \rangle + \langle \mathbf{b,H} \rangle + [\alpha] B, \\ & V = [\langle \mathbf{a,b} \rangle]G + [\gamma]B \end{aligned} \right\} + \end{align*} + \begin{alertblock}{Problem} + Proof size is linear in $n$ as the prover should send evaluated vectors $\mathbf{l_u}, \mathbf{r_u}$. We will adress this problem in the next section: \textbf{inner-product argument}. + \end{alertblock} +\end{frame} + +\section{Inner-product argument} + +\begin{frame}{Motivation: Inner-product Argument} + The heart of \textbf{bulletproofs} is \textbf{inner-product argument(IPA)} which allows to soundly prove inner-product relation between two vectors $\mathbf{a}, \mathbf{b} \in \mathbb{F}_p^n$: + $$\mathcal{R}_{ip} = \{ (\mathbf{G,H}, P, c; \mathbf{a,b}) \vert P = \langle \mathbf{a,G} \rangle + \langle \mathbf{b,H} \rangle \wedge \langle \mathbf{a,b} \rangle = c \}$$ + Firstly, let's combine statements $P = \langle \mathbf{a,G} \rangle + \langle \mathbf{b,H} \rangle \wedge \langle \mathbf{a,b} \rangle = c$ into a single statement by multiplying the second one by a random challenge $r \in \mathbb{F}_p$ and some orthogonal generator $B \in \mathbb{G}$, summing up: + $$ + \mathcal{R}'_{ip} = \{ (\mathbf{G,H}, Q, P'; \mathbf{a,b}) \vert P' = \langle \mathbf{a,G} \rangle + \langle \mathbf{b,H} \rangle + [\langle \mathbf{a,b} \rangle]Q \} + $$ + Where $Q=[r]B, P' = P + [cr]B = P+[c]Q$ +\end{frame} + +\begin{frame}{Comression step} + Assuming $n = 2^d$ define by $\mathbf{G_{lo}} = (G_1, \dots, G_{n/2}), \mathbf{G_{hi}} = (G_{n/2+1},\dots, G_n) \in \mathbb{G}^{n/2}$ -- lower and higher halves of vector $\mathbf{G}$ and $\mathbf{a_{lo}} = (a_1, \dots, a_{n/2}), \mathbf{a_{hi}} = (a_{n/2+1},\dots,a_n) \in \mathbb{F}_p^{n/2}$ -- lower and higher halves of $\mathbf{a} \in \mathbb{F}_p^{n}$. + + Let $u_k \in \mathbb{F}_p$ - be challenge scalar, define compressed vectors: + \begin{align*} + \mathbf{a}^{(k-1)} &= \mathbf{a_{lo}} \cdot u_k + u_k^{-1} \cdot \mathbf{a_{hi}} \\ + \mathbf{b}^{(k-1)} &= \mathbf{b_{lo}} \cdot u_k^{-1} + u_k \cdot \mathbf{b_{hi}} \\ + \mathbf{G}^{(k-1)} &= \mathbf{G_{lo}} \cdot u_k^{-1} + u_k \cdot \mathbf{G_{hi}} \\ + \mathbf{H}^{(k-1)} &= \mathbf{H_{lo}} \cdot u_k + u_k^{-1} \cdot \mathbf{H_{hi}} + \end{align*} + \vspace{-1em} + \begin{alertblock}{Note} + If $n$ is not a power of two we could pad vectors with zeroes to the next power of two. + \end{alertblock} +\end{frame} + +\begin{frame}{Compression step: illustration} + \begin{figure}[h!] + \centering + \includegraphics[width=0.8\textwidth]{images/lecture_17/compressed.png} + \label{fig:compression} + \end{figure} +\end{frame} + +\begin{frame}{Commitment compression} + Define $P_k \gets P' = \langle \mathbf{a,G} \rangle + \langle \mathbf{b,H} \rangle + [\langle \mathbf{a,b} \rangle]Q$ -- current commitment to vectors $\mathbf{a,b}$ and define $P_{k-1}$ using compressed vectors to have the same form as $P_k$, but in new basis $(\mathbf{G}^{(k-1)}, \mathbf{H}^{(k-1)})$: + \begin{equation*} + P_{k-1} = \langle \mathbf{a}^{(k-1)}, \mathbf{G}^{(k-1)} \rangle + \langle \mathbf{b}^{(k-1)}, \mathbf{H}^{(k-1)} \rangle + [\langle \mathbf{a}^{(k-1)}, \mathbf{b}^{(k-1)} \rangle]Q + \end{equation*} + Substituting compressed vectors and applying bilinearity property of inner product we get: +\begin{align*} + P_{k-1} = & \textcolor{Blue}{\langle \mathbf{a_{lo}}, \mathbf{G_{lo}}\rangle + \langle \mathbf{a_{hi}}, \mathbf{G_{hi}}\rangle} &+ \textcolor{Peach}{u_k^2\langle \mathbf{a_{lo}}, \mathbf{G_{hi}}\rangle} + \textcolor{ForestGreen}{u_k^{-2}\langle \mathbf{a_{hi}}, \mathbf{G_{lo}}\rangle} + \\ + & \textcolor{Blue}{\langle \mathbf{b_{lo}}, \mathbf{H_{lo}}\rangle + \langle \mathbf{b_{hi}}, \mathbf{H_{hi}}\rangle} &+ \textcolor{Peach}{u_k^2\langle \mathbf{b_{hi}}, \mathbf{H_{lo}}\rangle} + \textcolor{ForestGreen}{u_k^{-2}\langle \mathbf{b_{lo}}, \mathbf{H_{hi}}\rangle} + \\ + & \textcolor{Blue}{[\langle \mathbf{a_{lo}}, \mathbf{b_{lo}}\rangle + \langle \mathbf{a_{hi}}, \mathbf{b_{hi}}\rangle]Q} &+ \textcolor{Peach}{[u_k^2\langle \mathbf{a_{lo}}, \mathbf{b_{hi}}\rangle} + \textcolor{ForestGreen}{u_k^{-2}\langle \mathbf{a_{hi}}, \mathbf{b_{lo}}\rangle]Q} +\end{align*} +The first two columns precisely represent current commitment $P_k$, for the last two columns we define $L_k, R_k$ as commitments to cross terms +\end{frame} + +\begin{frame}{Commitment compression} + So that we represent new commitment $P_{k-1}$ from the old one $P_k$ and cross terms $L_k, R_k$: + \begin{align*} + P_{k-1} &= \textcolor{Blue}{P_k} + \textcolor{Peach}{[u_k^2] L_k} + \textcolor{ForestGreen}{[u_k^{-2}] R_k} \\ + L_{k} &= \langle \mathbf{a_{lo}}, \mathbf{G_{hi}}\rangle + \langle \mathbf{b_{hi}}, \mathbf{H_{lo}}\rangle + [\langle \mathbf{a_{lo}}, \mathbf{b_{hi}}\rangle]Q \\ + R_{k} &= \langle \mathbf{a_{hi}}, \mathbf{G_{lo}}\rangle + \langle \mathbf{b_{lo}}, \mathbf{H_{hi}}\rangle + [\langle \mathbf{a_{hi}}, \mathbf{b_{lo}}\rangle]Q + \end{align*} + So the basic logic of compression step: + \begin{itemize} + \item Verifier draws challenge $u_k \xleftarrow{R} \mathbb{F}_p$ and sends it to prover + \item Prover computes $\mathbf{a}^{(k-1)}, \mathbf{b}^{(k-1)}$ and $L_k, R_k$ and sends them to verifier + \item Verifier reconstructs $P_{k-1}$ using $\mathbf{a}^{(k-1)}, \mathbf{b}^{(k-1)}$ and checks: + $$P_{k-1} = P_k + [u_k^2] L_k + [u_k^{-2}] R_k$$ + \end{itemize} + +\end{frame} + +\begin{frame}{Recursive compression} + \begin{block}{Remark} + We wish not send $\mathbf{a}^{(k-1)}, \mathbf{b}^{(k-1)}$ directly as this's inefficient due to still linear sizes, instead we apply recursion to compress this vectors to just one element. + \end{block} + Here we come up with some kind of statement compression algorithm reducing size of all vectors in half per compression step. Repeating compression algorithm $k$ times we end up with sending vectors $\mathbf{a}^{(0)}, \mathbf{b}^{(0)}$ each of length one and $P_0$ containing all accumulated cross-terms: + \begin{align*} + P_0 &= [a_1^{(0)}]G_1^{(0)} + [b_1^{(0)}]H_1^{(0)} + [a_1^{(0)}b_1^{(0)}]Q \\ + P_0 &= P_k + \sum_{i=1}^k ([u_i^2]L_i + [u_i^{-2}]R_i) + \end{align*} + Verifier compares this $P_0$s and asserts inner-product correctness. +\end{frame} + +\begin{frame}{Inner-product argument protocol} + Here we describe the \textbf{inner-product argument} protocol between prover $\mathcal{P}$ and verifier $\mathcal{V}$ for relation $\mathcal{R}'_{ip} = \{ (\mathbf{G,H}, Q, P'; \mathbf{a,b}) \vert P' = \langle \mathbf{a,G} \rangle + \langle \mathbf{b,H} \rangle + [\langle \mathbf{a,b} \rangle]Q \}$ from scratch. + \begin{itemize} + \item Prover $\mathcal{P}$ sets $$(k, \mathbf{a}^{(k)}, \mathbf{b}^{(k)}, \mathbf{G}^{(k)}, \mathbf{H}^{(k)}, P_k) \gets (d, \mathbf{a,b,G,H},P')$$ + \item Verifier $\mathcal{V}$ sets $$(k, \mathbf{G}^{(k)}, \mathbf{H}^{(k)}, P_k) \gets (d, \mathbf{G,H},P')$$ + \item While $k > 0$ then parties involve in \textbf{compression step protocol} + \end{itemize} +\end{frame} + +\begin{frame}{Compression step protocol} + \begin{itemize} + \item Prover $\mathcal{P}$ computes and sends to $\mathcal{V}$ + \begin{align*} + L_{k} &= \langle \mathbf{a_{lo}}^{(k)}, \mathbf{G_{hi}}^{(k)}\rangle + \langle \mathbf{b_{hi}}^{(k)}, \mathbf{H_{lo}}^{(k)}\rangle + [\langle \mathbf{a_{lo}}^{(k)}, \mathbf{b_{hi}}^{(k)}\rangle]Q \\ + R_{k} &= \langle \mathbf{a_{hi}}^{(k)}, \mathbf{G_{lo}}^{(k)}\rangle + \langle \mathbf{b_{lo}}^{(k)}, \mathbf{H_{hi}}^{(k)}\rangle + [\langle \mathbf{a_{hi}}^{(k)}, \mathbf{b_{lo}}^{(k)}\rangle]Q + \end{align*} + \item $\mathcal{V}$ draws challenge $u_k \xleftarrow{R} \mathbb{F}_p$ and sends it to $\mathcal{P}$ + \item Both $\mathcal{P}$ and $\mathcal{V}$ compute: + \begin{align*} + \mathbf{G}^{(k-1)} &= \mathbf{G_{lo}}^{(k)} \cdot u_k^{-1} + u_k \cdot \mathbf{G_{hi}}^{(k)} \\ + \mathbf{H}^{(k-1)} &= \mathbf{H_{lo}}^{(k)} \cdot u_k + u_k^{-1} \cdot \mathbf{H_{hi}}^{(k)} + \end{align*} + \item $\mathcal{P}$ computes: + \begin{align*} + \mathbf{a}^{(k-1)} &= \mathbf{a_{lo}}^{(k)} \cdot u_k + u_k^{-1} \cdot \mathbf{a_{hi}}^{(k)} \\ + \mathbf{b}^{(k-1)} &= \mathbf{b_{lo}}^{(k)} \cdot u_k^{-1} + u_k \cdot \mathbf{b_{hi}}^{(k)} \\ + \end{align*} + \end{itemize} +\end{frame} + +\begin{frame}{Final step} + At the final step when $k=0$ parties perform final check: + \begin{itemize} + \item Prover $\mathcal{P}$ sends $(a,b) \gets (\mathbf{a}_1^{(0)}, \mathbf{b}_1^{(0)})$ to verifier $\mathcal{V}$ + \item Verifier performs final check: + $$P' + \sum_{i=1}^d ([u_i^2]L_i + [u_i^{-2}]R_i) = [a]G_1^{(0)} + [b]H_1^{(0)} + [ab]Q$$ + outputs \textbf{accept} if equality holds and \textbf{reject} otherwise. + \end{itemize} +\end{frame} + +\begin{frame}{Inner-product argument: illustration} + \begin{figure}[h!] + \centering + \includegraphics[width=0.8\textwidth]{images/lecture_17/ipa.png} + \label{fig:ipa} + \end{figure} +\end{frame} + +\begin{frame}{Inner-product argument: security \& performance} + \begin{block}{Remark} + Overall communication complexity of \textbf{inner-product argument} is $2\log_2 n$ group elements plus $2$ field elements so we come up with logarithmic proof size for our inner-product relation $\mathcal{R}_{ip}$. + \end{block} + \begin{theorem}[Inner-Product Argument] + The \textbf{inner-product argument} for relation $\mathcal{R}_{ip}$ has \textit{perfect completeness and statistical witness-extended emulation} for either extracting a non-trivial discrete logarithm relation between $\mathbf{G,H}, Q$ or extracting valid witness $\mathbf{a,b}$. + \end{theorem} + \begin{alertblock}{Note} + \textit{Zero-knowledge} doesn't hold as if $n=1$ then $\mathcal{P}$ sends witness pair $a,b$ directly. We'll later compile efficient \textbf{inner-product argument} with zero-knowledge \textbf{zk-mul} protocol to achieve efficient zero-knowledge proofs for range proofs and arithmetic circuits. + \end{alertblock} +\end{frame} + +\begin{frame}{What's next?} + \begin{figure}[h!] + \centering + \includegraphics[width=0.5\textwidth]{images/lecture_17/ipa_meme.jpg} + \end{figure} +\end{frame} + + +\end{document} \ No newline at end of file diff --git a/presentations/images/lecture_17/compressed.png b/presentations/images/lecture_17/compressed.png new file mode 100644 index 0000000000000000000000000000000000000000..e231f28f560565795dcca0d0565a25421abde1c3 GIT binary patch literal 49844 zcmZsCWmr^E*ES#`C=C+QAtBP;Al)Dx3QBkP(2WB_BaNg=gLKynjevABq?AK9-$CE! zdEX!3bsc^%XP>qAioMp__gXtjT~!Y2>5HdGNJv-;^3rdSkWe7N-(z$%;Fp?a;$9>q zY9s||NiDDUds&Y^Q%qh)=tn)HdGiL8C_&w+=!Eq{&zbpssYKONoiA2ZZ=6}xnGOuB z#)!_*qEXSY`q2_3@abuA7Z-1q9R;@TEDzr=m$(KTi)^K(am%$Gj|}lIlb7kMW}rs~ 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znhS$at1U@&;;x|Ls#z@VS>OCO)0YqzIqMe$$;_cyG8|kFsSu^L7TXGD5fOP7OK|Z8 zZkFfAP!(>r<$Y>0-CbLn`jwzH#s2^%8=}K*&r!E}m7_Ko57DfZ9VS>)A^3}7Lu|3Xk z(k%@wEM0nEsMuBEHq2uTHV;>pih->?)j%4qbN~N z`!^P=syOBa8Z#(KvgKK3Kb6$j4l>@NQNHC8j-?$+I+S%N>QU4J^$^X8o0&zEB1uUr zUHW=D5dIJP3bKIz2aqasA_xD zwpAH@9BF{)dP#cm)8%PyddQZKr31Q&F!Hke^5^of~Zm~#@M Nj=zh!tNpY1|Je$Gn*aa+ literal 0 HcmV?d00001 From cfa80d4f41f6e5ef4e3e1365070b80c16b6e9e9a Mon Sep 17 00:00:00 2001 From: Yevhen Hrubiian Date: Fri, 25 Jul 2025 12:12:22 +0300 Subject: [PATCH 17/25] added citations --- bibliography.bib | 23 +++++++++++++++++++++++ lecture-notes-148x210.tex | 3 +++ lectures/2-9-bulletproofs.tex | 19 +++++++++---------- preface/intro_words/part3.pdf | Bin 32757 -> 32757 bytes preface/intro_words/part3.tex | 4 +++- presentations/17-bulletproofs.pdf | Bin 633824 -> 633858 bytes presentations/17-bulletproofs.tex | 2 +- 7 files changed, 39 insertions(+), 12 deletions(-) diff --git a/bibliography.bib b/bibliography.bib index fad5696..1513fba 100644 --- a/bibliography.bib +++ b/bibliography.bib @@ -65,3 +65,26 @@ @article{Saniee_2007_LagrangeInterpolation title = {A Simple Expression for Multivariate Lagrange Interpolation}, year = {2007}, } + +@misc{bulletproofs, + author = {Benedikt Bünz and Jonathan Bootle and Dan Boneh and Andrew Poelstra and Pieter Wuille and Greg Maxwell}, + title = {Bulletproofs: Short Proofs for Confidential Transactions and More}, + howpublished = {Cryptology {ePrint} Archive, Paper 2017/1066}, + year = {2017}, + url = {https://eprint.iacr.org/2017/1066} +} + +@misc{dalek_bulletproofs, + author = {Cathie Yun, Henry de Valence, Oleg Andreev}, + title = {bulletproofs rust crate}, + year = {2019}, + note = {Available at: \url{https://doc-internal.dalek.rs/bulletproofs/index.html}}, +} + +@book{thaler, + title = {Proofs, Arguments, and Zero-Knowledge}, + author = {Justin Thaler}, + year = {2023}, + note = {Available at: \url{https://people.cs.georgetown.edu/jthaler/ProofsArgsAndZK.pdf}}, +} + \ No newline at end of file diff --git a/lecture-notes-148x210.tex b/lecture-notes-148x210.tex index 4b43754..3464355 100644 --- a/lecture-notes-148x210.tex +++ b/lecture-notes-148x210.tex @@ -110,6 +110,9 @@ \section{Basics of STARKs}\label{section:stark} \subfile{lectures/2-8-stark} + \section{Bulletproofs}\label{section:bulletproofs} + \subfile{lectures/2-9-bulletproofs} + % --- Solutions --- \part{Concluding Remarks}\label{section:solutions} \subsection*{Solutions to Exercises} diff --git a/lectures/2-9-bulletproofs.tex b/lectures/2-9-bulletproofs.tex index 7e19a30..4abc888 100644 --- a/lectures/2-9-bulletproofs.tex +++ b/lectures/2-9-bulletproofs.tex @@ -1,5 +1,4 @@ \documentclass[../lecture-notes-148x210.tex]{subfiles} -\usepackage{systeme} \begin{document} @@ -16,7 +15,7 @@ \subsection{Introduction} -\textbf{Bulletproofs} is a zero-knowledge proof protocol with logarithmically sized proofs without a trusted setup. Originally, \textbf{bulletproofs} was developed to provide efficient range proofs in application to confidential transactions, but it applies also to arbitrary arithmetic circuit (possibly encoded in R1CS). In the heart of protocol lays \textbf{inner-product argument} which we describe in details. Technically, the protocol is built in an interactive fashion (like $\Sigma$-protocols), but one could make it non-interactive with a Fiat-Shamir transform. One key feature that differs it from $\Sigma$-protocols is the number of challenges from a verifier $\mathcal{V}$ -- in $\Sigma$-protocols there is only one challenge, while \textbf{bulletproofs} implies a logarithmic in circuit size number of queries. +\textbf{Bulletproofs} is a zero-knowledge proof protocol with logarithmically sized proofs without a trusted setup. Originally, \textbf{bulletproofs} was developed to provide efficient range proofs in application to confidential transactions, but it applies also to arbitrary arithmetic circuit (possibly encoded in R1CS). In the heart of protocol lays \textbf{inner-product argument} which we describe in details. Technically, the protocol is built in an interactive fashion (like $\Sigma$-protocols from \Cref{section:sigma}), but one could make it non-interactive with a Fiat-Shamir transform. One key feature that differs it from $\Sigma$-protocols is the number of challenges from a verifier $\mathcal{V}$ -- in $\Sigma$-protocols there is only one challenge, while \textbf{bulletproofs} implies a logarithmic in circuit size number of queries. Also, \textbf{bulletproofs}' \textbf{inner-product argument} could be used to build various polynomial commitment schemes -- crucial building block of proving systems built with \textit{IOP} framework (\textit{Halo, Nova, etc}). @@ -30,7 +29,7 @@ \subsection{Zero-knowledge multiplication} Consider the first-degree polynomials $l(x) = a + s_L x, r(x) = b + s_R x \in \mathbb{F}_p[x]$. Let $t(x) = l(x)r(x)$ and relation $$\mathcal{R}_{mul} = \{ (\bot;l(x),r(x),t(x)) \vert t(x) = l(x)r(x)\}$$ -Firstly, observe that proving $t(x) = l(x)r(x)$ may be reduced to evaluation check at some challenge point $u \in \mathbb{F}_p$: $t(u) = l(u)r(u)$, due to the \textit{Schwartz-Zippel lemma}: +Firstly, observe that proving $t(x) = l(x)r(x)$ may be reduced to evaluation check at some challenge point $u \in \mathbb{F}_p$: $t(u) = l(u)r(u)$, due to the \textit{Schwartz-Zippel lemma}(\Cref{lemma:one-sz}): $$\mathsf{Pr}[l(u)r(u) = t(u) \vert l(x)r(x) \neq t(x)] \le \frac{max(\deg(l(x)r(x)), \deg(t(x)))}{p} = \frac{2}{p}$$ is typically negligible function from security level which makes this check \textit{sound}. @@ -42,7 +41,7 @@ \subsubsection{Naїve polynomial multiplication protocol} $$t(x) = l(x)r(x) = (a+s_L x)(b+s_R x) = ab + (as_R + bs_L) + s_L s_R x^2$$ \item Prover $\mathcal{P}$ draws blinding factors $\alpha_0, \alpha_1, \beta_0, \beta_1, \tau_0, \tau_1, \tau_2 \xleftarrow{R} \mathbb{F}_p$ forming blinding polynomials $$\alpha(x) = \alpha_0 + \alpha_1 x, \beta(x) = \beta_0 + \beta_1 x, \tau(x) = \tau_0 + \tau_1 x + \tau_2 x^2$$ - and sends to $\mathcal{V}$ Pedersen commitments for each coefficient of $l(x), r(x), t(x)$: + and sends to $\mathcal{V}$ Pedersen commitments (\Cref{section:commitment-schemes}) for each coefficient of $l(x), r(x), t(x)$: \begin{equation} \begin{aligned} L_0 &= [a]G + [\alpha_0]B & R_0 &= [b]G + [\beta_0]B\\ @@ -187,10 +186,10 @@ \subsubsection{Zero-knowledge multiplication protocol} The protocol obviously has \textit{perfect completeness, special soundness, perfect honest-verifier zero-knowledge} due to the \Cref{th:poly_mul}. \begin{remark} %todo: add reference for Chaum-Pedersen - Now curious reader may wonder why build so overwhelmingly complicated protocol for simple multiplication and not just use classic \textit{Chaum-Pedersen protocol} for DH-triplets? Indeed, it definitely could establish that for given group elements $[a]G, [b]G, [c]G$ equality $c=ab$ holds, but unfortunately commitments $[a]G, [b]G, [c]G$ do not have a \textit{perfect hiding} property (though preserving \textit{computational binding} property) so an adversary could potentially learn $a,b,c$ values if they are not uniformly distributed. + Now curious reader may wonder why build so overwhelmingly complicated protocol for simple multiplication and not just use classic \textit{Chaum-Pedersen protocol} for DH-triplets? Indeed, it definitely could establish that for given group elements $[a]G, [b]G, [c]G$ equality $c=ab$ holds, but unfortunately commitments $[a]G, [b]G, [c]G$ do not have a \textit{perfect hiding} property (though preserving \textit{computational binding} property) so an adversary could potentially learn $a,b,c$ values especially if they are small or have non-uniform distribution. \end{remark} -Also, there's a folklore version of very similar protocol for establishing product relationship between Pedersen committed values described in \href{https://people.cs.georgetown.edu/jthaler/ProofsArgsAndZK.pdf}{section 12.3 of Thaler's book} +Also, there's a folklore version of very similar protocol for establishing product relationship between Pedersen committed values described in \cite[section 12]{thaler}. \subsubsection{Zero-knowledge inner-product protocol} @@ -519,7 +518,7 @@ \subsubsection{Proving $\mathcal{R}'_{ip}$} We need the fourth rewinding to assert equality of inner product: $$\langle \mathbf{a}^{(1)}, \mathbf{b}^{(1)} \rangle = \sum_{i=1}^3 v_i \langle \mathbf{a}_i^{(0)}, \mathbf{b}_i^{(0)} \rangle$$ - We won't describe it fully since it takes some unwieldy technical details and refer a reader to the original \textit{bulletproofs} paper, where the full proof of extraction is described in Theorem 1. + We won't describe it fully since it takes some unwieldy technical details and refer a reader to the original \textit{bulletproofs} paper \cite{bulletproofs}, where the full proof of extraction is described in Theorem 1. \item The extractor $\mathcal{E}'_{ip}$ recursively extracts $\mathbf{a}^{(k+1)}, \mathbf{b}^{(k+1)}$ from $ \mathbf{a}^{(k)}, \mathbf{b}^{(k)}$ using the method described in steps 2-5 until it reaches the final witness $\mathbf{a}^{(d)}, \mathbf{b}^{(d)} = \mathbf{a}, \mathbf{b}$ for which the relation $\mathcal{R}'_{ip}$ holds: $$P' = \langle \mathbf{a,G} \rangle + \langle \mathbf{b,H} \rangle + [\langle \mathbf{a,b} \rangle]Q$$ \end{enumerate} @@ -547,7 +546,7 @@ \subsubsection{Proving $\mathcal{R}'_{ip}$} Hence $c = \langle \mathbf{a,b} \rangle$. \end{enumerate} -To formally finalize a proof of \textit{witness-extended emulation} we also need to apply so-called \textit{the forking lemma}, we again refer a reader to the original \textit{bulletproofs} paper $\quad \square$. +To formally finalize a proof of \textit{witness-extended emulation} we also need to apply so-called \textit{the forking lemma}, we again refer a reader to the original \textit{bulletproofs} paper \cite{bulletproofs} $\quad \square$. \subsection{Inner-product based polynomial commitment scheme} @@ -772,7 +771,7 @@ \subsection{Range proofs} \begin{remark} Range proofs could be efficiently aggregated: e.g. one could prove the relation using slightly modified range proof protocol $\Pi_{rp}$ $$\mathcal{R}_{rpm} = \{ (G, B, \vec{V}, n; \vec{v}, \vec{\gamma}) \vert \forall i \in 1..m: V_i = [v_i]G + [\gamma_i]B, v_i \in [0, 2^n) \}$$ - Where $\vec{v} = (v_1, v_2, \dots, v_m)$ and $\vec{\gamma} = (\gamma_1, \gamma_2, \dots, \gamma_m)$ -- respectively secrets and blinding factors. Detail explanation of aggregation protocol could be found in \href{https://eprint.iacr.org/2017/1066.pdf}{original bulletroofs paper} + Where $\vec{v} = (v_1, v_2, \dots, v_m)$ and $\vec{\gamma} = (\gamma_1, \gamma_2, \dots, \gamma_m)$ -- respectively secrets and blinding factors. Detail explanation of aggregation protocol could be found in original bulletproofs paper \cite{bulletproofs} \end{remark} \begin{example} @@ -788,7 +787,7 @@ \subsection{Arithmetic circuits proofs} \subsubsection{Arithmetization} -\textbf{Bulletproofs} arithmetization slightly differs from the classic R1CS we studied before, however it could be transformed vice-versa easily. Also \textbf{bulletproofs} arithmetization is more convenient and human-friendly for encoding most of the arithmetic circuits than the R1CS. +\textbf{Bulletproofs} arithmetization slightly differs from the classic R1CS we studied before at \Cref{section:r1cs}, however it could be transformed vice-versa easily. Also \textbf{bulletproofs} arithmetization is more convenient and human-friendly for encoding most of the arithmetic circuits than the R1CS. There are two types of variables in \textit{bulletproofs} constraint system: \textit{low-level} and \textit{high-level}. Typically \textit{high-level} variables are provided with Pedersen commitments $\mathbf{V} \in \mathbb{G}^m$ as the private witness inputs $\mathbf{v} \in \mathbb{F}_p^m$ to the circuit, while \textit{low-level} variables $\mathbf{a}_L, \mathbf{a}_R, \mathbf{a}_O \in \mathbb{F}_p^n$ are the intermediate witness values of computation. 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Arithmetization \\ \hdashline \rowcolor{purple!20}\ref{section:stark} & STARK & FRI, Hash-based proving system, Example \\ + \hdashline + \rowcolor{purple!10}\ref{section:bulletproofs} & Bulletproofs & Inner-product argument, Range proofs, Arithmetic circuits proofs \\ \Xhline{3\arrayrulewidth} \end{tabularx} \caption{Topics covered in Part III} @@ -36,7 +38,7 @@ \end{table} While currently book features only $\Sigma$-proofs, zk-SNARKs, and STARKs, we -plan to extend it with more topics in the future (such as Bulletproofs). +plan to extend it with more topics in the future. \end{document} \ No newline at end of file diff --git a/presentations/17-bulletproofs.pdf b/presentations/17-bulletproofs.pdf index 49b96aab44d14caf8f12d169969aac2bf9a8d6bf..e2be6fb3e33daa53de07c706c61daef05615c4d5 100644 GIT binary patch delta 6176 zcmbtX2T)U6x4v{v2p~ugRp}rQr7D7SQIMt*>E!|bF~;&eBd$@@~FGh0+&p%u()YSKI7;4V&yEub#vQvV+g>ZdfhfxIL%=Hfg~GbC{#W-xONf`6OI_jC4R$I 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z>-fVx*s~*4+lloLdQ$~I=1>1_u@0-nfhE*Ij{H@J`mfT+k)t9199b8&J}>$2)M}@v zuaR`|>k$f7be%IJLPb^_?*`gxVaQvZs{3@M9gr0>jb7*jtQXhiDV(Ul+bO1PyvfH#8ei$v@q5_}R4`9HcX^U#Ei zaBcF$QLKo&t@6ZCKixJ%>J+F49$&^54Dq9IP;7w=KYqFwsbjBk+S$A@)HG;)wb&o5 zvYT^&+ZYNRgw1zh3!-dAT1x%FqSWgzWpMG~RWy*T*F+IHN;At&fK1#-&{aCRrmbKi zvj)1RQ;P+nMr-An({7>!ZjP~l4wf1NDL>Z8mQcey+DJZCA Hs7?7lAj$V) diff --git a/presentations/17-bulletproofs.tex b/presentations/17-bulletproofs.tex index f0e35ca..a05f5ac 100644 --- a/presentations/17-bulletproofs.tex +++ b/presentations/17-bulletproofs.tex @@ -74,7 +74,7 @@ \section{Zero-knowledge multiplication} $$\mathcal{R}'_{abc} = \{ (P, Q_a, Q_b, Q_c \in \mathbb{G};a,b) \mid Q_c = [a]Q_b, Q_a = [a]P, Q_b = [b]P \}$$ \begin{alertblock}{Problem} - Prover does not hide $a,b,c$ values so that adversary could potentially learn them if they are not uniformly distributed. + Prover does not hide $a,b,c$ values so that adversary could potentially learn them if they are values especially if they are small or have non-uniform distribution. \end{alertblock} \end{frame} From 7a600a3c87fd3ad11329311dc241c8118d3fe9e6 Mon Sep 17 00:00:00 2001 From: Yevhen Hrubiian Date: Fri, 25 Jul 2025 15:27:38 +0300 Subject: [PATCH 18/25] fix soundness proof of zkmul --- lectures/2-9-bulletproofs.tex | 15 +++++++++++---- 1 file changed, 11 insertions(+), 4 deletions(-) diff --git a/lectures/2-9-bulletproofs.tex b/lectures/2-9-bulletproofs.tex index 4abc888..1cf4366 100644 --- a/lectures/2-9-bulletproofs.tex +++ b/lectures/2-9-bulletproofs.tex @@ -76,7 +76,7 @@ \subsubsection{Naїve polynomial multiplication protocol} \end{equation} \end{itemize} \begin{theorem} - Naїve \textbf{polynomial multiplication} protocol $\Pi'_{mul}$ has \textit{perfect completeness, special soundness, perfect honest-verifier zero-knowledge} + Naїve \textbf{polynomial multiplication} protocol $\Pi'_{mul}$ has \textit{perfect completeness, 3-special soundness, perfect honest-verifier zero-knowledge} \label{th:poly_mul_naive} \end{theorem} \textbf{Proof idea.} \textit{Perfect completeness} holds due to: @@ -98,9 +98,9 @@ \subsubsection{Naїve polynomial multiplication protocol} Proving \textit{honest-verifier zero-knowledge} is a bit complicated due to proper building of a simulator and proving indistinguishability of distributions, so we briefly describe the idea behind it: each commitment sent in the first phase by $\mathcal{P}$ is a Pedersen commitment which is hiding by design, every second phase response of $\mathcal{P}$ is an evaluation of some first or second degree polynomial at chosen point so there's not enough information for interpolation and polynomial reconstruction, moreover it could be easily simulated. -To prove \textit{special soundness} we need to build a knowledge extractor $\mathcal{E}$: +To prove \textit{3-special soundness} we need to build a knowledge extractor $\mathcal{E}$ which extracts knowledge of witness polynomials $l(x), r(x), t(x)$ such that $l(x)r(x) = t(x)$ using 3 accepting transcripts: \begin{enumerate} - \item $\mathcal{E}$ runs $\mathcal{P}$ to the end and rewinds back the second phase of $\mathcal{P}$, getting two non-equal challenges $u_1, u_2 \in \mathbb{F}_p$ and two prover responses $(l_{u_1}, r_{u_1}), (l_{u_2}, r_{u_2})$ + \item $\mathcal{E}$ runs $\mathcal{P}$ to the end and rewinds back the second phase of $\mathcal{P}$, getting three non-equal challenges $u_1, u_2, u_3 \in \mathbb{F}_p$ and three prover responses $(l_{u_i}, r_{u_i}, t_{u_i})_{i=1}^3$ \item $\mathcal{E}$ solves the following systems of linear equations: \begin{equation*} \begin{cases} @@ -112,8 +112,15 @@ \subsubsection{Naїve polynomial multiplication protocol} r_{u_1} = b + s_R u_1 \\ r_{u_2} = b + s_R u_2 \end{cases} + \qquad + \begin{cases} + t_{u_1} = t_0 + t_1 u_1 + t_2 u_1^2 \\ + t_{u_2} = t_0 + t_1 u_2 + t_2 u_2^2 \\ + t_{u_3} = t_0 + t_1 u_3 + t_2 u_3^2 + \end{cases} \end{equation*} - and gets the coefficients of witness polynomials: $(a, s_L, b, s_R) \quad \square$ + and gets the coefficients of witness polynomials: $(a, s_L, b, s_R, t_0, t_1, t_2)$ + \item To prove that $l(x)r(x) = t(x)$ we apply Schwartz-Zippel lemma which asserts polynomial equality with high probability since for random challenges $u_i$ for honest prover we have $l(u_i)r(u_i) = t(u_i) \quad \square$. \end{enumerate} \subsubsection{Optimized polynomial multiplication protocol} From e90bc94a1d599baebd3781acccf8fd2eb3628328 Mon Sep 17 00:00:00 2001 From: Yevhen Hrubiian Date: Tue, 29 Jul 2025 18:23:42 +0300 Subject: [PATCH 19/25] added alternative zk extension of ipa --- bibliography.bib | 9 ++++++++- lectures/2-9-bulletproofs.tex | 25 +++++++++++++++++++++++-- 2 files changed, 31 insertions(+), 3 deletions(-) diff --git a/bibliography.bib b/bibliography.bib index 1513fba..5b76914 100644 --- a/bibliography.bib +++ b/bibliography.bib @@ -87,4 +87,11 @@ @book{thaler year = {2023}, note = {Available at: \url{https://people.cs.georgetown.edu/jthaler/ProofsArgsAndZK.pdf}}, } - \ No newline at end of file + +@misc{pvss, + author = {Craig Gentry and Shai Halevi and Vadim Lyubashevsky}, + title = {Practical Non-interactive Publicly Verifiable Secret Sharing with Thousands of Parties}, + howpublished = {Cryptology {ePrint} Archive, Paper 2021/1397}, + year = {2021}, + url = {https://eprint.iacr.org/2021/1397} +} \ No newline at end of file diff --git a/lectures/2-9-bulletproofs.tex b/lectures/2-9-bulletproofs.tex index 1cf4366..96fdf51 100644 --- a/lectures/2-9-bulletproofs.tex +++ b/lectures/2-9-bulletproofs.tex @@ -555,12 +555,33 @@ \subsubsection{Proving $\mathcal{R}'_{ip}$} To formally finalize a proof of \textit{witness-extended emulation} we also need to apply so-called \textit{the forking lemma}, we again refer a reader to the original \textit{bulletproofs} paper \cite{bulletproofs} $\quad \square$. +\subsubsection{Zero-knowledge extension of inner-product argument} + +The main approach to make \textit{inner-product argument} \textit{zero-knowledge} is to use $\Pi_{zkip}$ protocol which original bulletproofs \cite{bulletproofs} does for \textit{range proofs} and \textit{arithmetic circuits satisfiability}. However, there exists an alternative elegant construction based on tweaking \textit{inner-product argument} itself described in \cite[Appendix E.2]{pvss}. + +The key idea is to bring a blinding factor $\delta \in \mathbb{F}_p$ with a verifier-provided random element $S \in \mathbb{G}$ to the commitment $P_k$: +$$ P_k = \langle \mathbf{a}^{(k)}, \mathbf{G}^{(k)} \rangle + \langle \mathbf{b}^{(k)}, \mathbf{H}^{(k)} \rangle + [\langle \mathbf{a}^{(k)}, \mathbf{b}^{(k)} \rangle]Q + [\delta^{(k)}]S $$ +To get $L_k, R_k$ prover draws $\delta^{(k)}_L, \delta^{(k)}_R \xleftarrow{R} \mathbb{F}_p$ and sets: +\begin{align*} + P_{k-1} &= P_k + [u_k^2] L_k + [u_k^{-2}] R_k \\ + L_{k} &= \langle \mathbf{a_{lo}}^{(k)}, \mathbf{G_{hi}}^{(k)}\rangle + \langle \mathbf{b_{hi}}^{(k)}, \mathbf{H_{lo}}^{(k)}\rangle + [\langle \mathbf{a_{lo}}^{(k)}, \mathbf{b_{hi}}^{(k)}\rangle]Q + [\delta^{(k)}_L]S \\ + R_{k} &= \langle \mathbf{a_{hi}}^{(k)}, \mathbf{G_{lo}}^{(k)}\rangle + \langle \mathbf{b_{lo}}^{(k)}, \mathbf{H_{hi}}^{(k)}\rangle + [\langle \mathbf{a_{hi}}^{(k)}, \mathbf{b_{lo}}^{(k)}\rangle]Q + [\delta^{(k)}_R]S +\end{align*} +Then $\mathcal{P}$ updates next-round $\delta$ using verifier-provided challenge $u_k$: +$$\delta^{(k-1)} = \delta^{(k)} + u_k^2\delta^{(k)}_L + u_k^{-2}\delta^{(k)}_R$$ + +On the last step prover must prove that he possesses $a^{(0)}, b^{(0)}$ such that: +\begin{equation*} + P_0 = P_k + \sum_{i=1}^k ([u_i^2]L_i + [u_i^{-2}]R_i) = [a^{(0)}]G^{(0)} + [b^{(0)}]H^{(0)} + [a^{(0)}b^{(0)}]Q + [\delta^{(0)}]S +\end{equation*} + +$P_0$ is a Pedersen commitment to $a^{(0)}, b^{(0)}, a^{(0)} \cdot b^{(0)}$ with blinding factor $\delta^{(0)}$ and could easily be computed by verifier. One could prove opening to that commitment using protocol similar to $\Pi_{zkip}$ \subsection{Inner-product based polynomial commitment scheme} Here we describe one of the main theoretical applications of the \textit{inner-product argument} -- \textbf{polynomial commitment scheme} that relies only on \textit{discrete logarithm} assumption, while studied before \textit{KZG} commitment scheme needs bilinear pairings. \begin{definition} - The inner-product polynomial commitment scheme $\mathcal{C}_{ip} = (\mathsf{Commit, Open, VerifyOpen})$ is defined as follows. Let $f(x) = \sum_{i=0}^{n-1} a_i x^i \in \mathbb{F}_p[x]$ be a polynomial of degree $n-1$ and let $\mathbf{G} = (G_1, \dots, G_n)$ be independent group generators. + The inner-product non-hiding polynomial commitment scheme $\mathcal{C}_{ip} = (\mathsf{Commit, Open, VerifyOpen})$ is defined as follows. Let $f(x) = \sum_{i=0}^{n-1} a_i x^i \in \mathbb{F}_p[x]$ be a polynomial of degree $n-1$ and let $\mathbf{G} = (G_1, \dots, G_n)$ be independent group generators. \begin{itemize} \item $\mathsf{Commit}$ returns polynomial commitment $\mathsf{Com}(f) = \langle \mathbf{f}, \mathbf{G} \rangle$ where $\mathbf{f} = (a_0, \dots, a_{n-1})$ \item $\mathsf{Open}$ given evaluation point $u \in \mathbb{F}_p$ computes $\mathbf{u^n} = (1, u, u^2, \dots, u^{n-1})$, obtains $f(u) = \langle \mathbf{f, u^n} \rangle$ and runs \textit{inner-product argument} $\Pi_{ip}$ non-interactively setting @@ -571,7 +592,7 @@ \subsection{Inner-product based polynomial commitment scheme} \end{definition} \begin{remark} - As the second vector $\mathbf{b} = \mathbf{u^n}$ is known to the verifier, the prover don't have to commit to it using vector $\mathbf{H}$, so the parties might adjust all the steps eliminating vector $\mathbf{H}$ and $\mathbf{b}$ vector compression as well. The full scheme is described \href{https://www.zkdocs.com/docs/zkdocs/commitments/ipa-pcs/}{here}. + As the second vector $\mathbf{b} = \mathbf{u^n}$ is known to the verifier, the prover don't have to commit to it using vector $\mathbf{H}$, so the parties might adjust all the steps eliminating vector $\mathbf{H}$ and $\mathbf{b}$ vector compression as well. The full scheme is described \href{https://www.zkdocs.com/docs/zkdocs/commitments/ipa-pcs/}{here}. Note that described scheme is not zero-knowledge as classic \textit{inner-product} argument is not zero-knowledge. But we could make it zero-knowledge using $\Pi_{zkip}$ protocol or using alternative construction from \Cref{subsection:alternative-zk-ip}. \end{remark} \subsection{Range proofs} From a5fdeb602ac8a49d6ee1c6faaaace174973d3b66 Mon Sep 17 00:00:00 2001 From: Yevhen Hrubiian Date: Wed, 30 Jul 2025 18:38:58 +0300 Subject: [PATCH 20/25] wip on 2nd bp lecture --- lectures/2-9-bulletproofs.tex | 43 +++- ...etproofs.pdf => 17-bulletproofs-intro.pdf} | Bin 633858 -> 633858 bytes ...etproofs.tex => 17-bulletproofs-intro.tex} | 4 +- presentations/18-bulletproofs.pdf | Bin 0 -> 408301 bytes presentations/18-bulletproofs.tex | 228 ++++++++++++++++++ 5 files changed, 262 insertions(+), 13 deletions(-) rename presentations/{17-bulletproofs.pdf => 17-bulletproofs-intro.pdf} (99%) rename presentations/{17-bulletproofs.tex => 17-bulletproofs-intro.tex} (99%) create mode 100644 presentations/18-bulletproofs.pdf create mode 100644 presentations/18-bulletproofs.tex diff --git a/lectures/2-9-bulletproofs.tex b/lectures/2-9-bulletproofs.tex index 96fdf51..9e127a0 100644 --- a/lectures/2-9-bulletproofs.tex +++ b/lectures/2-9-bulletproofs.tex @@ -555,34 +555,53 @@ \subsubsection{Proving $\mathcal{R}'_{ip}$} To formally finalize a proof of \textit{witness-extended emulation} we also need to apply so-called \textit{the forking lemma}, we again refer a reader to the original \textit{bulletproofs} paper \cite{bulletproofs} $\quad \square$. -\subsubsection{Zero-knowledge extension of inner-product argument} +\subsubsection{Zero-knowledge extension of inner-product argument}\label{subsection:alternative-zk-ip} -The main approach to make \textit{inner-product argument} \textit{zero-knowledge} is to use $\Pi_{zkip}$ protocol which original bulletproofs \cite{bulletproofs} does for \textit{range proofs} and \textit{arithmetic circuits satisfiability}. However, there exists an alternative elegant construction based on tweaking \textit{inner-product argument} itself described in \cite[Appendix E.2]{pvss}. +The main approach to make \textit{inner-product argument} \textit{zero-knowledge} is to use $\Pi_{zkip}$ protocol which original bulletproofs \cite{bulletproofs} does for \textit{range proofs}(\Cref{subsection:bulletproofs-range-proofs}) and \textit{arithmetic circuits satisfiability}(\Cref{subsection:bulletproofs-arithmetic-circuits}). However, there exists an alternative elegant construction based on tweaking \textit{inner-product argument} itself described in \cite[Appendix E.2]{pvss}. -The key idea is to bring a blinding factor $\delta \in \mathbb{F}_p$ with a verifier-provided random element $S \in \mathbb{G}$ to the commitment $P_k$: -$$ P_k = \langle \mathbf{a}^{(k)}, \mathbf{G}^{(k)} \rangle + \langle \mathbf{b}^{(k)}, \mathbf{H}^{(k)} \rangle + [\langle \mathbf{a}^{(k)}, \mathbf{b}^{(k)} \rangle]Q + [\delta^{(k)}]S $$ +The key idea is to bring a blinding factor $\delta \in \mathbb{F}_p$ with a verifier-provided random element $D \in \mathbb{G}$ to the commitment $P_k$: +$$ P_k = \langle \mathbf{a}^{(k)}, \mathbf{G}^{(k)} \rangle + \langle \mathbf{b}^{(k)}, \mathbf{H}^{(k)} \rangle + [\langle \mathbf{a}^{(k)}, \mathbf{b}^{(k)} \rangle]Q + [\delta^{(k)}]D $$ To get $L_k, R_k$ prover draws $\delta^{(k)}_L, \delta^{(k)}_R \xleftarrow{R} \mathbb{F}_p$ and sets: \begin{align*} P_{k-1} &= P_k + [u_k^2] L_k + [u_k^{-2}] R_k \\ - L_{k} &= \langle \mathbf{a_{lo}}^{(k)}, \mathbf{G_{hi}}^{(k)}\rangle + \langle \mathbf{b_{hi}}^{(k)}, \mathbf{H_{lo}}^{(k)}\rangle + [\langle \mathbf{a_{lo}}^{(k)}, \mathbf{b_{hi}}^{(k)}\rangle]Q + [\delta^{(k)}_L]S \\ - R_{k} &= \langle \mathbf{a_{hi}}^{(k)}, \mathbf{G_{lo}}^{(k)}\rangle + \langle \mathbf{b_{lo}}^{(k)}, \mathbf{H_{hi}}^{(k)}\rangle + [\langle \mathbf{a_{hi}}^{(k)}, \mathbf{b_{lo}}^{(k)}\rangle]Q + [\delta^{(k)}_R]S + L_{k} &= \langle \mathbf{a_{lo}}^{(k)}, \mathbf{G_{hi}}^{(k)}\rangle + \langle \mathbf{b_{hi}}^{(k)}, \mathbf{H_{lo}}^{(k)}\rangle + [\langle \mathbf{a_{lo}}^{(k)}, \mathbf{b_{hi}}^{(k)}\rangle]Q + [\delta^{(k)}_L]D \\ + R_{k} &= \langle \mathbf{a_{hi}}^{(k)}, \mathbf{G_{lo}}^{(k)}\rangle + \langle \mathbf{b_{lo}}^{(k)}, \mathbf{H_{hi}}^{(k)}\rangle + [\langle \mathbf{a_{hi}}^{(k)}, \mathbf{b_{lo}}^{(k)}\rangle]Q + [\delta^{(k)}_R]D \end{align*} Then $\mathcal{P}$ updates next-round $\delta$ using verifier-provided challenge $u_k$: $$\delta^{(k-1)} = \delta^{(k)} + u_k^2\delta^{(k)}_L + u_k^{-2}\delta^{(k)}_R$$ -On the last step prover must prove that he possesses $a^{(0)}, b^{(0)}$ such that: +On the last step prover must prove that he possesses $a = a^{(0)}, b = b^{(0)}, \delta = \delta^{(0)}$ such that for $G = G^{(0)}, H = H^{(0)}$ equality holds: \begin{equation*} - P_0 = P_k + \sum_{i=1}^k ([u_i^2]L_i + [u_i^{-2}]R_i) = [a^{(0)}]G^{(0)} + [b^{(0)}]H^{(0)} + [a^{(0)}b^{(0)}]Q + [\delta^{(0)}]S + P_0 = P_k + \sum_{i=1}^k ([u_i^2]L_i + [u_i^{-2}]R_i) = [a]G + [b]H + [a\cdot b]Q + [\delta]D \end{equation*} -$P_0$ is a Pedersen commitment to $a^{(0)}, b^{(0)}, a^{(0)} \cdot b^{(0)}$ with blinding factor $\delta^{(0)}$ and could easily be computed by verifier. One could prove opening to that commitment using protocol similar to $\Pi_{zkip}$ +$P_0 = P_k + \sum_{i=1}^k ([u_i^2]L_i + [u_i^{-2}]R_i)$ is a Pedersen commitment to $a, b, a \cdot b$ with blinding factor $\delta$ and could easily be computed by verifier. One could prove knowledge of an opening $(a,b,\delta)$ to that commitment using protocol similar to $\Pi_{zkip}$. +\begin{enumerate} + \item $\mathcal{P}$ draws blinders $u,v,r,s \xleftarrow{R} \mathbb{F}_p$ and sends to $\mathcal{V}$ commitments: + \begin{align*} + A &= [r]G + [s]H + [as + br]Q + [u]D \\ + T &= [rs]Q + [v]D + \end{align*} + \item $\mathcal{V}$ draws challenge $c \xleftarrow{R} \mathbb{F}_p$ and sends it to $\mathcal{P}$ + \item $\mathcal{P}$ evaluates and sends to $\mathcal{V}$: + \begin{align*} + a' &= a + rc \\ + b' &= b + sc \\ + \delta' &= \delta + uc + vc^2 + \end{align*} + \item $\mathcal{V}$ performs check: + $$[a']G + [b']H + [a'b']Q + [\delta']D \stackrel{\text{?}}{=} P_0 + [c]A + [c^2]T$$ +\end{enumerate} + +Protocol is \textit{knowledge-sound} as value $\delta$ could easily be extracted from three accepting transcripts. To argue \textit{zero-knowledge} we stress that an adversary couldn't learn anything from transcript and each prover's message $(L_k, R_k)$ could be easily simulated by random element, at the base of recursion simulator simulates zero-knowledge proof of opening $(a,b,\delta)$ to commitment $P_0$. \subsection{Inner-product based polynomial commitment scheme} Here we describe one of the main theoretical applications of the \textit{inner-product argument} -- \textbf{polynomial commitment scheme} that relies only on \textit{discrete logarithm} assumption, while studied before \textit{KZG} commitment scheme needs bilinear pairings. \begin{definition} - The inner-product non-hiding polynomial commitment scheme $\mathcal{C}_{ip} = (\mathsf{Commit, Open, VerifyOpen})$ is defined as follows. Let $f(x) = \sum_{i=0}^{n-1} a_i x^i \in \mathbb{F}_p[x]$ be a polynomial of degree $n-1$ and let $\mathbf{G} = (G_1, \dots, G_n)$ be independent group generators. + The inner-product non-hiding polynomial commitment scheme $\mathcal{C}_{ip} = (\mathsf{Setup, Commit, Open, VerifyOpen})$ is defined as follows. Let $f(x) = \sum_{i=0}^{n-1} a_i x^i \in \mathbb{F}_p[x]$ be a polynomial of degree $n-1$. \begin{itemize} + \item $\mathsf{Setup}$ returns a vector of independent generators $\mathbf{G} = (G_1, \dots, G_n)$. \item $\mathsf{Commit}$ returns polynomial commitment $\mathsf{Com}(f) = \langle \mathbf{f}, \mathbf{G} \rangle$ where $\mathbf{f} = (a_0, \dots, a_{n-1})$ \item $\mathsf{Open}$ given evaluation point $u \in \mathbb{F}_p$ computes $\mathbf{u^n} = (1, u, u^2, \dots, u^{n-1})$, obtains $f(u) = \langle \mathbf{f, u^n} \rangle$ and runs \textit{inner-product argument} $\Pi_{ip}$ non-interactively setting $$\mathbf{a} = \mathbf{f}, \mathbf{b} = \mathbf{u^n}, P = \mathsf{Com}(f), c = f(u)$$ @@ -595,7 +614,7 @@ \subsection{Inner-product based polynomial commitment scheme} As the second vector $\mathbf{b} = \mathbf{u^n}$ is known to the verifier, the prover don't have to commit to it using vector $\mathbf{H}$, so the parties might adjust all the steps eliminating vector $\mathbf{H}$ and $\mathbf{b}$ vector compression as well. The full scheme is described \href{https://www.zkdocs.com/docs/zkdocs/commitments/ipa-pcs/}{here}. Note that described scheme is not zero-knowledge as classic \textit{inner-product} argument is not zero-knowledge. But we could make it zero-knowledge using $\Pi_{zkip}$ protocol or using alternative construction from \Cref{subsection:alternative-zk-ip}. \end{remark} -\subsection{Range proofs} +\subsection{Range proofs}\label{subsection:bulletproofs-range-proofs} Let $G,B\in \mathbb{G}$ -- independent group generators. Let's consider the relation: $$\mathcal{R}_{rp} = \{ (G, B, V, n; v, \gamma) \vert V = [v]G + [\gamma]B, v \in [0, 2^n) \}$$ @@ -809,7 +828,7 @@ \subsection{Range proofs} Therefore, \textbf{bulletproofs range proof} protocol is capable to prove a knowledge of witness to any $NP$-problem as they all could be reduced to the $\textbf{subset-sum problem}$ \end{example} -\subsection{Arithmetic circuits proofs} +\subsection{Arithmetic circuits proofs}\label{subsection:bulletproofs-arithmetic-circuits} \textbf{Bulletproofs} presents not only range proofs, but also efficient proofs for arithmetic circuits satisfiability. As we could see before, inner-product relation is quite powerful tool and could be used to prove a knowledge of witness to any $NP$-problem. 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e^c^cd9qI1Jmg^L*&N*?+PJL&op8%q96 zfgFk`YDe8DP!JdM5J$U%KFw7{Q}IF4!kG(Xhn~hyE2#CI#}x5z4y}_;`;#Bkli&RG z^AG;)IoPm)$={Y_(ggG?HW^iwIc}JJeRe8Ld~xvzMW0!PS4V@{=au zZ^RKT#Ygn^f*bA5X zCeCfi`jg&L#N9?$GMGT_Tql?8iyNwcT$(jt->scBulyvN-Q$WKl>a^pTHsj+?=6T9jP2h%jaQH#ar4lOn=`Hl zaHzT~P3%@o6Ngq%^IMhcJLdkR{CRTUI@x&}DBo@Tw>|*B-150iCWwN^IxGBc^L-m} zMBHYTXR*fyO58l-gIp zG|hNUlc1_tR>VC|vs|#<+1t&8D1N~`PrLjUA+A}gx-+)j(VNah7$%hD`(zhHHM3Q$h972r5Fl=kdU*?d-T=>r?*vc#1Z)cO z#MNa^_JVH(?2Pba1U#99)+)d^-eSOY`IGIlJP6JY%OjrkdOd;4n&`6A!~w_+pSE|| z3L5Z_kaYviO0lDBVI<5spSIP&sR#?w70s<9T<(v8WUn+n&N3yQ?_128@%^w`>-1wp zYL|jn8j!^vZoj_paS@TY+C#UN*W*hJTE1RSK6Ish2KB$6=5KmidwHY1C{OQF5>gO3 NDOy26^{X1R{{`Ce9KZko diff --git a/presentations/18-bulletproofs.tex b/presentations/18-bulletproofs.tex index 7c30d34..12ee5f1 100644 --- a/presentations/18-bulletproofs.tex +++ b/presentations/18-bulletproofs.tex @@ -47,6 +47,26 @@ \section{Introduction} \end{block} \end{frame} +\begin{frame}{Recap: zkmul} + Consider relation $R_{mul} = \{ (\bot; l(x), r(x), t(x)) \vert t(x) = l(x)r(x) \}$ where $l(x) = a + s_L x, r(x) = b + s_R x, t(x) = l(x)r(x)$. Protocol \textbf{zk-mul} is defined as follows: + + \begin{itemize} + \item Prover computes and sends to $\mathcal{V}$ commitments to $l(x), r(x), t(x)$: + \begin{align*} + A &= [a]G + [b]H + [\alpha]B & T_0 &= [ab]G + [\tau_0]B\\ + S &= [s_L]G + [s_R]H + [\beta]B &T_1 &= [s_L + s_R]G + [\tau_1]B\\ + &&T_2 &= [s_Ls_R]G + [\tau_2]B + \end{align*} + \item Verifier draws random challenge $u \in \mathbb{F}_p$ and sends it to prover + \item Prover evaluates and sends to Verifier $(l_u, r_u, t_u, \alpha_u, \tau_u)$: + \begin{align*} + l_u =l(u), r_u = r(u), t_u = l_u \cdot r_u, + \alpha_u = \alpha + \beta u, \tau_u = \tau_0 + \tau_1 u + \tau_2 u^2 + \end{align*} + \item Verifier checks: $A + [u]S \stackrel{?}{=} [l_u]G + [r_u]H + [\alpha_u]B$, $[t_u]G + [\tau_u]B \stackrel{?}{=} T_0 + [u]T_1 + [u^2]T_2$, $t_u \stackrel{?}{=} l_u r_u$ + \end{itemize} +\end{frame} + \section{IPA polynomial commitment scheme} \begin{frame}{Recap: Polynomial commitments} @@ -129,28 +149,89 @@ \section{Range proofs} $$\mathbf{a}_L' \gets \mathbf{a}_L + \mathbf{s}_L x \quad \mathbf{a}_R' \gets \mathbf{a}_R + \mathbf{s}_R x$$ Compute polynomials $\mathbf{l}(x) = \mathbf{l}_0 + \mathbf{l}_1 x, \quad \mathbf{r}(x) = \mathbf{r}_0 + \mathbf{r}_1 x$: \begin{align*} - \mathbf{l}(x) &= \mathbf{a}_L' - z \cdot \mathbf{1}^n = (\mathbf{a}_L + \mathbf{s}_L x) - z \cdot \mathbf{1}^n = \mathbf{a}_L - z \cdot \mathbf{1}^n + \mathbf{s}_L x\\ + \mathbf{l}(x) &= \mathbf{a}_L' - z \cdot \mathbf{1}^n = (\mathbf{a}_L + \mathbf{s}_L x) - z \cdot \mathbf{1}^n = \textcolor{teal}{\mathbf{a}_L - z \cdot \mathbf{1}^n} + \mathbf{s}_L x\\ \mathbf{r}(x) &= z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n + \mathbf{a}_R' \circ \mathbf{y}^n = z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n + (\mathbf{a}_R + \mathbf{s}_R x) \circ \mathbf{y}^n \\ - & = z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n + \mathbf{a}_R \circ \mathbf{y}^n + \mathbf{s}_R \circ \mathbf{y}^n x + & = \textcolor{orange}{z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n + \mathbf{a}_R \circ \mathbf{y}^n} + \mathbf{s}_R \circ \mathbf{y}^n x \end{align*} \end{frame} -\begin{frame}{Bulletproofs range proof: Example} +\begin{frame}{Compiling range proof into inner-product} + $$t(x) = \langle \mathbf{l}(x), \mathbf{r}(x) \rangle = t_0 + t_1 x + t_2 x^2$$ + Now $\mathcal{P}$ needs to apply \textbf{zk-mul} for proving: $$t_0 = \langle \textcolor{teal}{\mathbf{a}_L - z \cdot \mathbf{1}^n}, \textcolor{orange}{z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n + \mathbf{a}_R \circ \mathbf{y}^n}\rangle = z^2v + \delta(y,z)$$ + \textbf{Note}: $\mathcal{V}$ could compute commitment $Com(t_0)$ using $V=Com(v)$ + \begin{alertblock}{Remark} + We couldn't apply raw \textbf{zk-mul} as $\mathbf{l}_0$ depends on verifier-provided challenges, instead $\mathcal{P}$ firstly commits to $\mathbf{a}_L, \mathbf{a}_R$ and blinders $\mathbf{s}_L, \mathbf{s}_R$, obtaints challenges $y,z$ from $\mathcal{V}$ and computes rest of the commitments. + + During verification phase $\mathcal{V}$ should adjust commitments to $\mathbf{l}(x), \mathbf{r}(x)$ by himself using homomorphic proterties of Pedersen commitment scheme. + \end{alertblock} +\end{frame} + +\begin{frame}{Range proofs: building the protocol} \begin{itemize} - \item $v = 19$, $n=5$, $\mathbf{a}_L = (1,1,0,0,1)$ - \item $v = 1\cdot1 + 2\cdot1 + 4\cdot0 + 8\cdot0 + 16\cdot1 = 19$ - \item Prove $\mathbf{a}_L \in \{0,1\}^5$ and $v = \langle \mathbf{a}_L, (1,2,4,8,16) \rangle$ - \item All done in zero-knowledge + \item $\mathsf{Setup}$ returns independent generators $\mathbf{G}, \mathbf{H} \in \mathbb{G}^n$ + \item Prover does bit decomposition of $v$: $\mathbf{a}_L \gets \mathbf{v}, \mathbf{b_L} \gets \mathbf{a}_L - \mathbf{1}^n$, choses blinding terms $\mathbf{s}_L, \mathbf{s}_R \in \mathbb{F}_p^n, \alpha, \beta \in \mathbb{F}_p$, sends commitments: + \begin{align*} + A = \langle \mathbf{a}_L, \mathbf{G} \rangle + \langle \mathbf{b_L}, \mathbf{H} \rangle + [\alpha]B && + S = \langle \mathbf{s}_L, \mathbf{G} \rangle + \langle \mathbf{s}_R, \mathbf{H} \rangle + [\beta]B + \end{align*} + \item Verifier $\mathcal{V}$ samples challenges $y, z \xleftarrow{R} \mathbb{F}_p$ and sends them to $\mathcal{P}$ + \item Prover $\mathcal{P}$ reconstructs polynomials $\mathbf{l}(x), \mathbf{r}(x), \mathbf{t}(x)$: + \begin{equation*} + \begin{aligned} + \mathbf{l}(x) &= \mathbf{a}_L - z \cdot \mathbf{1}^n + \mathbf{s}_L x\\ + \mathbf{r}(x) &= z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n + \mathbf{a}_R \circ \mathbf{y}^n + \mathbf{s}_R \circ \mathbf{y}^n x\\ + t(x) &= \langle \mathbf{l}(x), \mathbf{r}(x) \rangle = t_0 + t_1 x + t_2 x^2 + \end{aligned} + \end{equation*} + \begin{equation*} + \begin{aligned} + t_0 &= \langle \textcolor{teal}{\mathbf{a}_L - z \cdot \mathbf{1}^n}, \textcolor{orange}{z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n + \mathbf{a}_R \circ \mathbf{y}^n}\rangle = z^2v + \delta(y,z)\\ + t_1 &= \langle \mathbf{a}_L - z \cdot \mathbf{1}^n, \mathbf{y}^n \circ \mathbf{s}_R \rangle + \langle \mathbf{y}^n \circ (\mathbf{a}_R + z\cdot \mathbf{1}^n ) + z^2 \cdot \mathbf{2}^n, \mathbf{s}_L\rangle \\ + t_2 & = \langle \mathbf{s}_L, \mathbf{y}^n \circ \mathbf{s}_R \rangle + \end{aligned} + \end{equation*} \end{itemize} \end{frame} -\begin{frame}{Bulletproofs range proof: Performance} +\begin{frame}{Range proofs: proving} \begin{itemize} - \item Proof size: $2\log_2 n + 9$ group elements (for $n$-bit range) - \item No trusted setup - \item Batch proofs: multiple values in one proof - \item Efficient for client-side proving + \item Prover $\mathcal{P}$ draws blinding factors $\tau_1, \tau_2 \xleftarrow{R} \mathbb{F}_p$ and sends to $\mathcal{V}$ commitments for coefficients of $t(x)$: + \begin{equation*} + \begin{aligned} + T_1 &= [t_1]G + [\tau_1]B \\ + T_2 &= [t_2]G + [\tau_2]B + \end{aligned} + \end{equation*} + \textbf{Note:} prover does not have to send commitment to $t_0$ as it's the inner-product we want to prove and it could be computed from high-level commitment $V$. + \item Verifier $\mathcal{V}$ samples and sends to $\mathcal{P}$ evaluation point $u \xleftarrow{R} \mathbb{F}_p$ + \item Prover $\mathcal{P}$ evaluates polynomials at $u$: + \begin{equation*} + \begin{aligned} + \mathbf{l}_u & = \mathbf{l}(u) & \alpha_u &= \alpha + \beta u\\ + \mathbf{r}_u &= \mathbf{r}(u) & \tau_u &= z^2\gamma + \tau_1 u + \tau_2 u^2\\ + t_u &= t(u) = t_0 + t_1u + t_2u^2 & + \end{aligned} + \end{equation*} + and sends $(\mathbf{l}_u, \mathbf{r}_u, t_u, \alpha_u, \tau_u)$ to $\mathcal{V}$. + \end{itemize} +\end{frame} + +\begin{frame}{Range proofs: verification} + \begin{itemize} + \item Verifier $\mathcal{V}$ checks: + \begin{equation*} + \begin{aligned} + A + [u]S + \langle -z \cdot \mathbf{1}^n, \mathbf{G} \rangle &+ \langle z \cdot \mathbf{y}^n + z^2 \cdot \mathbf{2}^n, \mathbf{y}^{-n} \circ \mathbf{H} \rangle \\ + &\stackrel{\text{?}}{=} \langle \mathbf{l}_u, \mathbf{G} \rangle + \langle \mathbf{r}_u, \mathbf{y}^{-n} \circ \mathbf{H} \rangle + [\alpha_u]B \\ + [t_u]G + [\tau_u]B &\stackrel{\text{?}}{=} [z^2]V + [\delta(y,z)]G + [u]T_1 + [u^2]T_2 \\ + t_u &\stackrel{\text{?}}{=} \langle \mathbf{l}_u \mathbf{r}_u \rangle + \end{aligned} + \end{equation*} \end{itemize} + \begin{block}{Remark} + To provide logarithmic size-proof instead of sending $\mathbf{l}_u, \mathbf{r}_u$ parties could run an inner-product argument \textbf{IPA} on inputs $(\mathbf{G},\mathbf{y}^{-n} \circ \mathbf{H}, P, t_u; \mathbf{l}_u, \mathbf{r}_u)$ where: + $$P = A + [u]S + \langle -z \cdot \mathbf{1}^n, \mathbf{G} \rangle + \langle z \cdot \mathbf{y}^n + z^2 \cdot \mathbf{2}^n, \mathbf{y}^{-n} \circ \mathbf{H} \rangle - [\alpha_u]B$$ + \end{block} \end{frame} \section{Arithmetic circuits} From ce5c2bf26c2b2ba6ef2e4d32e17f2d3230bb42dd Mon Sep 17 00:00:00 2001 From: Yevhen Hrubiian Date: Fri, 1 Aug 2025 14:30:24 +0300 Subject: [PATCH 22/25] finalize bp lecture --- presentations/18-bulletproofs.pdf | Bin 424476 -> 622381 bytes 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zanr?$6_r;M#jrx6{v6TC`+HQvjmnAdrRAs>Mg&1oU;%7k5RteKOb1XFXgdcdTSt_Y K)rWiBQU3v2*gcj2 diff --git a/presentations/14-bulletproofs.tex b/presentations/14-bulletproofs.tex index 88b026a..60989c6 100644 --- a/presentations/14-bulletproofs.tex +++ b/presentations/14-bulletproofs.tex @@ -27,7 +27,7 @@ \section{Introduction} \begin{frame}{Inner-product argument: illustration} \begin{figure}[h!] \centering - \includegraphics[width=0.8\textwidth]{images/lecture_17/ipa.png} + \includegraphics[width=0.8\textwidth]{images/lecture_14/ipa.png} \label{fig:ipa} \end{figure} \end{frame} @@ -70,7 +70,7 @@ \section{Introduction} \begin{frame}{What's next?} \begin{figure}[h!] \centering - \includegraphics[width=0.5\textwidth]{images/lecture_17/ipa_meme.jpg} + \includegraphics[width=0.5\textwidth]{images/lecture_14/ipa_meme.jpg} \end{figure} \end{frame} \section{IPA polynomial commitment scheme} @@ -566,7 +566,7 @@ \section{Arithmetic circuits} \begin{frame}{Questions?} \begin{figure} \centering - \includegraphics[width=0.44\textwidth]{images/lecture_17/circuit.png} + \includegraphics[width=0.44\textwidth]{images/lecture_14/circuit.png} \label{fig:np-completeness} \end{figure} \end{frame} diff --git a/presentations/images/lecture_17/circuit.png b/presentations/images/lecture_14/circuit.png similarity index 100% rename from presentations/images/lecture_17/circuit.png rename to presentations/images/lecture_14/circuit.png diff --git a/presentations/images/lecture_17/compressed.png b/presentations/images/lecture_14/compressed.png similarity index 100% rename from presentations/images/lecture_17/compressed.png rename to presentations/images/lecture_14/compressed.png diff --git a/presentations/images/lecture_17/ipa.png b/presentations/images/lecture_14/ipa.png similarity index 100% rename from presentations/images/lecture_17/ipa.png rename to presentations/images/lecture_14/ipa.png diff --git a/presentations/images/lecture_17/ipa_meme.jpg b/presentations/images/lecture_14/ipa_meme.jpg similarity index 100% rename from presentations/images/lecture_17/ipa_meme.jpg rename to presentations/images/lecture_14/ipa_meme.jpg diff --git a/presentations/images/lecture_17/main-qimg-c3bd2c20f632edf509ff6c41010dbd29-pjlq.jpeg b/presentations/images/lecture_14/main-qimg-c3bd2c20f632edf509ff6c41010dbd29-pjlq.jpeg similarity index 100% rename from presentations/images/lecture_17/main-qimg-c3bd2c20f632edf509ff6c41010dbd29-pjlq.jpeg rename to presentations/images/lecture_14/main-qimg-c3bd2c20f632edf509ff6c41010dbd29-pjlq.jpeg diff --git a/presentations/images/lecture_17/meme.jpg b/presentations/images/lecture_14/meme.jpg similarity index 100% rename from presentations/images/lecture_17/meme.jpg rename to presentations/images/lecture_14/meme.jpg From 1279812a9770f92133f3f0e427c865ff179505e1 Mon Sep 17 00:00:00 2001 From: Yevhen Hrubiian Date: Fri, 10 Apr 2026 13:45:53 +0300 Subject: [PATCH 24/25] refine the grammar in bulletproofs section --- lectures/2-9-bulletproofs.tex | 254 +++++++++++++++++----------------- 1 file changed, 127 insertions(+), 127 deletions(-) diff --git a/lectures/2-9-bulletproofs.tex b/lectures/2-9-bulletproofs.tex index 601085b..9e6dcb2 100644 --- a/lectures/2-9-bulletproofs.tex +++ b/lectures/2-9-bulletproofs.tex @@ -15,27 +15,27 @@ \subsection{Introduction} -\textbf{Bulletproofs} is a \textit{non-interactive zero-knowledge protocol(NIZK)} with logarithmically sized proofs without a trusted setup. Originally, \textbf{bulletproofs} was developed to provide efficient range proofs in application to confidential transactions, but it applies also to arbitrary arithmetic circuit (possibly encoded in R1CS). In the heart of protocol lays \textbf{inner-product argument} which we describe in details. Technically, the protocol is built in an interactive fashion (like $\Sigma$-protocols from \Cref{section:sigma}), but one could make it non-interactive with a Fiat-Shamir transform. One key feature that differs it from $\Sigma$-protocols is the number of challenges from a verifier $\mathcal{V}$ -- in $\Sigma$-protocols there is only one challenge, while \textbf{bulletproofs} implies a logarithmic in circuit size number of queries. +\textbf{Bulletproofs} is a \textit{non-interactive zero-knowledge protocol (NIZK)} with logarithmically sized proofs without a trusted setup. Originally, \textbf{bulletproofs} was developed to provide efficient range proofs in application to confidential transactions, but it also applies to arbitrary arithmetic circuits (possibly encoded in R1CS). At the heart of the protocol lies the \textbf{inner-product argument}, which we describe in detail. Technically, the protocol is built in an interactive fashion (like $\Sigma$-protocols from \Cref{section:sigma}), but one could make it non-interactive with a Fiat-Shamir transform. One key feature that differs it from $\Sigma$-protocols is the number of challenges from a verifier $\mathcal{V}$ -- in $\Sigma$-protocols there is only one challenge, while \textbf{bulletproofs} implies a number of queries logarithmic in circuit size. -Also, \textbf{bulletproofs}' \textbf{inner-product argument} could be used to build various polynomial commitment schemes -- crucial building block of proving systems built with \textit{IOP} framework (\textit{Halo, Nova, etc}). +Also, the \textbf{bulletproofs} \textbf{inner-product argument} could be used to build various polynomial commitment schemes -- a crucial building block of proving systems built with the \textit{IOP} framework (\textit{Halo, Nova, etc}). -The main advantages of \textbf{bulletproofs} are an absence of a trusted setup and security against eavesdropping that relies on the \textit{discrete-logarithm} assumption without any other auxiliary structures like bilinear pairings. Also it has quite fast prover for small circuits making it practically useful for client-side proving. However, the main disadvantage of \textbf{bulletproofs} is that it isn't a classic \textit{SNARK} due to linear in circuit size verification time, however still very efficient for small circuits. +The main advantages of \textbf{bulletproofs} are the absence of a trusted setup and security that relies on the \textit{discrete-logarithm} assumption without any other auxiliary structures like bilinear pairings. It also has a quite fast prover for small circuits, making it practically useful for client-side proving. However, the main disadvantage of \textbf{bulletproofs} is that it isn't a classic \textit{SNARK} due to verification time linear in circuit size, though it is still very efficient for small circuits. \subsection{Notation} -Let $\mathbb{G}$ - cyclic group of prime order $p$ written additively, $\mathbf{G} = (G_1, \dots, G_n), \mathbf{H} = (H_1, \dots, H_n) \in \mathbb{G}^n$ - vectors of independent generators. We denote by $\langle \mathbf{a,b} \rangle$ - inner product of vectors $\mathbf{a} = (a_1, \dots, a_n), \mathbf{b} = (b_1, \dots, b_n) \in \mathbb{F}_p^n$ and $\langle \mathbf{a,G} \rangle = \sum_{i=1}^n [a_i] G_i \in \mathbb{G}$ - inner product of vector $\mathbf{a}$ with vector of generators $\mathbf{G}$. Denote by $\mathbf{k}^n$ vector of $k$'s first $n$ powers: $\mathbf{k}^n = (1, k, k^2, \dots, k^{n-1})$, for example $\mathbf{0}^n, \mathbf{1}^n$ represents vectors of zeros and ones respectively, while $\mathbf{2}^{n} = (1, 2, 4, \dots, 2^{n-1})$ +Let $\mathbb{G}$ be a cyclic group of prime order $p$ written additively, and $\mathbf{G} = (G_1, \dots, G_n), \mathbf{H} = (H_1, \dots, H_n) \in \mathbb{G}^n$ be vectors of independent generators. We denote by $\langle \mathbf{a,b} \rangle$ the inner product of vectors $\mathbf{a} = (a_1, \dots, a_n), \mathbf{b} = (b_1, \dots, b_n) \in \mathbb{F}_p^n$, and by $\langle \mathbf{a,G} \rangle = \sum_{i=1}^n [a_i] G_i \in \mathbb{G}$ the inner product of vector $\mathbf{a}$ with the vector of generators $\mathbf{G}$. Denote by $\mathbf{k}^n$ the vector of the first $n$ powers of $k$: $\mathbf{k}^n = (1, k, k^2, \dots, k^{n-1})$; for example, $\mathbf{0}^n, \mathbf{1}^n$ represent vectors of zeros and ones respectively, while $\mathbf{2}^{n} = (1, 2, 4, \dots, 2^{n-1})$ \subsection{Zero-knowledge multiplication} -Let $a,b,c \in \mathbb{F}_p$. Here we build a zero-knowledge protocol for relation $\mathcal{R}_{abc} = \{ (\bot;c,a,b) \vert c=ab \}$. We use well-known $\Sigma$-protocol framework for that, but firstly we make very useful generalization that could allow us to prove much larger class of relations. +Let $a,b,c \in \mathbb{F}_p$. Here we build a zero-knowledge protocol for the relation $\mathcal{R}_{abc} = \{ (\bot;c,a,b) \vert c=ab \}$. We use the well-known $\Sigma$-protocol framework for that, but firstly we make a very useful generalization that allows us to prove a much larger class of relations. -Consider the first-degree polynomials $l(x) = a + s_L x, r(x) = b + s_R x \in \mathbb{F}_p[x]$. Let $t(x) = l(x)r(x)$ and relation +Consider the first-degree polynomials $l(x) = a + s_L x, r(x) = b + s_R x \in \mathbb{F}_p[x]$. Let $t(x) = l(x)r(x)$ and consider the relation $$\mathcal{R}_{mul} = \{ (\bot;l(x),r(x),t(x)) \vert t(x) = l(x)r(x)\}$$ -Firstly, observe that proving $t(x) = l(x)r(x)$ may be reduced to evaluation check at some challenge point $u \in \mathbb{F}_p$: $t(u) = l(u)r(u)$, due to the \textit{Schwartz-Zippel lemma}(\Cref{lemma:one-sz}): +Firstly, observe that proving $t(x) = l(x)r(x)$ may be reduced to an evaluation check at some challenge point $u \in \mathbb{F}_p$: $t(u) = l(u)r(u)$, due to the \textit{Schwartz-Zippel lemma} (\Cref{lemma:one-sz}): $$\mathsf{Pr}[l(u)r(u) = t(u) \vert l(x)r(x) \neq t(x)] \le \frac{max(\deg(l(x)r(x)), \deg(t(x)))}{p} = \frac{2}{p}$$ -is typically negligible function from security level which makes this check \textit{sound}. +which is typically a negligible function of the security level, making this check \textit{sound}. \subsubsection{Naїve polynomial multiplication protocol} -Let's describe naїve unoptimized version of \textbf{polynomial multiplication} protocol $\Pi'_{mul} = (\mathsf{Setup},\mathcal{P,V})$ for relation $\mathcal{R}_{mul}$. -During $\mathsf{Setup}$ parties agree on group elements $G,B \in \mathbb{G}$. After that parties involve in the following protocol: +Let's describe a naїve unoptimized version of the \textbf{polynomial multiplication} protocol $\Pi'_{mul} = (\mathsf{Setup},\mathcal{P,V})$ for the relation $\mathcal{R}_{mul}$. +During $\mathsf{Setup}$, parties agree on group elements $G,B \in \mathbb{G}$. After that, the parties engage in the following protocol: \begin{itemize} \item Prover $\mathcal{P}$ computes: $$t(x) = l(x)r(x) = (a+s_L x)(b+s_R x) = ab + (as_R + bs_L) + s_L s_R x^2$$ @@ -53,7 +53,7 @@ \subsubsection{Naїve polynomial multiplication protocol} \end{aligned} \end{equation} \begin{remark} - Each commitment could be also seen as a Pedersen commitment to a reciprocal blinding polynomial coefficient as well. + Each commitment can also be seen as a Pedersen commitment to the corresponding blinding polynomial coefficient. \end{remark} \item Verifier $\mathcal{V}$ samples and sends to $\mathcal{P}$ random evaluation point $u \leftarrowS \mathbb{F}_p$ \item Prover $\mathcal{P}$ evaluates $l(x),r(x),t(x)$ and $\alpha(x), \beta(x), \tau(x)$ at $u$: @@ -96,11 +96,11 @@ \subsubsection{Naїve polynomial multiplication protocol} \end{aligned} \end{equation*} -Proving \textit{honest-verifier zero-knowledge} is a bit complicated due to proper building of a simulator and proving indistinguishability of distributions, so we briefly describe the idea behind it: each commitment sent in the first phase by $\mathcal{P}$ is a Pedersen commitment which is hiding by design, every second phase response of $\mathcal{P}$ is an evaluation of some first or second degree polynomial at chosen point so there's not enough information for interpolation and polynomial reconstruction, moreover it could be easily simulated. +Proving \textit{honest-verifier zero-knowledge} is a bit complicated, as it requires properly building a simulator and proving indistinguishability of distributions, so we briefly describe the idea behind it: each commitment sent in the first phase by $\mathcal{P}$ is a Pedersen commitment, which is hiding by design; every second-phase response of $\mathcal{P}$ is an evaluation of some first- or second-degree polynomial at a chosen point, so there is not enough information for interpolation and polynomial reconstruction; moreover, it can easily be simulated. -To prove \textit{3-special soundness} we need to build a knowledge extractor $\mathcal{E}$ which extracts knowledge of witness polynomials $l(x), r(x), t(x)$ such that $l(x)r(x) = t(x)$ using 3 accepting transcripts: +To prove \textit{3-special soundness} we need to build a knowledge extractor $\mathcal{E}$ which extracts the witness polynomials $l(x), r(x), t(x)$ such that $l(x)r(x) = t(x)$ using 3 accepting transcripts: \begin{enumerate} - \item $\mathcal{E}$ runs $\mathcal{P}$ to the end and rewinds back the second phase of $\mathcal{P}$, getting three non-equal challenges $u_1, u_2, u_3 \in \mathbb{F}_p$ and three prover responses $(l_{u_i}, r_{u_i}, t_{u_i})_{i=1}^3$ + \item $\mathcal{E}$ runs $\mathcal{P}$ to the end and rewinds back the second phase of $\mathcal{P}$, obtaining three distinct challenges $u_1, u_2, u_3 \in \mathbb{F}_p$ and three prover responses $(l_{u_i}, r_{u_i}, t_{u_i})_{i=1}^3$ \item $\mathcal{E}$ solves the following systems of linear equations: \begin{equation*} \begin{cases} @@ -120,16 +120,16 @@ \subsubsection{Naїve polynomial multiplication protocol} \end{cases} \end{equation*} and gets the coefficients of witness polynomials: $(a, s_L, b, s_R, t_0, t_1, t_2)$ - \item To prove that $l(x)r(x) = t(x)$ we apply Schwartz-Zippel lemma which asserts polynomial equality with high probability since for random challenges $u_i$ for honest prover we have $l(u_i)r(u_i) = t(u_i) \quad \square$. + \item To prove that $l(x)r(x) = t(x)$ we apply the Schwartz-Zippel lemma, which asserts polynomial equality with high probability, since for random challenges $u_i$ and an honest prover we have $l(u_i)r(u_i) = t(u_i) \quad \square$. \end{enumerate} \subsubsection{Optimized polynomial multiplication protocol} -We could optimize our \textbf{polynomial multiplication protocol} furthermore. Note that we could simply apply vector Pedersen commitment for constant and linear terms using one more group element $H \in \mathbb{G}$. +We can optimize our \textbf{polynomial multiplication protocol} further. Note that we can simply apply a vector Pedersen commitment for the constant and linear terms using one more group element $H \in \mathbb{G}$. \begin{definition} The \textbf{polynomial multiplication protocol} $\Pi_{mul} = (\mathsf{Setup}, \mathcal{P,V})$ for the relation $$\mathcal{R}_{mul} = \{ (\bot;l(x),r(x),t(x)) \vert t(x) = l(x)r(x)\}$$ with prover $\mathcal{P}$ and verifier $\mathcal{V}$ is defined as follows: \begin{itemize} - \item $\mathsf{Setup}$ returns triple of group generators with unknown discrete log relations $G, H, B \in \mathbb{G}$ + \item $\mathsf{Setup}$ returns a triple of group generators with unknown discrete log relations $G, H, B \in \mathbb{G}$ \item Parties $\mathcal{P, V}$ run the following protocol: \begin{itemize} \item Prover $\mathcal{P}$ computes: @@ -171,11 +171,11 @@ \subsubsection{Optimized polynomial multiplication protocol} The \textbf{polynomial multiplication} protocol $\Pi_{mul}$ has \textit{perfect completeness, special soundness, perfect honest-verifier zero-knowledge} \label{th:poly_mul} \end{theorem} -\textbf{Proof}. We left to a reader proof of the theorem in the sake of brevity because it's very similar to the proof of \Cref{th:poly_mul_naive} $\quad \square$ +\textbf{Proof}. We leave the proof of the theorem to the reader for the sake of brevity, since it is very similar to the proof of \Cref{th:poly_mul_naive} $\quad \square$ \subsubsection{Zero-knowledge multiplication protocol} -Finally, we could easily build the protocol for the zk-multiplication relation where each witness element presented in statement as a Pedersen commitment: +Finally, we can easily build the protocol for the zk-multiplication relation where each witness element is presented in the statement as a Pedersen commitment: $$\mathcal{R}_{abc} = \left\{\begin{array}{l} (G,H,B,A,T_0;a,b, \alpha, \tau_0) \vert \\ A = [a]G + [b]H + [\alpha]B \wedge \\ @@ -187,20 +187,20 @@ \subsubsection{Zero-knowledge multiplication protocol} The \textbf{multiplication protocol} $\Pi_{abc} = (\mathcal{P,V})$ for the relation $\mathcal{R}_{abc}$ with prover $\mathcal{P}$ and verifier $\mathcal{V}$ is defined as follows: \begin{itemize} \item Prover $\mathcal{P}$ draws random $s_L, s_R \leftarrowS \mathbb{F}_p$ and defines polynomials: $$l(x)=a+s_L x,\quad r(x)=b+s_R x,\quad t(x)=l(x)r(x)$$ - \item Parties run $\Pi_{mul}$ on inputs $(l(x),r(x),t(x))$ along with provided a-priori setup $G,H,B \in \mathbb{G}$ and commitments $A, T_0$ + \item Parties run $\Pi_{mul}$ on inputs $(l(x),r(x),t(x))$ along with the a-priori provided setup $G,H,B \in \mathbb{G}$ and commitments $A, T_0$ \end{itemize} \end{definition} -The protocol obviously has \textit{perfect completeness, special soundness, perfect honest-verifier zero-knowledge} due to the \Cref{th:poly_mul}. +The protocol obviously has \textit{perfect completeness, special soundness, perfect honest-verifier zero-knowledge} due to \Cref{th:poly_mul}. \begin{remark} %todo: add reference for Chaum-Pedersen - Now curious reader may wonder why build so overwhelmingly complicated protocol for simple multiplication and not just use classic \textit{Chaum-Pedersen protocol} for DH-triplets? Indeed, it definitely could establish that for given group elements $[a]G, [b]G, [c]G$ equality $c=ab$ holds, but unfortunately commitments $[a]G, [b]G, [c]G$ do not have a \textit{perfect hiding} property (though preserving \textit{computational binding} property) so an adversary could potentially learn $a,b,c$ values especially if they are small or have non-uniform distribution. + A curious reader may wonder why we build such an overwhelmingly complicated protocol for simple multiplication and not just use the classic \textit{Chaum-Pedersen protocol} for DH-triplets. Indeed, it could establish that for given group elements $[a]G, [b]G, [c]G$ the equality $c=ab$ holds, but unfortunately the commitments $[a]G, [b]G, [c]G$ do not have the \textit{perfect hiding} property (though they do preserve the \textit{computational binding} property), so an adversary could potentially learn the values $a,b,c$, especially if they are small or have a non-uniform distribution. \end{remark} -Also, there's a folklore version of very similar protocol for establishing product relationship between Pedersen committed values described in \cite[section 12]{thaler2022proofs}. +Also, there is a folklore version of a very similar protocol for establishing a product relationship between Pedersen-committed values described in \cite[section 12]{thaler2022proofs}. \subsubsection{Zero-knowledge inner-product protocol} -We could extend our $\Pi_{mul}$ protocol even further to provide zero-knowledge proof for the inner-product of vectors: $\langle \mathbf{a, b} \rangle = v$. The main trick is to substitute polynomials $l(x), r(x) \in \mathbb{F}_p[x]$ by vector polynomials $\mathbf{l}(x), \mathbf{r}(x) \in \mathbb{F}_p^n[x]$ where constant terms are equal to $\mathbf{a}$ and $\mathbf{b}$ respectively, taking inner-product $\langle \mathbf{l}(x), \mathbf{r}(x) \rangle$ results in polynomial with scalar coefficients where constant term is equal to $\langle \mathbf{a, b} \rangle$. +We can extend our $\Pi_{mul}$ protocol even further to provide a zero-knowledge proof for the inner product of vectors: $\langle \mathbf{a, b} \rangle = v$. The main trick is to substitute the polynomials $l(x), r(x) \in \mathbb{F}_p[x]$ with vector polynomials $\mathbf{l}(x), \mathbf{r}(x) \in \mathbb{F}_p^n[x]$ whose constant terms are equal to $\mathbf{a}$ and $\mathbf{b}$ respectively. Taking the inner product $\langle \mathbf{l}(x), \mathbf{r}(x) \rangle$ results in a polynomial with scalar coefficients whose constant term is equal to $\langle \mathbf{a, b} \rangle$. \begin{example} Let $\mathbf{a} = (a_1, a_2)$ and $\mathbf{b} = (b_1, b_2)$ be vectors in $\mathbb{F}_p^2$. Consider vector polynomials with vector coefficients: @@ -226,16 +226,16 @@ \subsubsection{Zero-knowledge inner-product protocol} \begin{align*} \mathcal{R}_{zkip} = \left\{ \begin{aligned} (\mathbf{G,H},G,B,A,V;\mathbf{a,b},\alpha, \gamma) \vert & A = \langle \mathbf{a,G} \rangle + \langle \mathbf{b,H} \rangle + [\alpha] B, \\ & V = [\langle \mathbf{a,b} \rangle]G + [\gamma]B \end{aligned} \right\} \end{align*} - where $\mathbf{G,H} \in \mathbb{G}^n, G,B \in \mathbb{G}$ -- independent group generators with prover $\mathcal{P}$ and verifier $\mathcal{V}$ is defined as follows: + where $\mathbf{G,H} \in \mathbb{G}^n$ and $G,B \in \mathbb{G}$ are independent group generators, with prover $\mathcal{P}$ and verifier $\mathcal{V}$, is defined as follows: \begin{itemize} - \item Prover $\mathcal{P}$ choses blinding vectors $\mathbf{s}_L, \mathbf{s}_R \in \mathbb{F}_p^n$ and computes polynomials: + \item Prover $\mathcal{P}$ chooses blinding vectors $\mathbf{s}_L, \mathbf{s}_R \in \mathbb{F}_p^n$ and computes polynomials: \begin{align*} \mathbf{l}(x) &= \mathbf{a} + \mathbf{s}_L x \\ \mathbf{r}(x) &= \mathbf{b} + \mathbf{s}_R x \\ t(x) &= \langle \mathbf{l}(x), \mathbf{r}(x) \rangle = \langle \mathbf{a}, \mathbf{b} \rangle + (\langle \mathbf{a}, \mathbf{s}_R \rangle + \langle \mathbf{s}_L, \mathbf{b} \rangle) x + \langle \mathbf{s}_L, \mathbf{s}_R \rangle x^2 \end{align*} - \item Prover $\mathcal{P}$ draws blinding factors $\beta, \tau_1, \tau_2 \leftarrowS \mathbb{F}_p$ and sends to $\mathcal{V}$ the following commitments for coefficients of $\mathbf{l}(x), \mathbf{r}(x), \mathbf{t}(x)$: + \item Prover $\mathcal{P}$ draws blinding factors $\beta, \tau_1, \tau_2 \leftarrowS \mathbb{F}_p$ and sends to $\mathcal{V}$ the following commitments to the coefficients of $\mathbf{l}(x), \mathbf{r}(x), \mathbf{t}(x)$: \begin{equation} \begin{aligned} S &= \langle \mathbf{s}_L, \mathbf{G} \rangle + \langle \mathbf{s}_R, \mathbf{H} \rangle + [\beta]B\\ @@ -264,23 +264,23 @@ \subsubsection{Zero-knowledge inner-product protocol} \end{itemize} \end{definition} -The protocol also has \textit{perfect completeness, special soundness, perfect honest-verifier zero-knowledge} due to the \Cref{th:poly_mul}, however building the extractor needs some vector equations we omit for the sake of brevity. Also note that transcript size is linear in size of vectors $\mathbf{l}_u, \mathbf{r}_u$ which is extremely inefficient when vectors are large. So in the next section we present so called \textbf{inner-product argument} which is summoned to reduce conversational complexity to logarithmic in vector length making the last check $t_u \stackrel{\text{?}}{=} \langle \mathbf{l}_u \mathbf{r}_u \rangle$ quite efficient. +The protocol also has \textit{perfect completeness, special soundness, perfect honest-verifier zero-knowledge} due to \Cref{th:poly_mul}; however, building the extractor requires some vector equations which we omit for the sake of brevity. Also note that the transcript size is linear in the size of the vectors $\mathbf{l}_u, \mathbf{r}_u$, which is extremely inefficient when the vectors are large. So in the next section we present the so-called \textbf{inner-product argument}, which is designed to reduce the communication complexity to logarithmic in vector length, making the last check $t_u \stackrel{\text{?}}{=} \langle \mathbf{l}_u \mathbf{r}_u \rangle$ quite efficient. \subsection{Inner-product argument} -Here we describe the further generalization of $\Pi_{mul}$ -- efficient protocol for the \textbf{inner-product argument} - core component of the \textbf{bulletproofs} protocol. After that we will apply it to range proofs and arithmetic circuits. We have already seen that inner-products are the main ingredients for R1CS language because any R1CS relation could be seen as a batch of inner-products though it's not the most efficient representation and we'll see how to amortize all the constraints into inner-products more efficiently. +Here we describe a further generalization of $\Pi_{mul}$ -- an efficient protocol for the \textbf{inner-product argument}, the core component of the \textbf{bulletproofs} protocol. After that, we will apply it to range proofs and arithmetic circuits. We have already seen that inner products are the main ingredient of the R1CS language, because any R1CS relation can be seen as a batch of inner products, though this is not the most efficient representation, and we will see how to amortize all the constraints into inner products more efficiently. -The \textbf{inner-product argument} allows to prove that two vectors $\mathbf{a,b} \in \mathbb{F}_p^n$ satisfy the relation: +The \textbf{inner-product argument} allows one to prove that two vectors $\mathbf{a,b} \in \mathbb{F}_p^n$ satisfy the relation: $$\mathcal{R}_{ip} = \{ (\mathbf{G,H}, P, c; \mathbf{a,b}) \vert P = \langle \mathbf{a,G} \rangle + \langle \mathbf{b,H} \rangle \wedge \langle \mathbf{a,b} \rangle = c \}$$ We refer to $P \in \mathbb{G}$ as a binding Pedersen vector commitment to $\mathbf{a,b}$. -One way to prove the relation is to use $\Pi_{zkip}$, but as we've seen it's not efficient due to linear in $n$ size of the proof. We want to build an argument system for the relation $\mathcal{R}_{ip}$ with logarithmic in $n$ size of the proof. +One way to prove the relation is to use $\Pi_{zkip}$, but as we have seen it is not efficient due to the proof size being linear in $n$. We want to build an argument system for the relation $\mathcal{R}_{ip}$ with proof size logarithmic in $n$. -Firtsly, let's combine statements $P = \langle \mathbf{a,G} \rangle + \langle \mathbf{b,H} \rangle \wedge \langle \mathbf{a,b} \rangle = c$ into a single statement by multiplying the second one by a random $r \in \mathbb{F}_p$ and some orthogonal generator $B \in \mathbb{G}$, summing up: +Firstly, let us combine the statements $P = \langle \mathbf{a,G} \rangle + \langle \mathbf{b,H} \rangle \wedge \langle \mathbf{a,b} \rangle = c$ into a single statement by multiplying the second one by a random $r \in \mathbb{F}_p$ and some orthogonal generator $B \in \mathbb{G}$, and summing up: $$ \mathcal{R}'_{ip} = \{ (\mathbf{G,H}, Q, P'; \mathbf{a,b}) \vert P' = \langle \mathbf{a,G} \rangle + \langle \mathbf{b,H} \rangle + [\langle \mathbf{a,b} \rangle]Q \} $$ -Where $P' = P + [cr]B, Q=[r]B$. Intuitively, if prover $\mathcal{P}$ can prove $\mathcal{R}'_{ip}$ for all $r \in \mathbb{F}_p$, then it can prove $\mathcal{R}_{ip}$ for any valid witness. We use such transformation to compress each vector in half and arrive to the same form of commitment +where $P' = P + [cr]B, Q=[r]B$. Intuitively, if prover $\mathcal{P}$ can prove $\mathcal{R}'_{ip}$ for all $r \in \mathbb{F}_p$, then it can prove $\mathcal{R}_{ip}$ for any valid witness. We use such a transformation to compress each vector in half and arrive at the same form of commitment $$P' = \langle \mathbf{a,G} \rangle + \langle \mathbf{b,H} \rangle + [\langle \mathbf{a,b} \rangle]Q$$ \begin{definition} @@ -295,10 +295,10 @@ \subsection{Inner-product argument} \subsubsection{Inner-product compression} -Here we describe \textbf{inner-product compression} algorithm -- main building block of the interactive \textbf{inner-product} protocol. -Firstly, assuming that $n = 2^d$ define by $\mathbf{G_{lo}} = (G_1, \dots, G_{n/2}), \mathbf{G_{hi}} = (G_{n/2+1},\dots, G_n) \in \mathbb{G}^{n/2}$ -- lower and higher halves of vector $\mathbf{G}$ and $\mathbf{a_{lo}} = (a_1, \dots, a_{n/2}), \mathbf{a_{hi}} = (a_{n/2+1},\dots,a_n) \in \mathbb{F}_p^{n/2}$ -- lower and higher halves of $\mathbf{a} \in \mathbb{F}_p^{n}$. +Here we describe the \textbf{inner-product compression} algorithm -- the main building block of the interactive \textbf{inner-product} protocol. +Firstly, assuming that $n = 2^d$, define $\mathbf{G_{lo}} = (G_1, \dots, G_{n/2}), \mathbf{G_{hi}} = (G_{n/2+1},\dots, G_n) \in \mathbb{G}^{n/2}$ to be the lower and higher halves of vector $\mathbf{G}$, and $\mathbf{a_{lo}} = (a_1, \dots, a_{n/2}), \mathbf{a_{hi}} = (a_{n/2+1},\dots,a_n) \in \mathbb{F}_p^{n/2}$ to be the lower and higher halves of $\mathbf{a} \in \mathbb{F}_p^{n}$. -Let $u_k \in \mathbb{F}_p$ - be some scalar, define compressed vectors: +Let $u_k \in \mathbb{F}_p$ be some scalar. Define the compressed vectors: \begin{align*} \mathbf{a}^{(k-1)} &= \mathbf{a_{lo}} \cdot u_k + u_k^{-1} \cdot \mathbf{a_{hi}} \\ \mathbf{b}^{(k-1)} &= \mathbf{b_{lo}} \cdot u_k^{-1} + u_k \cdot \mathbf{b_{hi}} \\ @@ -306,7 +306,7 @@ \subsubsection{Inner-product compression} \mathbf{H}^{(k-1)} &= \mathbf{H_{lo}} \cdot u_k + u_k^{-1} \cdot \mathbf{H_{hi}} \end{align*} -Define $P_k \gets P' = \langle \mathbf{a,G} \rangle + \langle \mathbf{b,H} \rangle + [\langle \mathbf{a,b} \rangle]Q$ -- current commitment to vectors $\mathbf{a,b}$ and define $P_{k-1}$ using compressed vectors to have the same form as $P_k$, but in new basis $(\mathbf{G}^{(k-1)}, \mathbf{H}^{(k-1)})$: +Define $P_k \gets P' = \langle \mathbf{a,G} \rangle + \langle \mathbf{b,H} \rangle + [\langle \mathbf{a,b} \rangle]Q$ -- the current commitment to the vectors $\mathbf{a,b}$ -- and define $P_{k-1}$ using the compressed vectors to have the same form as $P_k$, but in the new basis $(\mathbf{G}^{(k-1)}, \mathbf{H}^{(k-1)})$: \begin{equation} P_{k-1} = \langle \mathbf{a}^{(k-1)}, \mathbf{G}^{(k-1)} \rangle + \langle \mathbf{b}^{(k-1)}, \mathbf{H}^{(k-1)} \rangle + [\langle \mathbf{a}^{(k-1)}, \mathbf{b}^{(k-1)} \rangle]Q @@ -329,7 +329,7 @@ \subsubsection{Inner-product compression} & [\langle \mathbf{a_{lo}}, \mathbf{b_{lo}}\rangle + \langle \mathbf{a_{hi}}, \mathbf{b_{hi}}\rangle]Q &+ [u_k^2\langle \mathbf{a_{lo}}, \mathbf{b_{hi}}\rangle + u_k^{-2}\langle \mathbf{a_{hi}}, \mathbf{b_{lo}}\rangle]Q \end{align*} -Note that $\langle \mathbf{a_{lo}}, \mathbf{G_{lo}}\rangle + \langle \mathbf{a_{hi}}, \mathbf{G_{hi}}\rangle = \langle \mathbf{a,G}\rangle$ so that the first two columns of $P_{k-1}$ definition precisecly contains $P_{k} = P'$: +Note that $\langle \mathbf{a_{lo}}, \mathbf{G_{lo}}\rangle + \langle \mathbf{a_{hi}}, \mathbf{G_{hi}}\rangle = \langle \mathbf{a,G}\rangle$, so the first two columns of the $P_{k-1}$ definition precisely contain $P_{k} = P'$: $$P_{k} = \langle \mathbf{a_{lo}}, \mathbf{G_{lo}}\rangle + \langle \mathbf{a_{hi}}, \mathbf{G_{hi}}\rangle + \langle \mathbf{b_{lo}}, \mathbf{H_{lo}}\rangle + \langle \mathbf{b_{hi}}, \mathbf{H_{hi}}\rangle + [\langle \mathbf{a_{lo}}, \mathbf{b_{lo}}\rangle + \langle \mathbf{a_{hi}}, \mathbf{b_{hi}}\rangle]Q$$ Define cross-terms $L_k, R_k$ of $P_{k-1}$ such that: @@ -339,9 +339,9 @@ \subsubsection{Inner-product compression} R_{k} &= \langle \mathbf{a_{hi}}, \mathbf{G_{lo}}\rangle + \langle \mathbf{b_{lo}}, \mathbf{H_{hi}}\rangle + [\langle \mathbf{a_{hi}}, \mathbf{b_{lo}}\rangle]Q \end{align*} -The first equation $P_{k-1} = P_k + [u_k^2] L_k + [u_k^{-2}] R_k$ could be used as a check for asserting correctness of next commitment $P_{k-1}$ given cross-terms $L_k, R_k$, half-sized vectors $\mathbf{a}^{(k-1)}, \mathbf{b}^{(k-1)}$ from which $P_{k-1}$ was computed (\ref{eq:p_k_1_from_new_basis}) using updated basis $\mathbf{G}^{(k-1)}, \mathbf{H}^{(k-1)}$ and current commitment value $P_k$. +The first equation $P_{k-1} = P_k + [u_k^2] L_k + [u_k^{-2}] R_k$ can be used as a check for asserting the correctness of the next commitment $P_{k-1}$ given the cross-terms $L_k, R_k$, the half-sized vectors $\mathbf{a}^{(k-1)}, \mathbf{b}^{(k-1)}$ from which $P_{k-1}$ was computed (\ref{eq:p_k_1_from_new_basis}) using the updated basis $\mathbf{G}^{(k-1)}, \mathbf{H}^{(k-1)}$, and the current commitment value $P_k$. -But we wish not send $\mathbf{a}^{(k-1)}, \mathbf{b}^{(k-1)}$ directly as this's inefficient due to still linear sizes, instead we apply recursion to compress this vectors to just one element. Here we come up with some kind of statement compression algorithm reducing size of all vectors in half per compression step. Repeating compression algorithm $k$ times we end up with sending vectors $\mathbf{a}^{(0)}, \mathbf{b}^{(0)}$ each of length one and $P_0$ containing all accumulated cross-terms: +But we do not wish to send $\mathbf{a}^{(k-1)}, \mathbf{b}^{(k-1)}$ directly, as this is inefficient due to their still linear sizes; instead, we apply recursion to compress these vectors to just one element. Here we come up with a kind of statement compression algorithm, reducing the size of all vectors by half per compression step. Repeating the compression algorithm $k$ times, we end up with sending vectors $\mathbf{a}^{(0)}, \mathbf{b}^{(0)}$, each of length one, and $P_0$ containing all accumulated cross-terms: \begin{align*} P_0 &= [a_1^{(0)}]G_1^{(0)} + [b_1^{(0)}]H_1^{(0)} + [a_1^{(0)}b_1^{(0)}]Q \\ P_0 &= P_k + \sum_{i=1}^k ([u_i^2]L_i + [u_i^{-2}]R_i) @@ -361,11 +361,11 @@ \subsubsection{Proving $\mathcal{R}'_{ip}$} Let's describe the \textbf{inner-product} protocol $\Pi'_{ip}$ for relation $\mathcal{R}'_{ip}$. \begin{definition} - The \textbf{inner-product} protocol $\Pi'_{ip} = (\mathcal{P}, \mathcal{V})$ for relation $\mathcal{R}'_{ip} = \{ (\mathbf{G,H}, Q, P'; \mathbf{a,b}) \vert P' = \langle \mathbf{a,G} \rangle + \langle \mathbf{b,H} \rangle + [\langle \mathbf{a,b} \rangle]Q \}$, where all vectors have length $n=2^d$ with prover $\mathcal{P}$, verifier $\mathcal{V}$ is defined as follows: + The \textbf{inner-product} protocol $\Pi'_{ip} = (\mathcal{P}, \mathcal{V})$ for the relation $\mathcal{R}'_{ip} = \{ (\mathbf{G,H}, Q, P'; \mathbf{a,b}) \vert P' = \langle \mathbf{a,G} \rangle + \langle \mathbf{b,H} \rangle + [\langle \mathbf{a,b} \rangle]Q \}$, where all vectors have length $n=2^d$, with prover $\mathcal{P}$ and verifier $\mathcal{V}$, is defined as follows: \begin{itemize} \item Prover $\mathcal{P}$ sets $$(k, \mathbf{a}^{(k)}, \mathbf{b}^{(k)}, \mathbf{G}^{(k)}, \mathbf{H}^{(k)}, P_k) \gets (d, \mathbf{a,b,G,H},P')$$ \item Verifier $\mathcal{V}$ sets $$(k, \mathbf{G}^{(k)}, \mathbf{H}^{(k)}, P_k) \gets (d, \mathbf{G,H},P')$$ - \item While $k > 0$ then: + \item While $k > 0$ do: \begin{itemize} \item Prover $\mathcal{P}$ computes \begin{align*} @@ -468,13 +468,13 @@ \subsubsection{Proving $\mathcal{R}'_{ip}$} \label{fig:interactive_ip} \end{figure} -As we can see, overall communication complexity of $\Pi_{ip}$ is $2\log_2 n$ group elements plus $2$ field elements so we come up with logarithmic proof size for our inner-product relation $\mathcal{R}_{ip}$. Now one might ask whether $\Pi_{ip}$ has any desired properties such as completeness, soundness and zero-knowledge. It appears that the first two holds under some generalizations needed for security proofs but not \textit{zero-knowledge} (indeed, if $n=1$ then $\mathcal{P}$ sends witness pair $a,b$ directly). We'll compile efficient \textbf{inner-product argument} $\Pi_{ip}$ with zero-knowledge $\Pi_{zkip}$ to achieve efficient zero-knowledge proofs for range proofs and arithmetic circuits. +As we can see, the overall communication complexity of $\Pi_{ip}$ is $2\log_2 n$ group elements plus $2$ field elements, so we come up with a logarithmic proof size for our inner-product relation $\mathcal{R}_{ip}$. Now one might ask whether $\Pi_{ip}$ has any desired properties such as completeness, soundness and zero-knowledge. It appears that the first two hold under some generalizations needed for security proofs, but not \textit{zero-knowledge} (indeed, if $n=1$ then $\mathcal{P}$ sends the witness pair $a,b$ directly). We will compile the efficient \textbf{inner-product argument} $\Pi_{ip}$ with the zero-knowledge $\Pi_{zkip}$ to achieve efficient zero-knowledge proofs for range proofs and arithmetic circuits. \begin{theorem}[Inner-Product Argument] The argument system $\Pi_{ip}$ for relation $\mathcal{R}_{ip}$ has \textit{perfect completeness and statistical witness-extended emulation} for either extracting a non-trivial discrete logarithm relation between $\mathbf{G,H}, Q$ or extracting valid witness $\mathbf{a,b}$. \end{theorem} -\textbf{Proof idea}. \textit{Perfect completeness} of $Pi_{ip}$ follows because $Pi_{ip}$ converts instance of $\mathcal{R}_{ip}$ to instance of $\mathcal{P}'_{ip}$ and $\Pi'_{ip}$ is trivially complete by construction due to \Cref{eq:ip-final-compressed}. Notation \textit{statistical witness-extended emulation} generalizes \textit{special soundness} in the way applicable for multi-stage complex argument systems where each step of the protocol could be rewound to extract part of the witness so that more accurate definition of protocol security is achieved despite \textit{special soundness} implies building the whole knowledge extractor which might has non-polynomial running time for multi-stage protocols. +\textbf{Proof idea}. \textit{Perfect completeness} of $\Pi_{ip}$ follows because $\Pi_{ip}$ converts an instance of $\mathcal{R}_{ip}$ into an instance of $\mathcal{R}'_{ip}$, and $\Pi'_{ip}$ is trivially complete by construction due to \Cref{eq:ip-final-compressed}. The notion of \textit{statistical witness-extended emulation} generalizes \textit{special soundness} in a way applicable to multi-stage complex argument systems, where each step of the protocol can be rewound to extract part of the witness, so that a more accurate definition of protocol security is achieved, whereas \textit{special soundness} implies building the whole knowledge extractor, which might have non-polynomial running time for multi-stage protocols. Here we briefly describe a knowledge extractor $\mathcal{E}'_{ip}$ for a witness $(\mathbf{a}, \mathbf{b})$ or non-trivial discrete logarithm relation for $(\mathbf{G, H}, Q)$ for $\Pi'_{ip}$. \begin{enumerate} @@ -488,7 +488,7 @@ \subsubsection{Proving $\mathcal{R}'_{ip}$} \label{eq:extractor_eq} \end{equation} - \item $\mathcal{E}'_{ip}$ choses $(v_1, v_2, v_3) \in \mathbb{F}_p^3 $ as a solution for the system of linear equations: + \item $\mathcal{E}'_{ip}$ chooses $(v_1, v_2, v_3) \in \mathbb{F}_p^3 $ as a solution to the system of linear equations: \begin{equation} \begin{cases} x_1^2 v_1 + x_2^2 v_2 + x_3^2 v_3 = 0 \\ @@ -517,21 +517,21 @@ \subsubsection{Proving $\mathcal{R}'_{ip}$} P_{1} = \langle \mathbf{a}^{(1)}, \mathbf{G}^{(1)} \rangle + \langle \mathbf{b}^{(1)}, \mathbf{H}^{(1)} \rangle + [\langle \mathbf{a}^{(1)}, \mathbf{b}^{(1)} \rangle]Q \label{eq:extractor_p1_alt} \end{equation} - \item Asserting equality (\ref{eq:extractor_p1})$=$(\ref{eq:extractor_p1_alt}) the extractor $\mathcal{E}'_{ip}$ sets: + \item Asserting equality (\ref{eq:extractor_p1})$=$(\ref{eq:extractor_p1_alt}), the extractor $\mathcal{E}'_{ip}$ sets: \begin{align} \mathbf{a}^{(1)} &= \sum_{i=1}^3 v_i x_i^{-1} \cdot \mathbf{a}_i^{(0)} + v_i x_i \cdot \mathbf{a}_i^{(0)} \\ \mathbf{b}^{(1)} &= \sum_{i=1}^3 v_i x_i \cdot \mathbf{b}_i^{(0)} + v_i x_i^{-1} \cdot \mathbf{b}_i^{(0)} \end{align} - We need the fourth rewinding to assert equality of inner product: + We need the fourth rewinding to assert equality of the inner product: $$\langle \mathbf{a}^{(1)}, \mathbf{b}^{(1)} \rangle = \sum_{i=1}^3 v_i \langle \mathbf{a}_i^{(0)}, \mathbf{b}_i^{(0)} \rangle$$ - We won't describe it fully since it takes some unwieldy technical details and refer a reader to the original \textit{bulletproofs} paper \cite{bulletproofs}, where the full proof of extraction is described in Theorem 1. + We will not describe it fully, since it involves some unwieldy technical details, and we refer the reader to the original \textit{bulletproofs} paper \cite{bulletproofs}, where the full proof of extraction is described in Theorem 1. \item The extractor $\mathcal{E}'_{ip}$ recursively extracts $\mathbf{a}^{(k+1)}, \mathbf{b}^{(k+1)}$ from $ \mathbf{a}^{(k)}, \mathbf{b}^{(k)}$ using the method described in steps 2-5 until it reaches the final witness $\mathbf{a}^{(d)}, \mathbf{b}^{(d)} = \mathbf{a}, \mathbf{b}$ for which the relation $\mathcal{R}'_{ip}$ holds: $$P' = \langle \mathbf{a,G} \rangle + \langle \mathbf{b,H} \rangle + [\langle \mathbf{a,b} \rangle]Q$$ \end{enumerate} -Building the extractor $\mathcal{E}_{ip}$ for $\Pi_{ip}$ is quite simple relatively to what we've done by now: +Building the extractor $\mathcal{E}_{ip}$ for $\Pi_{ip}$ is quite simple relative to what we have done by now: \begin{enumerate} - \item $\mathcal{E}_{ip}$ runs $\Pi_{ip}$ to the end and applies the extractor $\mathcal{E}'_{ip}$ for $Pi'_{ip}$ to extract the witness $\mathbf{a,b}$ such that the following holds for $\mathcal{V}$'s challenge $r_1 \in \mathbb{F}_p$: + \item $\mathcal{E}_{ip}$ runs $\Pi_{ip}$ to the end and applies the extractor $\mathcal{E}'_{ip}$ for $\Pi'_{ip}$ to extract the witness $\mathbf{a,b}$ such that the following holds for $\mathcal{V}$'s challenge $r_1 \in \mathbb{F}_p$: \begin{equation} P + [r_1c]B = \langle \mathbf{a,G} \rangle + \langle \mathbf{b,H} \rangle + [r_1 \cdot \langle \mathbf{a,b} \rangle]B \label{eq:extractor_r1} @@ -542,7 +542,7 @@ \subsubsection{Proving $\mathcal{R}'_{ip}$} \label{eq:extractor_r2} \end{equation} - \item $\mathcal{E}_{ip}$ substitute (\ref{eq:extractor_r1}) from (\ref{eq:extractor_r2}) to get: + \item $\mathcal{E}_{ip}$ subtracts (\ref{eq:extractor_r1}) from (\ref{eq:extractor_r2}) to get: \begin{equation} [c(r_1 - r_2)]B = \langle \mathbf{a-a',G} \rangle + \langle \mathbf{b-b',H} \rangle + [r_1 \cdot \langle \mathbf{a,b} \rangle - r_2 \cdot \langle \mathbf{a',b'} \rangle]B \end{equation} @@ -553,29 +553,29 @@ \subsubsection{Proving $\mathcal{R}'_{ip}$} Hence $c = \langle \mathbf{a,b} \rangle$. \end{enumerate} -To formally finalize a proof of \textit{witness-extended emulation} we also need to apply so-called \textit{the forking lemma}, we again refer a reader to the original \textit{bulletproofs} paper \cite{bulletproofs} $\quad \square$. +To formally finalize the proof of \textit{witness-extended emulation} we also need to apply the so-called \textit{forking lemma}; we again refer the reader to the original \textit{bulletproofs} paper \cite{bulletproofs} $\quad \square$. \subsubsection{Zero-knowledge extension of inner-product argument}\label{subsection:alternative-zk-ip} -The main approach to make \textit{inner-product argument} \textit{zero-knowledge} is to use $\Pi_{zkip}$ protocol which original bulletproofs \cite{bulletproofs} does for \textit{range proofs}(\Cref{subsection:bulletproofs-range-proofs}) and \textit{arithmetic circuits satisfiability}(\Cref{subsection:bulletproofs-arithmetic-circuits}). However, there exists an alternative elegant construction based on tweaking \textit{inner-product argument} itself described in \cite[Appendix E.2]{pvss}. +The main approach to making the \textit{inner-product argument} \textit{zero-knowledge} is to use the $\Pi_{zkip}$ protocol, which the original bulletproofs paper \cite{bulletproofs} does for \textit{range proofs} (\Cref{subsection:bulletproofs-range-proofs}) and \textit{arithmetic circuits satisfiability} (\Cref{subsection:bulletproofs-arithmetic-circuits}). However, there exists an alternative elegant construction based on tweaking the \textit{inner-product argument} itself, described in \cite[Appendix E.2]{pvss}. The key idea is to bring a blinding factor $\delta \in \mathbb{F}_p$ with a verifier-provided random element $D \in \mathbb{G}$ to the commitment $P_k$: $$ P_k = \langle \mathbf{a}^{(k)}, \mathbf{G}^{(k)} \rangle + \langle \mathbf{b}^{(k)}, \mathbf{H}^{(k)} \rangle + [\langle \mathbf{a}^{(k)}, \mathbf{b}^{(k)} \rangle]Q + [\delta^{(k)}]D $$ -To get $L_k, R_k$ prover draws $\delta^{(k)}_L, \delta^{(k)}_R \leftarrowS \mathbb{F}_p$ and sets: +To get $L_k, R_k$, the prover draws $\delta^{(k)}_L, \delta^{(k)}_R \leftarrowS \mathbb{F}_p$ and sets: \begin{align*} P_{k-1} &= P_k + [u_k^2] L_k + [u_k^{-2}] R_k \\ L_{k} &= \langle \mathbf{a_{lo}}^{(k)}, \mathbf{G_{hi}}^{(k)}\rangle + \langle \mathbf{b_{hi}}^{(k)}, \mathbf{H_{lo}}^{(k)}\rangle + [\langle \mathbf{a_{lo}}^{(k)}, \mathbf{b_{hi}}^{(k)}\rangle]Q + [\delta^{(k)}_L]D \\ R_{k} &= \langle \mathbf{a_{hi}}^{(k)}, \mathbf{G_{lo}}^{(k)}\rangle + \langle \mathbf{b_{lo}}^{(k)}, \mathbf{H_{hi}}^{(k)}\rangle + [\langle \mathbf{a_{hi}}^{(k)}, \mathbf{b_{lo}}^{(k)}\rangle]Q + [\delta^{(k)}_R]D \end{align*} -Then $\mathcal{P}$ updates next-round $\delta$ using verifier-provided challenge $u_k$: +Then $\mathcal{P}$ updates the next-round $\delta$ using the verifier-provided challenge $u_k$: $$\delta^{(k-1)} = \delta^{(k)} + u_k^2\delta^{(k)}_L + u_k^{-2}\delta^{(k)}_R$$ -On the last step prover must prove that he possesses $a = a^{(0)}, b = b^{(0)}, \delta = \delta^{(0)}$ such that for $G = G^{(0)}, H = H^{(0)}$ equality holds: +On the last step the prover must prove that he possesses $a = a^{(0)}, b = b^{(0)}, \delta = \delta^{(0)}$ such that for $G = G^{(0)}, H = H^{(0)}$ the equality holds: \begin{equation*} P_0 = P_k + \sum_{i=1}^k ([u_i^2]L_i + [u_i^{-2}]R_i) = [a]G + [b]H + [a\cdot b]Q + [\delta]D \end{equation*} -$P_0 = P_k + \sum_{i=1}^k ([u_i^2]L_i + [u_i^{-2}]R_i)$ is a Pedersen commitment to $a, b, a \cdot b$ with blinding factor $\delta$ and could easily be computed by verifier. One could prove knowledge of an opening $(a,b,\delta)$ to that commitment using protocol similar to $\Pi_{zkip}$. +$P_0 = P_k + \sum_{i=1}^k ([u_i^2]L_i + [u_i^{-2}]R_i)$ is a Pedersen commitment to $a, b, a \cdot b$ with blinding factor $\delta$, and can easily be computed by the verifier. One can prove knowledge of an opening $(a,b,\delta)$ to that commitment using a protocol similar to $\Pi_{zkip}$. \begin{enumerate} \item $\mathcal{P}$ draws blinders $u,v,r,s \leftarrowS \mathbb{F}_p$ and sends to $\mathcal{V}$ commitments: \begin{align*} @@ -593,10 +593,10 @@ \subsubsection{Zero-knowledge extension of inner-product argument}\label{subsect $$[a']G + [b']H + [a'b']Q + [\delta']D \stackrel{\text{?}}{=} P_0 + [c]A + [c^2]T$$ \end{enumerate} -Protocol is \textit{knowledge-sound} as value $\delta$ could easily be extracted from three accepting transcripts. To argue \textit{zero-knowledge} we stress that an adversary couldn't learn anything from transcript and each prover's message $(L_k, R_k)$ could be easily simulated by random element, at the base of recursion simulator simulates zero-knowledge proof of opening $(a,b,\delta)$ to commitment $P_0$. +The protocol is \textit{knowledge-sound}, as the value $\delta$ can easily be extracted from three accepting transcripts. To argue \textit{zero-knowledge}, we stress that an adversary cannot learn anything from the transcript and each prover's message $(L_k, R_k)$ can easily be simulated by a random element; at the base of the recursion, the simulator simulates a zero-knowledge proof of opening $(a,b,\delta)$ to the commitment $P_0$. \subsection{Inner-product based polynomial commitment scheme} -Here we describe one of the main theoretical applications of the \textit{inner-product argument} -- \textbf{polynomial commitment scheme} that relies only on \textit{discrete logarithm} assumption, while studied before \textit{KZG} commitment scheme needs bilinear pairings. +Here we describe one of the main theoretical applications of the \textit{inner-product argument} -- a \textbf{polynomial commitment scheme} that relies only on the \textit{discrete logarithm} assumption, whereas the previously studied \textit{KZG} commitment scheme needs bilinear pairings. \begin{definition} The inner-product non-hiding polynomial commitment scheme $\mathcal{C}_{ip} = (\mathsf{Setup, Commit, Open, VerifyOpen})$ is defined as follows. Let $f(x) = \sum_{i=0}^{n-1} a_i x^i \in \mathbb{F}_p[x]$ be a polynomial of degree $n-1$. @@ -611,26 +611,26 @@ \subsection{Inner-product based polynomial commitment scheme} \end{definition} \begin{remark} - As the second vector $\mathbf{b} = \mathbf{u^n}$ is known to the verifier, the prover don't have to commit to it using vector $\mathbf{H}$, so the parties might adjust all the steps eliminating vector $\mathbf{H}$ and $\mathbf{b}$ vector compression as well. The full scheme is described \href{https://www.zkdocs.com/docs/zkdocs/commitments/ipa-pcs/}{here}. Note that described scheme is not zero-knowledge as classic \textit{inner-product} argument is not zero-knowledge. But we could make it zero-knowledge using $\Pi_{zkip}$ protocol or using alternative construction from \Cref{subsection:alternative-zk-ip}. + As the second vector $\mathbf{b} = \mathbf{u^n}$ is known to the verifier, the prover does not have to commit to it using the vector $\mathbf{H}$, so the parties can adjust all the steps, eliminating the vector $\mathbf{H}$ and the $\mathbf{b}$ vector compression as well. The full scheme is described \href{https://www.zkdocs.com/docs/zkdocs/commitments/ipa-pcs/}{here}. Note that the described scheme is not zero-knowledge, as the classic \textit{inner-product} argument is not zero-knowledge. But we can make it zero-knowledge using the $\Pi_{zkip}$ protocol or using the alternative construction from \Cref{subsection:alternative-zk-ip}. \end{remark} \subsection{Range proofs}\label{subsection:bulletproofs-range-proofs} -Let $G,B\in \mathbb{G}$ -- independent group generators. Let's consider the relation: -$$\mathcal{R}_{rp} = \{ (G, B, V, n; v, \gamma) \vert V = [v]G + [\gamma]B, v \in [0, 2^n) \}$$ -This relation is often called the \textbf{range proof} relation. It asserts that committed value $v$ lays in the interval $[0, 2^n)$. Range proofs have very significant applications in various privacy \textit{blockchain} protocols since them usually imply proving that transaction inputs or outputs are valid, e.g. have positive value or satisfy other relations between them. +Let $G,B\in \mathbb{G}$ be independent group generators. Let's consider the relation: +$$\mathcal{R}_{rp} = \{ (G, B, V, n; v, \gamma) \vert V = [v]G + [\gamma]B, v \in [0, 2^n) \}$$ +This relation is often called the \textbf{range proof} relation. It asserts that the committed value $v$ lies in the interval $[0, 2^n)$. Range proofs have very significant applications in various privacy \textit{blockchain} protocols, since they usually imply proving that transaction inputs or outputs are valid, e.g.\ have positive value or satisfy other relations between them. -For the first view it seems very inconspicuous why \textbf{inner-product argument} is useful for proving the range proof relation, but we'll show it ab initio. +At first glance it seems quite inconspicuous why the \textbf{inner-product argument} is useful for proving the range proof relation, but we will show it ab initio. -Firstly, write $v$ in base-2 representation: $v = \sum_{i=0}^{\lfloor \log_2 v \rfloor} 2^i v_i$ and $\mathbf{a}_L = (v_0, v_1, \dots, v_{n-1})$ be the vector of bits padded with zeroes to length $n$, so the range validation that $v$ lays in $[0, 2^n)$ implies two checks: +Firstly, write $v$ in base-2 representation: $v = \sum_{i=0}^{\lfloor \log_2 v \rfloor} 2^i v_i$, and let $\mathbf{a}_L = (v_0, v_1, \dots, v_{n-1})$ be the vector of bits padded with zeroes to length $n$, so that the range validation that $v$ lies in $[0, 2^n)$ implies two checks: \begin{itemize} \item Each bit $v_i$ must be either $0$ or $1$ \item The following inner-product equality holds: $\langle \mathbf{a}_L, \mathbf{2}^n \rangle = v$ \end{itemize} -We already know how to prove the second one inner product equality -- simply by taking evaluation point $u \gets 2$ in \textit{inner-product based polynomial commitment scheme}. +We already know how to prove the second inner-product equality -- simply by taking the evaluation point $u \gets 2$ in the \textit{inner-product based polynomial commitment scheme}. -The first relation is a bit more tricky to check algebraically, but still we'll manage to do that, note that binary check for $v_i$ takes form $v_i(v_i - 1) = 0$, or in vector form: +The first relation is a bit trickier to check algebraically, but we will still manage to do so. Note that the binary check for $v_i$ takes the form $v_i(v_i - 1) = 0$, or in vector form: \begin{align*} \mathbf{a}_R = \mathbf{a}_L - \mathbf{1}^n \Leftrightarrow \mathbf{a}_L - \mathbf{a}_R - \mathbf{1}^n = \mathbf{0}^n \\ \mathbf{a}_L \circ \mathbf{a}_R = \mathbf{0}^n @@ -641,26 +641,26 @@ \subsection{Range proofs}\label{subsection:bulletproofs-range-proofs} $\mathbf{a}_L \circ \mathbf{a}_R = (0, 0, 0, 0)$ \end{example} -This two checks imply verification that some vector is zero vector, for that we use some challenge $y \in \mathbb{F}_p$ and check inner-product equalities $$\langle \mathbf{a}_L \circ \mathbf{a}_R, \mathbf{y}^n \rangle = 0 \text{ and } \langle \mathbf{a}_L - \mathbf{a}_R - \mathbf{1}^n, \mathbf{y}^n \rangle = 0$$ -This checks are sound because the prover doesn't know challenge $y$ in advance. +These two checks imply verification that some vector is the zero vector; for that we use some challenge $y \in \mathbb{F}_p$ and check the inner-product equalities $$\langle \mathbf{a}_L \circ \mathbf{a}_R, \mathbf{y}^n \rangle = 0 \text{ and } \langle \mathbf{a}_L - \mathbf{a}_R - \mathbf{1}^n, \mathbf{y}^n \rangle = 0$$ +These checks are sound because the prover does not know the challenge $y$ in advance. -Note that $\langle \mathbf{a}_L \circ \mathbf{a}_R, \mathbf{y}^n \rangle = \langle \mathbf{a}_L, \mathbf{a}_R \circ \mathbf{y}^n \rangle$ so the prover could commit to $\mathbf{a_L, a_R}$ and verifier will adjust commitment for $\mathbf{a}_R$ using modified generators $\mathbf{H} \circ \mathbf{y}^{-n}$. +Note that $\langle \mathbf{a}_L \circ \mathbf{a}_R, \mathbf{y}^n \rangle = \langle \mathbf{a}_L, \mathbf{a}_R \circ \mathbf{y}^n \rangle$, so the prover can commit to $\mathbf{a_L, a_R}$ and the verifier will adjust the commitment for $\mathbf{a}_R$ using modified generators $\mathbf{H} \circ \mathbf{y}^{-n}$. -Here we came up with three inner-product checks: +Here we have arrived at three inner-product checks: \begin{enumerate} \item $\langle \mathbf{a}_L, \mathbf{2}^n \rangle = v$ \item $\langle \mathbf{a}_L, \mathbf{a}_R \circ \mathbf{y}^n \rangle = 0$ \item $\langle \mathbf{a}_L - \mathbf{a}_R - \mathbf{1}^n, \mathbf{y}^n \rangle = 0$ \end{enumerate} -Here we could soundly combine all three checks into one using random linear combination with some verifier-provided challenge $z \in \mathbb{F}_p$: +Here we can soundly combine all three checks into one using a random linear combination with some verifier-provided challenge $z \in \mathbb{F}_p$: $$z^2 \cdot \langle \mathbf{a}_L, \mathbf{2}^n \rangle + z \cdot \langle \mathbf{a}_L - \mathbf{a}_R - \mathbf{1}^n, \mathbf{y}^n \rangle + \langle \mathbf{a}_L, \mathbf{a}_R \circ \mathbf{y}^n \rangle = z^2v$$ \begin{remark} - Naїve check $\langle \mathbf{a}_L, \mathbf{2}^n \rangle + \langle \mathbf{a}_L, \mathbf{a}_R \circ \mathbf{y}^n \rangle + \langle \mathbf{a}_L - \mathbf{a}_R - \mathbf{1}^n, \mathbf{y}^n \rangle = v$ is not sound as Prover could adjust vectors to be non-zero but still satisfy the check. + The naїve check $\langle \mathbf{a}_L, \mathbf{2}^n \rangle + \langle \mathbf{a}_L, \mathbf{a}_R \circ \mathbf{y}^n \rangle + \langle \mathbf{a}_L - \mathbf{a}_R - \mathbf{1}^n, \mathbf{y}^n \rangle = v$ is not sound, as the prover could adjust the vectors to be non-zero but still satisfy the check. \end{remark} -Now simplify this expression having only one inner-product check: +Now simplify this expression to have only one inner-product check: \begin{equation*} \begin{aligned} & z^2 \cdot \langle \mathbf{a}_L, \mathbf{2}^n \rangle + z \cdot \langle \mathbf{a}_L - \mathbf{a}_R - \mathbf{1}^n, \mathbf{y}^n \rangle + \langle \mathbf{a}_L, \mathbf{a}_R \circ \mathbf{y}^n \rangle = z^2v \\ @@ -683,7 +683,7 @@ \subsection{Range proofs}\label{subsection:bulletproofs-range-proofs} \end{aligned} \end{equation*} -Where $\delta(y,z)$ could easily be computed by verifier: +where $\delta(y,z)$ can easily be computed by the verifier: $$\delta(y,z) = \langle z \cdot \mathbf{1}^n, \mathbf{y}^n \rangle + \langle z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n, - z \cdot \mathbf{1}^n \rangle = (z-z^2)\langle \mathbf{1}^n, \mathbf{y}^n \rangle - z^3 \langle \mathbf{1}^n, \mathbf{2}^n \rangle $$ Now we have only one inner-product check left: @@ -692,7 +692,7 @@ \subsection{Range proofs}\label{subsection:bulletproofs-range-proofs} \label{eq:inner_product_check} \end{equation} -We will use a technique presented in $\Pi_{zkip}$ to provide zero-knowledge and \textbf{inner-product argument} to achieve logarithmic size-proof. One key problem is that the verifier must adjust commitments to compensate auxiliary terms. +We will use the technique presented in $\Pi_{zkip}$ to provide zero-knowledge, and the \textbf{inner-product argument} to achieve a logarithmic-size proof. One key problem is that the verifier must adjust the commitments to compensate for auxiliary terms. Firstly, construct the blinding polynomials for $\mathbf{a}_L$ and $\mathbf{a}_R$ with substitution: $$\mathbf{a}_L' \gets \mathbf{a}_L + \mathbf{s}_L x \quad \mathbf{a}_R' \gets \mathbf{a}_R + \mathbf{s}_R x$$ @@ -702,12 +702,12 @@ \subsection{Range proofs}\label{subsection:bulletproofs-range-proofs} \mathbf{r}(x) &= z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n + \mathbf{a}_R' \circ \mathbf{y}^n = z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n + (\mathbf{a}_R + \mathbf{s}_R x) \circ \mathbf{y}^n \\ & = z^2 \cdot \mathbf{2}^n + z \cdot \mathbf{y}^n + \mathbf{a}_R \circ \mathbf{y}^n + \mathbf{s}_R \circ \mathbf{y}^n x \end{align*} -So that $\langle \mathbf{l}_0, \mathbf{r}_0 \rangle = z^2v + \delta(y,z)$ -- inner product that we want to prove using a bit modified $\Pi_{zkip}$. +So that $\langle \mathbf{l}_0, \mathbf{r}_0 \rangle = z^2v + \delta(y,z)$ -- the inner product that we want to prove using a slightly modified $\Pi_{zkip}$. \begin{definition} The \textbf{range proof protocol} $\Pi_{rp} = (\mathsf{Setup}, \mathcal{P,V})$ for the relation $\mathcal{R}_{rp} = \{ (G, B, V, n; v, \gamma) \vert V = [v]G + [\gamma]B, v \in [0, 2^n) \}$ with prover $\mathcal{P}$ and verifier $\mathcal{V}$ is defined as follows: \begin{itemize} \item $\mathsf{Setup}$ returns vectors of group generators with unknown discrete log relations $\mathbf{G, H} \in \mathbb{G}^n$ - \item Prover does bit decomposition of $v$ to obtain vectors $\mathbf{a}_L \gets \mathbf{v}, \mathbf{b_L} \gets \mathbf{a}_L - \mathbf{1}^n$ and choses blinding terms $\mathbf{s}_L, \mathbf{s}_R \in \mathbb{F}_p^n, \alpha, \beta \in \mathbb{F}_p$ computing and sending commitments: + \item Prover does the bit decomposition of $v$ to obtain vectors $\mathbf{a}_L \gets \mathbf{v}, \mathbf{b_L} \gets \mathbf{a}_L - \mathbf{1}^n$, and chooses blinding terms $\mathbf{s}_L, \mathbf{s}_R \in \mathbb{F}_p^n, \alpha, \beta \in \mathbb{F}_p$, then computes and sends the commitments: \begin{align*} A = \langle \mathbf{a}_L, \mathbf{G} \rangle + \langle \mathbf{b_L}, \mathbf{H} \rangle + [\alpha]B\\ S = \langle \mathbf{s}_L, \mathbf{G} \rangle + \langle \mathbf{s}_R, \mathbf{H} \rangle + [\beta]B @@ -736,7 +736,7 @@ \subsection{Range proofs}\label{subsection:bulletproofs-range-proofs} T_2 &= [t_2]G + [\tau_2]B \end{aligned} \end{equation} - \textbf{Note:} prover does not have to send commitment to $t_0$ as it's the inner-product we want to prove and it could be computed from high-level commitment $V$. + \textbf{Note:} the prover does not have to send a commitment to $t_0$, as it is the inner product we want to prove and it can be computed from the high-level commitment $V$. \item Verifier $\mathcal{V}$ samples and sends to $\mathcal{P}$ random evaluation point $u \leftarrowS \mathbb{F}_p$ \item Prover $\mathcal{P}$ evaluates polynomials at $u$: \begin{equation} @@ -760,13 +760,13 @@ \subsection{Range proofs}\label{subsection:bulletproofs-range-proofs} \end{definition} \begin{remark} - The last two steps of $\Pi_{rp}$ could be substituted with an inner-product argument $\Pi_{ip}$ to provide logarithmic size-proof with the following steps: + The last two steps of $\Pi_{rp}$ could be substituted with an inner-product argument $\Pi_{ip}$ to provide a logarithmic-size proof with the following steps: \begin{itemize} - \item After $\mathcal{P}$ evaluates polynomials at $u$ he sends $(t_u, \alpha_u, \tau_u$ to $\mathcal{V})$ and computes commitment: + \item After $\mathcal{P}$ evaluates polynomials at $u$ he sends $(t_u, \alpha_u, \tau_u)$ to $\mathcal{V}$ and computes the commitment: $$P = \langle \mathbf{l}_u, \mathbf{G} \rangle + \langle \mathbf{r}_u, \mathbf{y}^{-n} \circ \mathbf{H} \rangle$$ - \item $\mathcal{V}$ performs check $[t_u]G + [\tau_u]B \stackrel{\text{?}}{=} [z^2]V + [\delta(y,z)]G + [u]T_1 + [u^2]T_2$, halts if it fails and reconstructs commitment $P$ otherwise: + \item $\mathcal{V}$ performs the check $[t_u]G + [\tau_u]B \stackrel{\text{?}}{=} [z^2]V + [\delta(y,z)]G + [u]T_1 + [u^2]T_2$, halts if it fails, and otherwise reconstructs the commitment $P$: $$P = A + [u]S + \langle -z \cdot \mathbf{1}^n, \mathbf{G} \rangle + \langle z \cdot \mathbf{y}^n + z^2 \cdot \mathbf{2}^n, \mathbf{y}^{-n} \circ \mathbf{H} \rangle - [\alpha_u]B$$ - \item Parties run inner-product argument $\Pi_{ip}$ on $(\mathbf{G},\mathbf{y}^{-n} \circ \mathbf{H}, P, t_u; \mathbf{l}_u, \mathbf{r}_u)$ + \item The parties run the inner-product argument $\Pi_{ip}$ on $(\mathbf{G},\mathbf{y}^{-n} \circ \mathbf{H}, P, t_u; \mathbf{l}_u, \mathbf{r}_u)$ \end{itemize} \end{remark} @@ -797,7 +797,7 @@ \subsection{Range proofs}\label{subsection:bulletproofs-range-proofs} & \langle \mathbf{a}_R, \mathbf{H} \rangle + u\cdot \langle \mathbf{s}_R, \mathbf{H} \rangle + [\alpha + u\beta]B \end{aligned} \end{equation*} -We could see that \textit{LHS} is equal to \textit{RHS} so the first check pass. +We can see that the \textit{LHS} is equal to the \textit{RHS}, so the first check passes. Taking the $t_0$ correctness check: \begin{equation*} @@ -812,31 +812,31 @@ \subsection{Range proofs}\label{subsection:bulletproofs-range-proofs} \textit{Perfect honest-verifier zero-knowledge} follows from the zero-knowledge construction of $\Pi_{zkip}$ protocol as Verifier learns no information about $\mathbf{a}_L, \mathbf{a}_R$. -\textit{Computational extended witness emulation} implies building extractor that combines extractors for two subprotocols: $\mathcal{E}_{ip}$ extracts witness ($\mathbf{l}_u, \mathbf{r}_u$) from $\Pi_{ip}$ than the extractor $\mathcal{E}_{zkip}$ extracts high-level witness ($\mathbf{a}_L, \mathbf{a}_R$) from $\Pi_{zkip} \quad \square$ +\textit{Computational extended witness emulation} implies building an extractor that combines extractors for two subprotocols: $\mathcal{E}_{ip}$ extracts the witness ($\mathbf{l}_u, \mathbf{r}_u$) from $\Pi_{ip}$, then the extractor $\mathcal{E}_{zkip}$ extracts the high-level witness ($\mathbf{a}_L, \mathbf{a}_R$) from $\Pi_{zkip} \quad \square$ -The proof size of \textbf{range-proof} protocol is $2 \log_2n +4$ group $\mathbb{G}$ elements and $5$ field $\mathbb{F}_p$ elements. +The proof size of the \textbf{range-proof} protocol is $2 \log_2n +4$ group $\mathbb{G}$ elements and $5$ field $\mathbb{F}_p$ elements. \begin{remark} - Range proofs could be efficiently aggregated: e.g. one could prove the relation using slightly modified range proof protocol $\Pi_{rp}$ + Range proofs can be efficiently aggregated: e.g.\ one could prove the relation using a slightly modified range proof protocol $\Pi_{rp}$ $$\mathcal{R}_{rpm} = \{ (G, B, \vec{V}, n; \vec{v}, \vec{\gamma}) \vert \forall i \in 1..m: V_i = [v_i]G + [\gamma_i]B, v_i \in [0, 2^n) \}$$ - Where $\vec{v} = (v_1, v_2, \dots, v_m)$ and $\vec{\gamma} = (\gamma_1, \gamma_2, \dots, \gamma_m)$ -- respectively secrets and blinding factors. Detail explanation of aggregation protocol could be found in original bulletproofs paper \cite{bulletproofs} + where $\vec{v} = (v_1, v_2, \dots, v_m)$ and $\vec{\gamma} = (\gamma_1, \gamma_2, \dots, \gamma_m)$ are the secrets and blinding factors respectively. A detailed explanation of the aggregation protocol can be found in the original bulletproofs paper \cite{bulletproofs}. \end{remark} \begin{example} - One of the most famous \textit{NP-complete} problems is the \textbf{subset-sum problem}: given a set of numbers presented as vector $\mathbf{s}$ and number $v \in \mathbb{N}$, does a some subset sums up to $v$. It turns out that we could use our \textbf{range-proof} protocol for this problem. One could simply replace first inner-product check $\langle \mathbf{a}_L, \mathbf{2}^n \rangle = v$ with $\langle \mathbf{a}_L, \mathbf{s} \rangle = v$ where $\mathbf{a}_L$ is the secret vector of bits that encode positions of $\mathbf{s}$ that sum up to $v$. + One of the most famous \textit{NP-complete} problems is the \textbf{subset-sum problem}: given a set of numbers presented as a vector $\mathbf{s}$ and a number $v \in \mathbb{N}$, does some subset sum to $v$? It turns out that we can use our \textbf{range-proof} protocol for this problem. One can simply replace the first inner-product check $\langle \mathbf{a}_L, \mathbf{2}^n \rangle = v$ with $\langle \mathbf{a}_L, \mathbf{s} \rangle = v$, where $\mathbf{a}_L$ is the secret vector of bits that encodes the positions of $\mathbf{s}$ that sum to $v$. - For example take $\mathbf{s} = (6,8,2,3)$ and $v = 14$. Then setting $\mathbf{a}_L = (1,1,0,0)$ we could use $\Pi_{rp}$ to prove that there exists a subset of $\mathbf{s}$ that sums up to $v=14$ without disclosing that subset. + For example, take $\mathbf{s} = (6,8,2,3)$ and $v = 14$. Then, setting $\mathbf{a}_L = (1,1,0,0)$, we can use $\Pi_{rp}$ to prove that there exists a subset of $\mathbf{s}$ that sums to $v=14$ without disclosing that subset. - Therefore, \textbf{bulletproofs range proof} protocol is capable to prove a knowledge of witness to any $NP$-problem as they all could be reduced to the $\textbf{subset-sum problem}$ + Therefore, the \textbf{bulletproofs range proof} protocol is capable of proving knowledge of a witness to any $NP$ problem, as they can all be reduced to the $\textbf{subset-sum problem}$. \end{example} \subsection{Arithmetic circuits proofs}\label{subsection:bulletproofs-arithmetic-circuits} -\textbf{Bulletproofs} presents not only range proofs, but also efficient proofs for arithmetic circuits satisfiability. As we could see before, inner-product relation is quite powerful tool and could be used to prove a knowledge of witness to any $NP$-problem. But here we present a more convenient way to compile arithmetic circuits into inner-product relation. +\textbf{Bulletproofs} presents not only range proofs, but also efficient proofs for arithmetic circuit satisfiability. As we saw before, the inner-product relation is a quite powerful tool and can be used to prove knowledge of a witness to any $NP$ problem. But here we present a more convenient way to compile arithmetic circuits into an inner-product relation. \subsubsection{Arithmetization} -\textbf{Bulletproofs} arithmetization slightly differs from the classic R1CS we studied before at \Cref{section:r1cs}, however it could be transformed vice-versa easily. Also \textbf{bulletproofs} arithmetization is more convenient and human-friendly for encoding most of the arithmetic circuits than the R1CS. +\textbf{Bulletproofs} arithmetization slightly differs from the classic R1CS we studied before in \Cref{section:r1cs}; however, it can easily be transformed back and forth. Also, \textbf{bulletproofs} arithmetization is more convenient and human-friendly for encoding most arithmetic circuits than R1CS. -There are two types of variables in \textit{bulletproofs} constraint system: \textit{low-level} and \textit{high-level}. Typically \textit{high-level} variables are provided with Pedersen commitments $\mathbf{V} \in \mathbb{G}^m$ as the private witness inputs $\mathbf{v} \in \mathbb{F}_p^m$ to the circuit, while \textit{low-level} variables $\mathbf{a}_L, \mathbf{a}_R, \mathbf{a}_O \in \mathbb{F}_p^n$ are the intermediate witness values of computation. We will define circuit as a set of multiplication constraints operating with \textit{low-level} variables and set of linear constraints which links \textit{low-level} variables between each other and \textit{high-level} variables as well. +There are two types of variables in the \textit{bulletproofs} constraint system: \textit{low-level} and \textit{high-level}. Typically, \textit{high-level} variables are provided with Pedersen commitments $\mathbf{V} \in \mathbb{G}^m$ as the private witness inputs $\mathbf{v} \in \mathbb{F}_p^m$ to the circuit, while \textit{low-level} variables $\mathbf{a}_L, \mathbf{a}_R, \mathbf{a}_O \in \mathbb{F}_p^n$ are the intermediate witness values of the computation. We will define a circuit as a set of multiplication constraints operating on \textit{low-level} variables and a set of linear constraints that link \textit{low-level} variables between each other and to \textit{high-level} variables as well. Multiplication constraints are defined with one vector equation: $$ \mathbf{a}_L \circ \mathbf{a}_R = \mathbf{a}_O $$ @@ -844,7 +844,7 @@ \subsubsection{Arithmetization} Linear constraints are defined via: $$ \mathbf{W}_L \cdot \mathbf{a}_L + \mathbf{W}_R \cdot \mathbf{a}_R + \mathbf{W}_O \cdot \mathbf{a}_O = \mathbf{W}_V \cdot \mathbf{v} + \mathbf{c} $$ -Where $\mathbf{a_L, a_R, a_O}$ -- vectors of left and right inputs for multiplication gates and output values (all of them are low-level variables). $\mathbf{W_L, W_R, W_O} \in \mathbb{F}_p^{q \times n}, \mathbf{W}_V \in \mathbb{F}_p^{q \times m}$ -- public matrices of weights for linear constraints(obviously known to verifier). $\mathbf{c} \in \mathbb{F}_p^q$ -- public vector of constants. Typically they encode wiring of the circuit and other linear relations between variables. +Here $\mathbf{a_L, a_R, a_O}$ are the vectors of left and right inputs for the multiplication gates and the output values (all of them are low-level variables); $\mathbf{W_L, W_R, W_O} \in \mathbb{F}_p^{q \times n}, \mathbf{W}_V \in \mathbb{F}_p^{q \times m}$ are public matrices of weights for the linear constraints (obviously known to the verifier); and $\mathbf{c} \in \mathbb{F}_p^q$ is a public vector of constants. Typically they encode the wiring of the circuit and other linear relations between variables. \begin{example} Consider the following elliptic curve membership circuit. Here witness $(v_1, v_2)$ should satisfy elliptic curve equation: @@ -998,15 +998,15 @@ \subsubsection{Proving a circuit satisfiability} \end{aligned} \end{equation} Where $\mathbf{a}_L, \mathbf{a}_R, \mathbf{a}_O \in \mathbb{F}_p^n$, $\mathbf{v}, \mathbf{r} \in \mathbb{F}_p^m$, $\mathbf{W}_L, \mathbf{W}_R, \mathbf{W}_O \in \mathbb{F}_p^{q \times n}$, $\mathbf{W}_V \in \mathbb{F}_p^{q \times m}$, $\mathbf{c} \in \mathbb{F}_p^q$. -Informally this relation states that there exists a valid witness $\mathbf{v}$ that satisfies all constraints of the circuit. For the verifier witness is presented only as commitments vector $\mathbf{V}$. +Informally, this relation states that there exists a valid witness $\mathbf{v}$ that satisfies all constraints of the circuit. For the verifier, the witness is presented only as the commitment vector $\mathbf{V}$. -We could use similar to \textit{range-proofs} technique to compile constraints of the circuit into inner-product relation. For multiplicative constraints take random $y \in \mathbb{F}_p$ and apply zero check: +We can use a technique similar to the \textit{range-proofs} one to compile the constraints of the circuit into an inner-product relation. For the multiplicative constraints, take a random $y \in \mathbb{F}_p$ and apply a zero check: $$ \langle \mathbf{a}_L \circ \mathbf{a}_R - \mathbf{a}_O, \mathbf{y}^n \rangle = 0$$ -Do the same for linear constraints, but for different randomness $z \in \mathbb{F}_p$: +Do the same for the linear constraints, but with different randomness $z \in \mathbb{F}_p$: $$ \langle \mathbf{W}_L \cdot \mathbf{a}_L + \mathbf{W}_R \cdot \mathbf{a}_R + \mathbf{W}_O \cdot \mathbf{a}_O - \mathbf{W}_V \cdot \mathbf{v} - \mathbf{c}, \mathbf{z}^q \rangle = 0$$ -Combine this two checks to one using the same randomness $z$: +Combine these two checks into one using the same randomness $z$: \begin{equation*} \begin{aligned} \langle \mathbf{a}_L \circ \mathbf{a}_R - \mathbf{a}_O, \mathbf{y}^n \rangle + z \cdot \langle \mathbf{W}_L \cdot \mathbf{a}_L + \mathbf{W}_R \cdot \mathbf{a}_R + \mathbf{W}_O \cdot \mathbf{a}_O - \mathbf{W}_V \cdot \mathbf{v} - \mathbf{c}, \mathbf{z}^q \rangle = 0 \\ @@ -1014,9 +1014,9 @@ \subsubsection{Proving a circuit satisfiability} \end{aligned} \end{equation*} -This check is sound as typically a prover could not control values of $y,z$ before he commits to $\mathbf{a_L, a_R, a_O}$ and $\mathbf{v}$. +This check is sound, as typically a prover cannot control the values of $y,z$ before committing to $\mathbf{a_L, a_R, a_O}$ and $\mathbf{v}$. -Then split the second inner product and factor-out public terms to RHS: +Then split the second inner product and factor out the public terms to the RHS: \begin{equation*} \begin{aligned} \langle \mathbf{a}_L \circ \mathbf{a}_R - \mathbf{a}_O, \mathbf{y}^n \rangle + \langle z \cdot \mathbf{z}^q, \mathbf{W}_L \cdot \mathbf{a}_L \rangle + \langle z \cdot \mathbf{z}^q, \mathbf{W}_R \cdot \mathbf{a}_R \rangle + \langle z \cdot \mathbf{z}^q, \mathbf{W}_O \cdot \mathbf{a}_O \rangle \\ @@ -1026,7 +1026,7 @@ \subsubsection{Proving a circuit satisfiability} \end{aligned} \end{equation*} -Applying conjugation rule(if $A$ -- linear operator and $A^T$ -- its transpose(conjugate) then $\langle \mathbf{a}, A \mathbf{b} \rangle = \langle A^T \mathbf{a}, \mathbf{b} \rangle$) to the second inner product we get: +Applying the conjugation rule (if $A$ is a linear operator and $A^T$ is its transpose (conjugate), then $\langle \mathbf{a}, A \mathbf{b} \rangle = \langle A^T \mathbf{a}, \mathbf{b} \rangle$) to the second inner product, we get: \begin{equation*} \begin{aligned} \langle \mathbf{a}_L \circ \mathbf{a}_R - \mathbf{a}_O, \mathbf{y}^n \rangle + \langle \mathbf{W}_L^T \cdot (z \cdot \mathbf{z}^q), \mathbf{a}_L \rangle + \langle \mathbf{W}_R^T \cdot (z \cdot \mathbf{z}^q), \mathbf{a}_R \rangle + \\ @@ -1034,7 +1034,7 @@ \subsubsection{Proving a circuit satisfiability} \end{aligned} \end{equation*} -Denote $w_c = \langle z \cdot \mathbf{z}^q, \mathbf{c} \rangle$ and flattened linear constraints(still public and easily computed by verifier): +Denote $w_c = \langle z \cdot \mathbf{z}^q, \mathbf{c} \rangle$ and the flattened linear constraints (still public and easily computed by the verifier): \begin{equation*} \begin{aligned} \mathbf{w}_L = \mathbf{W}_L^T \cdot (z \cdot \mathbf{z}^q) \\ @@ -1051,7 +1051,7 @@ \subsubsection{Proving a circuit satisfiability} \end{aligned} \end{equation*} -Then rearrange and combine terms so that $\mathbf{a}_L$, $\mathbf{a}_O$ be on the left side and $\mathbf{a}_R$ on the right side: +Then rearrange and combine terms so that $\mathbf{a}_L$, $\mathbf{a}_O$ are on the left side and $\mathbf{a}_R$ is on the right side: \begin{equation*} \begin{aligned} \langle \mathbf{a}_L \circ \mathbf{a}_R, \mathbf{y}^n \rangle - \langle \mathbf{a}_O, \mathbf{y}^n \rangle + \langle \mathbf{w}_L, \mathbf{a}_L \rangle + \langle \mathbf{w}_R, \mathbf{a}_R \rangle + \langle \mathbf{w}_O, \mathbf{a}_O \rangle = \langle \mathbf{w}_V, \mathbf{v} \rangle + w_c \\ @@ -1079,7 +1079,7 @@ \subsubsection{Proving a circuit satisfiability} \langle \mathbf{a}_L + \mathbf{y}^{-n} \circ \mathbf{w}_R, \mathbf{y}^n \circ \mathbf{a}_R + \mathbf{w}_L \rangle + \langle \mathbf{a}_O, -\mathbf{y}^n + \mathbf{w}_O \rangle \end{aligned} $$ -Now it seems we are stuck as there is two separate inner products so one could not simply linearly blind witness parts, multiply corresponding polynomials and obtain desired inner product in constant term just as we did in the \textit{range-proof} case. But fortunately we could take polynomials of higher degree and obtain desired sum of inner-products as some coefficient of the product of polynomials: +Now it seems we are stuck, as there are two separate inner products, so one could not simply linearly blind witness parts, multiply the corresponding polynomials and obtain the desired inner product in the constant term just as we did in the \textit{range-proof} case. But fortunately we could take polynomials of higher degree and obtain the desired sum of inner-products as some coefficient of the product of polynomials: \begin{equation} \langle \mathbf{a}x + \mathbf{c}x^2, \mathbf{d} + \mathbf{b}x \rangle = s_1x + s_2x^2 + s_3x^3 = x \cdot \langle \mathbf{a}, \mathbf{d} \rangle + x^2 \cdot (\langle \mathbf{a}, \mathbf{b} \rangle + \langle \mathbf{c}, \mathbf{d} \rangle) + x^3 \cdot \langle \mathbf{c}, \mathbf{b} \rangle \label{eq:inner-product-circuit-1} @@ -1092,9 +1092,9 @@ \subsubsection{Proving a circuit satisfiability} \end{aligned} \label{eq:inner-product-circuit-assignment} \end{equation} -So than we could get desired sum of inner products as the second-degree coefficient $s_2$: +So then we could get the desired sum of inner products as the second-degree coefficient $s_2$: $$ w_c + \langle \mathbf{w}_V, \mathbf{v} \rangle + \delta(y, z) = s_2$$ -In order to obtain final polynomials $\mathbf{l}(x), \mathbf{r}(x)$ we must firstly blind $\mathbf{a}_L, \mathbf{a}_R$: +In order to obtain the final polynomials $\mathbf{l}(x), \mathbf{r}(x)$, we must first blind $\mathbf{a}_L, \mathbf{a}_R$: \begin{equation} \begin{aligned} \mathbf{a}_L \gets \mathbf{a}_L + \mathbf{s}_Lx^2 && \mathbf{a}_R \gets \mathbf{a}_R + \mathbf{s}_Rx^2 @@ -1102,12 +1102,12 @@ \subsubsection{Proving a circuit satisfiability} \label{eq:inner-product-circuit-blinding} \end{equation} \begin{remark} - We multiplied blinders $\mathbf{s}_L, \mathbf{s}_R$ with the second power of challenge $x^2$ because we want the blinding terms to do not interfere with other parts of inner-product. + We multiplied the blinders $\mathbf{s}_L, \mathbf{s}_R$ by the second power of the challenge $x^2$ because we want the blinding terms not to interfere with other parts of the inner-product. - $\mathbf{a}_O$ does not need separate blinding as it's located on the left side of the inner-product (\ref{eq:inner-product-circuit-1}) along with $\mathbf{a}_L$, which is already blinded by $\mathbf{s}_L$. + $\mathbf{a}_O$ does not need separate blinding, as it is located on the left side of the inner-product (\ref{eq:inner-product-circuit-1}) along with $\mathbf{a}_L$, which is already blinded by $\mathbf{s}_L$. \end{remark} -Now we could compute polynomials $\mathbf{l}(x), \mathbf{r}(x)$ from (\ref{eq:inner-product-circuit-1}) using assignments from (\ref{eq:inner-product-circuit-assignment}) and blinders from (\ref{eq:inner-product-circuit-blinding}): +Now we could compute the polynomials $\mathbf{l}(x), \mathbf{r}(x)$ from (\ref{eq:inner-product-circuit-1}) using the assignments from (\ref{eq:inner-product-circuit-assignment}) and the blinders from (\ref{eq:inner-product-circuit-blinding}): \begin{equation} \begin{aligned} {\mathbf{l}}(x) &= (\mathbf{a}_L + \mathbf{s}_L \cdot x^2) \cdot x + \mathbf{y}^{-n} \circ \mathbf{w}_R \cdot x + \mathbf{a}_O \cdot x^2 \\ @@ -1135,18 +1135,18 @@ \subsubsection{Proving a circuit satisfiability} \end{aligned} \label{eq:inner-product-circuit-polynomials-product} \end{equation} -Our proving strategy is the same as in the \textit{range-proof} case: +Our proving strategy is the same as in the \textit{range-proof} case: \begin{itemize} - \item Prove that $\mathbf{l}(x), \mathbf{r}(x)$ are correct using binding commitments to their coefficients. - \item Prove that $t_2$ is correct (as it's the inner-product we want to prove) using evaluation at challenge point $u$. - \item Apply inner-product argument to compress the proof of evaluation of $t(x)$ at challenge point $u$. + \item Prove that $\mathbf{l}(x), \mathbf{r}(x)$ are correct using binding commitments to their coefficients. + \item Prove that $t_2$ is correct (as it is the inner-product we want to prove) using evaluation at the challenge point $u$. + \item Apply the inner-product argument to compress the proof of evaluation of $t(x)$ at the challenge point $u$. \end{itemize} \begin{definition} The \textbf{arithmetic circuit satisfiability} protocol $\Pi_{sat} = (\mathsf{Setup}, \mathcal{P}, \mathcal{V})$ for the relation $\mathcal{R}_{sat}$ with prover $\mathcal{P}$ and verifier $\mathcal{V}$ is defined as follows: \begin{itemize} \item $\mathsf{Setup}$: returns vector of group generators with unknown discrete log relations $\mathbf{G,H} \in \mathbb{G}^n$. - \item Prover $\mathcal{P}$ choses blinding factors $\alpha, \beta, \gamma \in \mathbb{F}_p, \mathbf{s}_L, \mathbf{s}_R \in \mathbb{F}_p^n$ and sends the following commitments to $\mathcal{V}$: + \item Prover $\mathcal{P}$ chooses blinding factors $\alpha, \beta, \gamma \in \mathbb{F}_p, \mathbf{s}_L, \mathbf{s}_R \in \mathbb{F}_p^n$ and sends the following commitments to $\mathcal{V}$: \begin{equation*} \begin{aligned} A_I &= \langle \mathbf{a}_L, \mathbf{G} \rangle + \langle \mathbf{a}_R, \mathbf{H} \rangle + [\alpha]B\\ @@ -1155,7 +1155,7 @@ \subsubsection{Proving a circuit satisfiability} \end{aligned} \end{equation*} \item Verifier samples challenges $y,z \leftarrowS \mathbb{F}_p$ and sends them to $\mathcal{P}$. - \item Using provided challenges $y,z$ prover forms polynomials $\mathbf{l}(x), \mathbf{r}(x)$: + \item Using the provided challenges $y,z$, the prover forms polynomials $\mathbf{l}(x), \mathbf{r}(x)$: \begin{equation*} \begin{aligned} \mathbf{l}(x) &= \mathbf{s}_L \cdot x^3 + \mathbf{a}_O \cdot x^2 + (\mathbf{a}_L + \mathbf{y}^{-n} \circ \mathbf{w}_R) \cdot x \\ @@ -1163,7 +1163,7 @@ \subsubsection{Proving a circuit satisfiability} \end{aligned} \end{equation*} computes $$t(x) = \langle \mathbf{l}(x), \mathbf{r}(x) \rangle = t_1 x + t_2 x^2 + t_3 x^3 + t_4 x^4 + t_5 x^5 + t_6 x^6$$ - choses random blinding factors $\tau_1, \tau_3, \tau_4, \tau_5, \tau_6 \in \mathbb{F}_p$ and sends to $\mathcal{V}$ commitments to its coefficients: + chooses random blinding factors $\tau_1, \tau_3, \tau_4, \tau_5, \tau_6 \in \mathbb{F}_p$ and sends to $\mathcal{V}$ commitments to its coefficients: \begin{equation*} \begin{aligned} T_1 &= [t_1]G + [\tau_1]B\\ @@ -1173,13 +1173,13 @@ \subsubsection{Proving a circuit satisfiability} T_6 &= [t_6]G + [\tau_6]B \end{aligned} \end{equation*} - \textbf{Note:} Prover does not send separate commitment to $t_2$ as the verifier could derive it from $\mathbf{V}$ and the circuit public parameters: + \textbf{Note:} The prover does not send a separate commitment to $t_2$, as the verifier could derive it from $\mathbf{V}$ and the circuit public parameters: \begin{align*} t_2 &= w_c + \langle \mathbf{w}_V, \mathbf{v} \rangle + \delta(y, z) \\ T_2 &= \langle \mathbf{w}_V, \mathbf{V} \rangle + [\delta(y, z) + w_c]G \end{align*} - \item Verifier samples and sends to $\mathcal{P}$ random evaluation point $u \leftarrowS \mathbb{F}_p$. - \item Prover evaluates polynomials at $u$: + \item Verifier samples and sends to $\mathcal{P}$ a random evaluation point $u \leftarrowS \mathbb{F}_p$. + \item Prover evaluates the polynomials at $u$: \begin{equation*} \begin{aligned} \mathbf{l}_u &= \mathbf{l}(u) \\ @@ -1205,25 +1205,25 @@ \subsubsection{Proving a circuit satisfiability} \end{definition} \begin{remark} - As in the \textit{range proof} protocol case the last two steps of $\Pi_{sat}$ could be substituted with an inner-product argument $\Pi_{ip}$ to provide logarithmic size-proof with the following steps: + As in the \textit{range proof} protocol case, the last two steps of $\Pi_{sat}$ could be substituted with an inner-product argument $\Pi_{ip}$ to provide a logarithmic-size proof with the following steps: \begin{itemize} - \item After $\mathcal{P}$ evaluates polynomials at $u$ he sends $(t_u, \alpha_u, \tau_u$ to $\mathcal{V})$ and computes commitment: + \item After $\mathcal{P}$ evaluates the polynomials at $u$, it sends $(t_u, \alpha_u, \tau_u)$ to $\mathcal{V}$ and computes the commitment: $$P = \langle \mathbf{l}_u, \mathbf{G} \rangle + \langle \mathbf{r}_u, \mathbf{y}^{-n} \circ \mathbf{H} \rangle$$ - \item $\mathcal{V}$ performs check: + \item $\mathcal{V}$ performs the check: $$ [t_u]G + [\tau_u]B \stackrel{\text{?}}{=} \sum_{k=1, k\neq 2}^6 [u^k]T_k + u^2 \cdot (\langle \mathbf{w}_V, \mathbf{V} \rangle + [\delta(y,z) + w_c]G) $$ - halts if it fails and reconstructs commitment $P$ otherwise: + halts if it fails, and reconstructs the commitment $P$ otherwise: \begin{align*} P =& [u]A_I + [u^2]A_O + [u^3]S - \langle \mathbf{1}, \mathbf{H} \rangle +\\ &u \cdot (\langle \mathbf{y}^{-n} \circ \mathbf{w}_L, \mathbf{G} \rangle + \langle \mathbf{y}^{-n} \circ \mathbf{w}_R, \mathbf{H} \rangle) + \langle \mathbf{y}^{-n} \circ \mathbf{w}_O, \mathbf{H} \rangle - [\alpha_u]B \end{align*} - \item Parties run inner-product argument $\Pi_{ip}$ on $(\mathbf{G},\mathbf{y}^{-n} \circ \mathbf{H}, P, t_u; \mathbf{l}_u, \mathbf{r}_u)$ + \item The parties run the inner-product argument $\Pi_{ip}$ on $(\mathbf{G},\mathbf{y}^{-n} \circ \mathbf{H}, P, t_u; \mathbf{l}_u, \mathbf{r}_u)$ \end{itemize} \end{remark} \begin{theorem} - The \textbf{arithmetic circuit satisfiability protocol} $\Pi_{sat}$ has \textit{perfect completeness, computational extended witness emulation, perfect honest-verifier zero-knowledge} + The \textbf{arithmetic circuit satisfiability protocol} $\Pi_{sat}$ has \textit{perfect completeness, computational extended witness emulation, and perfect honest-verifier zero-knowledge}. \label{th:arithmetic_circuit_satisfiability} \end{theorem} \textbf{Proof idea}. \textit{Perfect completeness} @@ -1245,7 +1245,7 @@ \subsubsection{Proving a circuit satisfiability} &\textcolor{gray}{- \langle \mathbf{1}^n, \mathbf{H} \rangle + \langle \mathbf{w}_O, \mathbf{y}^{-n} \circ \mathbf{H} \rangle} + \textcolor{RubineRed}{[\alpha u]B + [\gamma u^2]B + [\beta u^3]B} \end{aligned} \end{equation*} - As we see $LHS=RHS$ so the check holds. + As we see, $LHS=RHS$, so the check holds. \item \textbf{$t_2$ correctness check}: \begin{equation*} \begin{aligned} From cae48efa5146e9173b568d07870e6b337bbbc3fb Mon Sep 17 00:00:00 2001 From: Yevhen Hrubiian Date: Fri, 10 Apr 2026 20:28:23 +0300 Subject: [PATCH 25/25] add citations, fix compilation issues --- .latexmkrc | 9 +++ bibliography.bib | 120 +++++++++++++++++++++++++++++++++++ lectures/1-8-commitments.tex | 2 +- lectures/2-1-intro-zk.tex | 4 +- lectures/2-10-1-sumcheck.tex | 4 +- lectures/2-10-2-gkr.tex | 4 +- lectures/2-12-lookup.tex | 4 +- lectures/2-13-ultragroth.tex | 8 +-- lectures/2-2-sigma.tex | 4 +- lectures/2-3-circuits.tex | 2 +- lectures/2-4-qap-pcp.tex | 4 +- lectures/2-5-groth.tex | 6 +- lectures/2-6-circom.tex | 2 +- lectures/2-7-plonk.tex | 6 +- 14 files changed, 153 insertions(+), 26 deletions(-) create mode 100644 .latexmkrc diff --git a/.latexmkrc b/.latexmkrc new file mode 100644 index 0000000..4dad4a6 --- /dev/null +++ b/.latexmkrc @@ -0,0 +1,9 @@ +# Use pdflatex (overrides system default of lualatex) +$pdf_mode = 1; +$pdflatex = 'pdflatex -shell-escape %O %S'; + +# Fix bibtex bootstrap failure when no .aux exists yet: +# 1.5 means "run bibtex only when .bib is found AND (.bbl already exists +# OR .aux already contains \bibdata). This prevents bibtex from running +# on the empty stub .aux that latexmk creates on the very first pass. +$bibtex_use = 1.5; diff --git a/bibliography.bib b/bibliography.bib index d6a5582..64b3b72 100644 --- a/bibliography.bib +++ b/bibliography.bib @@ -66,6 +66,126 @@ @article{Saniee_2007_LagrangeInterpolation year = {2007}, } +// part 2 -- ZK protocols + +@inproceedings{fiat1986, + author = {Amos Fiat and Adi Shamir}, + title = {How to Prove Yourself: Practical Solutions to Identification and Signature Problems}, + booktitle = {Advances in Cryptology -- CRYPTO 1986}, + series = {Lecture Notes in Computer Science}, + volume = {263}, + pages = {186--194}, + publisher = {Springer}, + year = {1986}, +} + +@inproceedings{schnorr1989, + author = {Claus-Peter Schnorr}, + title = {Efficient Identification and Signatures for Smart Cards}, + booktitle = {Advances in Cryptology -- CRYPTO 1989}, + series = {Lecture Notes in Computer Science}, + volume = {435}, + pages = {239--252}, + publisher = {Springer}, + year = {1989}, +} + +@inproceedings{pedersen1991, + author = {Torben Pryds Pedersen}, + title = {Non-Interactive and Information-Theoretic Secure Verifiable Secret Sharing}, + booktitle = {Advances in Cryptology -- CRYPTO 1991}, + series = {Lecture Notes in Computer Science}, + volume = {576}, + pages = {129--140}, + publisher = {Springer}, + year = {1991}, +} + +@misc{qap, + author = {Rosario Gennaro and Craig Gentry and Bryan Parno and Mariana Raykova}, + title = {Quadratic Span Programs and Succinct {NIZKs} without {PCPs}}, + howpublished = {Cryptology {ePrint} Archive, Paper 2012/215}, + year = {2012}, + url = {https://eprint.iacr.org/2012/215}, +} + +@inproceedings{pinocchio, + author = {Bryan Parno and Jon Howell and Craig Gentry and Mariana Raykova}, + title = {Pinocchio: Nearly Practical Verifiable Computation}, + booktitle = {IEEE Symposium on Security and Privacy}, + pages = {238--252}, + publisher = {IEEE}, + year = {2013}, +} + +@misc{groth16, + author = {Jens Groth}, + title = {On the Size of Pairing-Based Non-interactive Arguments}, + howpublished = {Cryptology {ePrint} Archive, Paper 2016/260}, + year = {2016}, + url = {https://eprint.iacr.org/2016/260}, +} + +@misc{plonk, + author = {Ariel Gabizon and Zachary J. Williamson and Oana Ciobotaru}, + title = {{PLONK}: Permutations over Lagrange-bases for Oecumenical Noninteractive arguments of Knowledge}, + howpublished = {Cryptology {ePrint} Archive, Paper 2019/953}, + year = {2019}, + url = {https://eprint.iacr.org/2019/953}, +} + +@article{sumcheck, + author = {Carsten Lund and Lance Fortnow and Howard Karloff and Noam Nisan}, + title = {Algebraic Methods for Interactive Proof Systems}, + journal = {Journal of the ACM}, + volume = {39}, + number = {4}, + pages = {859--868}, + year = {1992}, + publisher = {ACM}, +} + +@inproceedings{gkr, + author = {Shafi Goldwasser and Yael Tauman Kalai and Guy N. Rothblum}, + title = {Delegating Computation: Interactive Proofs for Muggles}, + booktitle = {Proceedings of the 40th Annual ACM Symposium on Theory of Computing (STOC)}, + pages = {113--122}, + publisher = {ACM}, + year = {2008}, +} + +@misc{plookup, + author = {Ariel Gabizon and Zachary J. Williamson}, + title = {plookup: A simplified polynomial protocol for lookup tables}, + howpublished = {Cryptology {ePrint} Archive, Paper 2020/315}, + year = {2020}, + url = {https://eprint.iacr.org/2020/315}, +} + +@misc{mirage, + author = {Ahmed Kosba and Cheng Feng and Zheng Yang and Xiaodong Lin and Man Ho Au and Qiuliang Xu}, + title = {{MIRAGE}: Succinct Arguments for Randomized Algorithms with Applications to Universal {zk-SNARKs}}, + howpublished = {Cryptology {ePrint} Archive, Paper 2020/278}, + year = {2020}, + url = {https://eprint.iacr.org/2020/278}, +} + +@misc{mirageplus, + author = {Alex Ozdemir and Evan Laufer and Dan Boneh}, + title = {Jolt-{zk}: Efficient {zk-SNARKs} via {MLE}-based Lookup Arguments}, + howpublished = {Cryptology {ePrint} Archive, Paper 2024/979}, + year = {2024}, + url = {https://eprint.iacr.org/2024/979}, +} + +@misc{petkus2019, + author = {Maksym Petkus}, + title = {Why and How {zk-SNARK} Works: Definitive Explanation}, + howpublished = {arXiv preprint arXiv:1906.07221}, + year = {2019}, + url = {https://arxiv.org/abs/1906.07221}, +} + @misc{bulletproofs, author = {Benedikt Bünz and Jonathan Bootle and Dan Boneh and Andrew Poelstra and Pieter Wuille and Greg Maxwell}, title = {Bulletproofs: Short Proofs for Confidential Transactions and More}, diff --git a/lectures/1-8-commitments.tex b/lectures/1-8-commitments.tex index 4e33e68..a729e74 100644 --- a/lectures/1-8-commitments.tex +++ b/lectures/1-8-commitments.tex @@ -110,7 +110,7 @@ \subsection{Hash-based commitments} \subsection{Pedersen commitments} -Pedersen commitments allow us to represent arbitrarily large vectors with a single elliptic curve point, while optionally hiding any information about the vector. Pedersen commitment uses a public group $\mathbb{G}$ of order $q$ and two random public generators $G$ and $U$: $U = [u]G$. Secret parameter $u$ should be unknown to anyone, otherwise the $\textit{Binding}$ property of the commitment scheme will be violated. +Pedersen commitments \cite{pedersen1991} allow us to represent arbitrarily large vectors with a single elliptic curve point, while optionally hiding any information about the vector. Pedersen commitment uses a public group $\mathbb{G}$ of order $q$ and two random public generators $G$ and $U$: $U = [u]G$. Secret parameter $u$ should be unknown to anyone, otherwise the $\textit{Binding}$ property of the commitment scheme will be violated. EC point $U$ is chosen randomly using ``Nothing-up-my-sleeve`` to assure no one knows the discrete logarithm of a selected point. \begin{remark} diff --git a/lectures/2-1-intro-zk.tex b/lectures/2-1-intro-zk.tex index afcf417..5728c57 100644 --- a/lectures/2-1-intro-zk.tex +++ b/lectures/2-1-intro-zk.tex @@ -589,9 +589,9 @@ \subsubsection{Fiat-Shamir Transformation} While different protocols use different ways to achieve this, one of the most popular methods (which, in particular, is used in STARKs) is the -\textbf{Fiat-Shamir heuristic}. The idea is the following: instead of verifier +\textbf{Fiat-Shamir heuristic} \cite{fiat1986}. The idea is the following: instead of verifier sending the challenges, we can replace them with the random oracle applied to -all the previous messages. +all the previous messages. Here how it goes. Suppose we have an interactive protocol $(\mathcal{P}, \mathcal{V})$ for the statement $\mathbbm{x}$. As previously defined, the diff --git a/lectures/2-10-1-sumcheck.tex b/lectures/2-10-1-sumcheck.tex index 82aeb25..a2d1dd7 100644 --- a/lectures/2-10-1-sumcheck.tex +++ b/lectures/2-10-1-sumcheck.tex @@ -149,8 +149,8 @@ \subsection{The Sum-Check Protocol} \subsubsection{Protocol Description} Suppose we are given the $v$-variate polynomial (possibly non-multilinear) $f: -\{0,1\}^v \to \mathbb{F}$ over a finite field $\mathbb{F}$. The main goal -of the Sum-Check protocol is to convince the verifier $\mathcal{V}$ that +\{0,1\}^v \to \mathbb{F}$ over a finite field $\mathbb{F}$. The main goal +of the Sum-Check protocol \cite{sumcheck} is to convince the verifier $\mathcal{V}$ that \begin{equation*} \sum_{b_1 \in \{0,1\}}\sum_{b_2 \in \{0,1\}} \dots \sum_{b_v \in \{0,1\}} f(b_1,\dots,b_v) = H \end{equation*} diff --git a/lectures/2-10-2-gkr.tex b/lectures/2-10-2-gkr.tex index 6e8455d..bc66109 100644 --- a/lectures/2-10-2-gkr.tex +++ b/lectures/2-10-2-gkr.tex @@ -10,9 +10,9 @@ \subsection{Motivation} should be able to verify the correctness of the claimed value $H$ in the logarithmic time. -Goldwasser, Kalai, and Rothblum (GKR) described a protocol which +Goldwasser, Kalai, and Rothblum (GKR) \cite{gkr} described a protocol which solves exactly this issue over the arithmetical circuits, which -we solved using QAP$\to$NILP reduction in Groth16. Here we take +we solved using QAP$\to$NILP reduction in Groth16. Here we take the Sum-Check approach. Suppose we are given the \textit{layered} arithmetical diff --git a/lectures/2-12-lookup.tex b/lectures/2-12-lookup.tex index 0ccbe06..3184325 100644 --- a/lectures/2-12-lookup.tex +++ b/lectures/2-12-lookup.tex @@ -68,7 +68,7 @@ \subsection{Motivation} \subsection{Plookup Protocol} One of the first lookup protocols that became practical in the zero-knowledge -world is the \textit{plookup protocol}. It is mostly used in Poly-IOPs but can +world is the \textit{plookup protocol} \cite{plookup}. It is mostly used in Poly-IOPs but can presumably be compiled to other types of protocols as well, as long as \textit{multiset equality} check can be implemented optimally. @@ -258,7 +258,7 @@ \subsubsection{plookup Precise Scheme} \end{proposition} Now, this fact is completely unobvious, and you can see the proof in the -\href{https://eprint.iacr.org/2020/315.pdf}{original plookup paper}. This +original plookup paper \cite{plookup}. This motivates us to formulate the following protocol. Without loss of generality, assume $d=n+1$ (if $d \leq n$, pad $\boldsymbol{t}$ with $n-d+1$ repetitions of the last element). diff --git a/lectures/2-13-ultragroth.tex b/lectures/2-13-ultragroth.tex index 9646879..e9ffcc3 100644 --- a/lectures/2-13-ultragroth.tex +++ b/lectures/2-13-ultragroth.tex @@ -29,11 +29,11 @@ \subsubsection{Historical Notes} \href{https://hackmd.io/@Merlin404/Hy_O2Gi-h}{Lev Soukhanov's post} in 2023. However, as we prepared the conference paper for Bionetta, we discovered that his construction is nothing but a generalization of -\href{https://eprint.iacr.org/2020/278}{$\textsf{MIRAGE}$ protocol}, posted back +the $\textsf{MIRAGE}$ protocol \cite{mirage}, posted back in 2020! (in subsequent notation, $\textsf{MIRAGE}$ is an UltraGroth protocol with $d=1$). Even more surprisingly, it seems that independently of Lev Soukhanov's construction, Alex Ozdemir, Evan Laufer, and Dan Boneh introduced -\href{https://eprint.iacr.org/2024/979}{$\textsf{MIRAGE+}$ protocol} in 2024, +the $\textsf{MIRAGE+}$ protocol \cite{mirageplus} in 2024, which was used for proving RAM computations correctness. This protocol, which is hard to believe, but also coincides with the Lev Soukharnov's construction (although I believe the former generalizes the construction a bit more @@ -44,9 +44,7 @@ \subsubsection{Historical Notes} All three papers (Bionetta included) proved completeness, soundness, and zero-knowledge of this construction, so the protocol that follows can be safely integrated into production systems. We, of course, drop formalities in this blog -and recommend checking -\href{https://eprint.iacr.org/2024/979}{$\textsf{MIRAGE+}$ protocol} paper for -proof specifics in case someone is interested. +and recommend checking \cite{mirageplus} for proof specifics in case someone is interested. Finally, our personal opinion is that this construction is vastly underestimated and it is surprising that it is not yet used in production. Indeed, integrating diff --git a/lectures/2-2-sigma.tex b/lectures/2-2-sigma.tex index a45edbf..3b69d3c 100644 --- a/lectures/2-2-sigma.tex +++ b/lectures/2-2-sigma.tex @@ -16,7 +16,7 @@ \subsection{Schnorr's Identification Protocol} First, let us start with the interactive version of the protocol. \begin{definition} - \textbf{The Schnorr interactive identification protocol} $\Pi_{\text{Sch}} = (\mathsf{Gen}, \mathcal{P}, \mathcal{V})$ with a generation function $\mathsf{Gen}$ and prover $\mathcal{P}$ and verifier $\mathcal{V}$ is defined as follows: + \textbf{The Schnorr interactive identification protocol} $\Pi_{\text{Sch}} = (\mathsf{Gen}, \mathcal{P}, \mathcal{V})$ \cite{schnorr1989} with a generation function $\mathsf{Gen}$ and prover $\mathcal{P}$ and verifier $\mathcal{V}$ is defined as follows: \begin{itemize} \item $\mathsf{Gen}(1^{\lambda})$: As with most public-key cryptosystems, we take $\alpha \leftarrowS \mathbb{Z}_q$ and $u \gets g^{\alpha}$. We output the \textit{verification key} as $\mathsf{vk} := u$, and the \textit{secret key} as $\mathsf{sk} := \alpha$. \item The protocol between $(\mathcal{P},\mathcal{V})$ is run as follows: @@ -121,7 +121,7 @@ \subsection{Schnorr's Signature Scheme} Now, turning the Schnorr's Identification Protocol into a signature scheme is quite straightforward. The only modification to the non-interactive proof described in the previous section is that we include the message $m \in \mathcal{M}$ instead of our statement $u \in \mathbb{G}$ in the computation of the challenge $e$. Additionally, suppose we use the hash function $H$ as a random oracle from the previous section. Now, let us give a formal definition. \begin{definition} - The \textbf{Schnorr Signature Scheme} $\Sigma_{\text{Sch}}$ is a tuple of algorithms $(\mathsf{Gen}, \mathsf{Sign}, \mathsf{Verify})$, where: + The \textbf{Schnorr Signature Scheme} $\Sigma_{\text{Sch}}$ \cite{schnorr1989} is a tuple of algorithms $(\mathsf{Gen}, \mathsf{Sign}, \mathsf{Verify})$, where: \begin{itemize} \item $\mathsf{Gen}(1^{\lambda})$: We take $\alpha \leftarrowS \mathbb{Z}_q$ and $u \gets g^{\alpha}$. The \textit{public key} is $\mathsf{pk} := u$, while the \textit{secret key} as $\mathsf{sk} := \alpha$. \item $\mathsf{Sign}(m,\mathsf{sk})$: The signer computes $r \gets \mathbb{Z}_q^{\times}, a \gets g^{r}, e \gets H(m, a), \sigma \gets r + \alpha e$ and outputs the signature $(a,\sigma)$. diff --git a/lectures/2-3-circuits.tex b/lectures/2-3-circuits.tex index 7cd8437..480bbd0 100644 --- a/lectures/2-3-circuits.tex +++ b/lectures/2-3-circuits.tex @@ -545,7 +545,7 @@ \subsubsection{More advanced examples} \end{tikzpicture} } \caption{Example of a circuit evaluating the \texttt{if} statement logic.} - \label{fig:multivariate-polynomial-circuit} + \label{fig:if-statement-circuit} \end{figure} Corresponding equations for the circuit are: diff --git a/lectures/2-4-qap-pcp.tex b/lectures/2-4-qap-pcp.tex index fc62ede..0c7bf03 100644 --- a/lectures/2-4-qap-pcp.tex +++ b/lectures/2-4-qap-pcp.tex @@ -354,7 +354,7 @@ \subsection{Putting All Together!} by $z_{\Omega}$ without remainder! In other words, there exists some polynomial $h$ such that $m=z_{\Omega}h$. We further drop index $\Omega$ for simplicity. -All in all, let us give the definition of a \textbf{Quadratic Arithmetic Program}. +All in all, let us give the definition of a \textbf{Quadratic Arithmetic Program} \cite{qap}. \begin{definition}[Quadratic Arithmetic Program] Suppose that $m$ R1CS constraints with a witness of size $n$ are written in a form @@ -523,7 +523,7 @@ \subsection{Probabilistically Checkable Proofs} \end{tikzpicture} } \caption{Illustration of an Interactive Oracle Proof (IOP). On each round $i$ ($1 \leq i \leq r$), $\mathcal{V}$ sends a message $m_i$, and $\mathcal{P}$ commits to a new oracle $\pi_i$, which $\mathcal{V}$ can query at $\mathbf{q}_i=(q_{i,1},\dots,q_{i,m})$.} - \label{fig:pcp} + \label{fig:iop} \end{figure} While IOPs will be later used for PLONK and zk-STARKs, we will focus on Linear diff --git a/lectures/2-5-groth.tex b/lectures/2-5-groth.tex index 40529a3..1beb770 100644 --- a/lectures/2-5-groth.tex +++ b/lectures/2-5-groth.tex @@ -538,7 +538,7 @@ \subsection{Real Protocols} \end{itemize} \end{proposition} -Now, this is not bad at all! In fact, this is already practical for many applications. However, we can do better by a more clever choice of constants and terms. This is exactly what is done by Bryan Parno and Craig Gentry in their research ``Pinocchio: Nearly Practical Verifiable Computation''. +Now, this is not bad at all! In fact, this is already practical for many applications. However, we can do better by a more clever choice of constants and terms. This is exactly what is done by Bryan Parno and Craig Gentry in their research ``Pinocchio: Nearly Practical Verifiable Computation'' \cite{pinocchio}. \subsection{Pinocchio Protocol} @@ -681,7 +681,7 @@ \subsection{Pinocchio Protocol} \subsection{Groth16 Protocol} -Finally, Groth16 allows to reduce the number of pairings \textbf{down to 3}! This is done through a technique called \textbf{Generic Group Model} (GGM for short). Simply put, GGM allows the adversary to only make oracle requests to compute the group operations. For example, having a set $\{g^{\alpha r_i(\tau)}\}_{i \in [d]}$, adversary can compute only linear combinations of these values. In the particular case of Groth16, instead of considering $\ell_i(X)$, $r_i(X)$, and $o_i(X)$ separately, we construct their linear combinations as $Q_i(X) := \beta \ell_i(X) + \alpha r_i(X) + o_i(X)$, where $\alpha$ and $\beta$ are toxic parameters. +Finally, Groth16 \cite{groth16} allows to reduce the number of pairings \textbf{down to 3}! This is done through a technique called \textbf{Generic Group Model} (GGM for short). Simply put, GGM allows the adversary to only make oracle requests to compute the group operations. For example, having a set $\{g^{\alpha r_i(\tau)}\}_{i \in [d]}$, adversary can compute only linear combinations of these values. In the particular case of Groth16, instead of considering $\ell_i(X)$, $r_i(X)$, and $o_i(X)$ separately, we construct their linear combinations as $Q_i(X) := \beta \ell_i(X) + \alpha r_i(X) + o_i(X)$, where $\alpha$ and $\beta$ are toxic parameters. Let us now concretely describe the Groth16 construction. @@ -718,7 +718,7 @@ \subsection{Groth16 Protocol} \subsection*{Acknowledgements} -This section was greatly inspired by +This section was greatly inspired by \cite{petkus2019} \href{https://arxiv.org/abs/1906.07221}{``Why and How zk-SNARK works''} by Maksym Petkus and \href{https://rdi.berkeley.edu/zk-learning/}{``ZK MOOC, Spring 2023''} Linear PCP lecture. diff --git a/lectures/2-6-circom.tex b/lectures/2-6-circom.tex index d938da7..d459f0a 100644 --- a/lectures/2-6-circom.tex +++ b/lectures/2-6-circom.tex @@ -737,7 +737,7 @@ \subsection{Generating and Verifying Proofs} each consisting of three pairs of prime field elements? The primary reason is that the most convenient way to construct $\mathbb{F}_{p^{12}}$ element is to use the so-called \textbf{tower of extensions}: we represent an element from $\mathbb{F}_{p^{12}}$ as a pair of two $\mathbb{F}_{p^6}$ elements, while each $\mathbb{F}_{p^6}$ - consists of a triplet of $\mathbb{F}_{p^2}$ elements. For more details,~see~\Cref{section:field_extensions} + consists of a triplet of $\mathbb{F}_{p^2}$ elements. For more details,~see~\Cref{section:finite-fields} \end{remark} Thus, we have covered all the information about the internal structure of the Circom files needed for proof generation and verification. diff --git a/lectures/2-7-plonk.tex b/lectures/2-7-plonk.tex index b088128..24cb929 100644 --- a/lectures/2-7-plonk.tex +++ b/lectures/2-7-plonk.tex @@ -22,7 +22,7 @@ for the large enough finite field $\mathbb{F}$, does not cause any issues. However, the complexity of interpolation in this case is not optimal. Let us see why. -Recall that the interpolation formula (see \Cref{section:math-crypto-2} for details) +Recall that the interpolation formula (see \Cref{section:polynomial-rings} for details) is given by: \begin{equation*} p(x) = \sum_{i=0}^{N-1} a_i \ell_i(x), \quad \ell_i(x) = \prod_{j=0, j \neq i}^{N-1} \frac{x-x_j}{x_i-x_j}. @@ -345,7 +345,7 @@ \subsubsection{Fast Polynomial Multiplication} \subsection{Plonk Arithmetization} Consider we have a certain relation $\mathcal{R}$, which we would like to write -down into a processing-prone format over the field $\mathbb{F}$. Plonk arithmetizes this relation into a set +down into a processing-prone format over the field $\mathbb{F}$. Plonk \cite{plonk} arithmetizes this relation into a set of \textit{8 polynomials}, which are then used to verify the witness knowledge. Let us start with the concrete example. @@ -951,7 +951,7 @@ \subsection{Plonk Prover and Verifier} these values are not actually chosen by the prover, but rather computed deterministically from the transcript using Fiat-Shamir heuristic. In case you are not familiar with Fiat-Shamir heuristic, we recommend you to revisit - the \Cref{section:zk}. + the \Cref{section:sigma}. \end{remark} \subsubsection{Gadgets}

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a/presentations/18-bulletproofs.tex +++ b/presentations/18-bulletproofs.tex @@ -36,11 +36,11 @@ \section{Introduction} \begin{itemize} \item \textbf{Goal:} Prove $\langle \mathbf{a}, \mathbf{b} \rangle = c$ with logarithmic proof size \item Commitment: $P' = \langle \mathbf{a}, \mathbf{G} \rangle + \langle \mathbf{b}, \mathbf{H} \rangle + [\langle \mathbf{a,b} \rangle]Q$ - \item Protocol recursively compresses vectors, sending $L_k, R_k$ at each step + \item Protocol recursively compresses vectors at each step \item Final check: $P' + \sum ([u_i^2]L_i + [u_i^{-2}]R_i) = [a]G + [b]H + [ab]Q$ \end{itemize} - \begin{alertblock}{Key property} - Proof size is $2\log_2 n$ group elements $+$ $2$ scalars. Protocol is \textit{knowledge sound} and \textit{perfect complete} but not \textit{zero-knowledge}. + \begin{alertblock}{Key properties} + Proof size is $O(\log_2 n)$, prover and verifier both run in $O(n)$. The protocol doesn't need a \textit{trusted setup}. Protocol is \textit{knowledge sound} and \textit{perfect complete} but not \textit{zero-knowledge}. \end{alertblock} \begin{block}{Idea} We could provide \textit{zero-knowledge} directly to \textit{inner-product argument} construction or use \textbf{zk-mul} protocol for outer construction. @@ -67,6 +67,12 @@ \section{Introduction} \end{itemize} \end{frame} +\begin{frame}{What's next?} + \begin{figure}[h!] + \centering + \includegraphics[width=0.5\textwidth]{images/lecture_17/ipa_meme.jpg} + \end{figure} +\end{frame} \section{IPA polynomial commitment scheme} \begin{frame}{Recap: Polynomial commitments} @@ -169,9 +175,9 @@ \section{Range proofs} \begin{frame}{Range proofs: building the protocol} \begin{itemize} \item $\mathsf{Setup}$ returns independent generators $\mathbf{G}, \mathbf{H} \in \mathbb{G}^n$ - \item Prover does bit decomposition of $v$: $\mathbf{a}_L \gets \mathbf{v}, \mathbf{b_L} \gets \mathbf{a}_L - \mathbf{1}^n$, choses blinding terms $\mathbf{s}_L, \mathbf{s}_R \in \mathbb{F}_p^n, \alpha, \beta \in \mathbb{F}_p$, sends commitments: + \item Prover does bit decomposition of $v$: $\mathbf{a}_L \gets \mathbf{v}, \mathbf{a}_R \gets \mathbf{a}_L - \mathbf{1}^n$, choses blinding terms $\mathbf{s}_L, \mathbf{s}_R \in \mathbb{F}_p^n, \alpha, \beta \in \mathbb{F}_p$, sends commitments: \begin{align*} - A = \langle \mathbf{a}_L, \mathbf{G} \rangle + \langle \mathbf{b_L}, \mathbf{H} \rangle + [\alpha]B && + A = \langle \mathbf{a}_L, \mathbf{G} \rangle + \langle \mathbf{a}_R, \mathbf{H} \rangle + [\alpha]B && S = \langle \mathbf{s}_L, \mathbf{G} \rangle + \langle \mathbf{s}_R, \mathbf{H} \rangle + [\beta]B \end{align*} \item Verifier $\mathcal{V}$ samples challenges $y, z \xleftarrow{R} \mathbb{F}_p$ and sends them to $\mathcal{P}$ @@ -234,76 +240,334 @@ \section{Range proofs} \end{block} \end{frame} +\begin{frame}{Range proofs: efficiency \& extensions} + \begin{theorem} + The \textbf{range proof protocol} $\Pi_{rp}$ has \textit{perfect completeness, computational extended witness emulation, perfect honest-verifier zero-knowledge} + \label{th:range_proof} + \end{theorem} + Note that protocol is efficient as it has logarithmic proof size. + \begin{block}{Remark} + The \textbf{range proof protocol} could be extended to support proving multiple range proofs at once with some efficiency improvements. + \end{block} +\end{frame} +\begin{frame}{Range proofs \& subset-sum NP-complete problem} + One of the most famous \textit{NP-complete} problems is the \textbf{subset-sum problem}: given a set of numbers presented as vector $\mathbf{s}$ and number $v \in \mathbb{N}$, does a some subset sums up to $v$. It turns out that we could use our \textbf{range-proof} protocol for this problem. One could simply replace first inner-product check $\langle \mathbf{a}_L, \mathbf{2}^n \rangle = v$ with $\langle \mathbf{a}_L, \mathbf{s} \rangle = v$ where $\mathbf{a}_L$ is the secret vector of bits that encode positions of $\mathbf{s}$ that sum up to $v$. + \begin{example} + Let $\mathbf{s} = (6,8,2,3)$ and $v = 14$. Then setting $\mathbf{a}_L = (1,1,0,0)$ we could use $\Pi_{rp}$ to prove that there exists a subset of $\mathbf{s}$ that sums up to $v=14$ without disclosing that subset. + \end{example} + Therefore, \textbf{bulletproofs range proof} protocol is capable to prove a knowledge of witness to any $NP$-problem as they all could be reduced to the $\textbf{subset-sum problem}$ +\end{frame} \section{Arithmetic circuits} -\begin{frame}{Proving arithmetic circuit satisfiability} +\begin{frame}{Bulletproofs for arithmetic circuits} \begin{itemize} - \item \textbf{Goal:} Prove knowledge of $\mathbf{w}$ s.t. $C(\mathbf{w}) = 0$ for circuit $C$ - \item Encode as R1CS: $A\mathbf{w} \circ B\mathbf{w} = C\mathbf{w}$ - \item Each constraint is a multiplication of linear forms + \item \textbf{Goal:} Prove that a circuit computes correctly without revealing inputs or intermediate values (\textit{circuit satisfiability problem}). + \item \textbf{Approach:} Use inner-product argument to prove correctness of arithmetic circuits + \item \textbf{Applications:} Privacy-preserving smart contracts, confidential computations, zero-knowledge proofs for complex computations \end{itemize} + \textbf{Bulletproofs} arithmetization slightly differs from the classic R1CS, however it could be transformed vice-versa easily. Also \textbf{bulletproofs} arithmetization is more convenient and human-friendly for encoding most of the arithmetic circuits than the R1CS. \end{frame} -\begin{frame}{Bulletproofs for circuits: Approach} +\begin{frame}{Arithmetic circuits: variables} + There is two types of variables in \textit{bulletproofs} constraint system: \begin{itemize} - \item Prover commits to witness $\mathbf{w}$ - \item Encodes all constraints as inner-products - \item Uses IPA protocol to prove all constraints in zero-knowledge + \item \textbf{High-level variables} $\mathbf{v} \in \mathbb{F}_p^m$ are the private witness inputs to the circuit, typically provided with Pedersen commitments $\mathbf{V} \in \mathbb{G}^m$. + \item \textbf{Low-level variables} $\mathbf{a}_L, \mathbf{a}_R, \mathbf{a}_O \in \mathbb{F}_p^n$ are the intermediate witness values of computation. \end{itemize} - \begin{block}{Batching} - All constraints can be aggregated into a single inner-product argument + We will define circuit as a set of multiplication constraints operating with \textit{low-level} variables and set of linear constraints which links \textit{low-level} variables between each other and \textit{high-level} variables as well. +\end{frame} + +\begin{frame}{Arithmetic circuits: constraints} + Multiplication constraints are defined with one vector equation: + $$ \mathbf{a}_L \circ \mathbf{a}_R = \mathbf{a}_O $$ + Linear constraints are defined via: + $$ \mathbf{W}_L \cdot \mathbf{a}_L + \mathbf{W}_R \cdot \mathbf{a}_R + \mathbf{W}_O \cdot \mathbf{a}_O = \mathbf{W}_V \cdot \mathbf{v} + \mathbf{c} $$ + Where $\mathbf{a_L, a_R, a_O}$ -- vectors of left and right inputs for multiplication gates and output values (all of them are low-level variables). $\mathbf{W_L, W_R, W_O} \in \mathbb{F}_p^{q \times n}, \mathbf{W}_V \in \mathbb{F}_p^{q \times m}$ -- public matrices of weights for linear constraints(obviously known to verifier). $\mathbf{c} \in \mathbb{F}_p^q$ -- public vector of constants. Typically they encode wiring of the circuit and other linear relations between variables. +\end{frame} + +\begin{frame}{Arithmetic circuits: example} + \begin{example} + Consider the following elliptic curve membership circuit. Here witness $(v_1, v_2)$ should satisfy elliptic curve equation: + \begin{equation*} + y^2 = x^3 + ax + b + \end{equation*} + + The arithmetization for this circuit is as follows: + + \textbf{Low-level variables:} + \begin{equation*} + \mathbf{a}_L = \begin{bmatrix} x \\ x \\ y \end{bmatrix}, \quad + \mathbf{a}_R = \begin{bmatrix} x \\ x^2 \\ y \end{bmatrix}, \quad + \mathbf{a}_O = \begin{bmatrix} x^2\\x^3 \\ y^2 \end{bmatrix} + \end{equation*} + + \textbf{High-level variables:} + \begin{equation*} + \mathbf{v} = \begin{bmatrix} v_1 \\ v_2 \end{bmatrix} + \end{equation*} + \end{example} +\end{frame} + +\begin{frame}{Arithmetic circuits: example} + \begin{example} + \textbf{Multiplication constraints:} + \begin{equation*} + \mathbf{a}_L \circ \mathbf{a}_R = \mathbf{a}_O \Rightarrow \begin{bmatrix} + x \cdot x = x^2 \\ + x \cdot x^2 = x^3 \\ + y \cdot y = y^2 + \end{bmatrix} + \end{equation*} + \textbf{Linear constraints:} + + \begin{equation*} + \begin{aligned} + \mathbf{a}_L^{(1)} &= v_1 & \mathbf{a}_R^{(1)} &= v_1 \\ + \mathbf{a}_L^{(2)} - \mathbf{a}_L^{(1)} &= 0 & \mathbf{a}_R^{(2)} - \mathbf{a}_O^{(1)} &= 0 \\ + \mathbf{a}_L^{(3)} &= v_2 & \mathbf{a}_R^{(3)} &= v_2 \\ + \mathbf{a}_O^{(3)} - \mathbf{a}_O^{(2)} - a \cdot \mathbf{a}_L^{(1)} &= b & + \end{aligned} + \end{equation*} + + \begin{equation*} + \mathbf{W}_L \cdot \mathbf{a}_L + \mathbf{W}_R \cdot \mathbf{a}_R + \mathbf{W}_O \cdot \mathbf{a}_O = \mathbf{W}_V \cdot \mathbf{v} + \mathbf{c} + \end{equation*} + \end{example} +\end{frame} + +\begin{frame}{Arithmetic circuits: example} + \begin{example} + \begin{align*} + \mathbf{W}_L = \begin{bmatrix} + 1 & 0 & 0 \\ + 0 & 0 & 0 \\ + -1 & 1 & 0 \\ + 0 & 0 & 0 \\ + 0 & 0 & 1 \\ + 0 & 0 & 0 \\ + -a & 0 & 0 + \end{bmatrix},\quad + \mathbf{W}_R = \begin{bmatrix} + 0 & 0 & 0 \\ + 1 & 0 & 0 \\ + 0 & 0 & 0 \\ + 0 & 1 & 0 \\ + 0 & 0 & 0 \\ + 0 & 0 & 1 \\ + 0 & 0 & 0 + \end{bmatrix},\quad + \mathbf{W}_O = \begin{bmatrix} + 0 & 0 & 0 \\ + 0 & 0 & 0 \\ + 0 & 0 & 0 \\ + -1 & 0 & 0 \\ + 0 & 0 & 0 \\ + 0 & 0 & 0 \\ + 0 & -1 & 1 + \end{bmatrix},\\ + \mathbf{W}_V = \begin{bmatrix} + 1 & 0 \\ + 1 & 0 \\ + 0 & 0 \\ + 0 & 0 \\ + 0 & 1 \\ + 0 & 1 \\ + 0 & 0 + \end{bmatrix},\quad + \mathbf{c} = \begin{bmatrix} + 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ b + \end{bmatrix} + \end{align*} + \end{example} +\end{frame} + +\begin{frame}{Bulletproofs for circuits: relation} + Consider the relation: + \begin{equation*} + \begin{aligned} + \mathcal{R}_{sat} = \left\{ + \begin{array}{l} + (G, B, \mathbf{V}, \mathbf{W}_L, \mathbf{W}_R, \mathbf{W}_O, \mathbf{W}_V, \mathbf{c}; \mathbf{a}_L, \mathbf{a}_R, \mathbf{a}_O, \mathbf{v}, \mathbf{r}) | \\ + \forall i=1..m: V_i = [v_i]G + [r_i]B \wedge \\ + \mathbf{a}_L \circ \mathbf{a}_R = \mathbf{a}_O \wedge \\ + \mathbf{W}_L \cdot \mathbf{a}_L + \mathbf{W}_R \cdot \mathbf{a}_R + \mathbf{W}_O \cdot \mathbf{a}_O = \mathbf{W}_V \cdot \mathbf{v} + \mathbf{c} + \end{array}\right\} + \end{aligned} + \end{equation*} + Where $\mathbf{a}_L, \mathbf{a}_R, \mathbf{a}_O \in \mathbb{F}_p^n$, $\mathbf{v}, \mathbf{r} \in \mathbb{F}_p^m$, $\mathbf{W}_L, \mathbf{W}_R, \mathbf{W}_O \in \mathbb{F}_p^{q \times n}$, $\mathbf{W}_V \in \mathbb{F}_p^{q \times m}$, $\mathbf{c} \in \mathbb{F}_p^q$. + \begin{block}{Note} + Informally this relation states that there exists a valid witness $\mathbf{v}$ that satisfies all constraints of the circuit. For the verifier witness is presented only as commitments vector $\mathbf{V}$. \end{block} \end{frame} -\begin{frame}{Bulletproofs for circuits: Example} - \begin{itemize} - \item Circuit: $w_1 \cdot w_2 = w_3$, $w_3 + w_1 = 7$ - \item Encode as R1CS, commit to $\mathbf{w}$ - \item Prove in zero-knowledge using bulletproofs - \end{itemize} +\begin{frame}{Arithmetic circuits: compiling into inner-product} +We could use similar to \textit{range-proofs} technique to compile constraints of the circuit into inner-product relation. For multiplicative constraints take random $y \in \mathbb{F}_p$ and apply zero check: +$$ \langle \mathbf{a}_L \circ \mathbf{a}_R - \mathbf{a}_O, \mathbf{y}^n \rangle = 0$$ +Same for linear constraints, but for different randomness $z \in \mathbb{F}_p$: +$$ \langle \mathbf{W}_L \cdot \mathbf{a}_L + \mathbf{W}_R \cdot \mathbf{a}_R + \mathbf{W}_O \cdot \mathbf{a}_O - \mathbf{W}_V \cdot \mathbf{v} - \mathbf{c}, \mathbf{z}^q \rangle = 0$$ +Combine this two checks to one using the same randomness $z$: +\begin{equation*} + \begin{aligned} + \langle \mathbf{a}_L \circ \mathbf{a}_R - \mathbf{a}_O, \mathbf{y}^n \rangle + \langle z \cdot \mathbf{z}^q, \mathbf{W}_L \cdot \mathbf{a}_L + \mathbf{W}_R \cdot \mathbf{a}_R + \mathbf{W}_O \cdot \mathbf{a}_O - \mathbf{W}_V \cdot \mathbf{v} - \mathbf{c} \rangle \\= 0 + \end{aligned} +\end{equation*} +This check is sound as typically a prover could not control values of $y,z$ before he commits to $\mathbf{a_L, a_R, a_O}$ and $\mathbf{v}$. \end{frame} -\begin{frame}{Bulletproofs for circuits: Performance} +\begin{frame}{Arithmetic circuits: compiling into inner-product} + Denote $w_c = \langle z \cdot \mathbf{z}^q, \mathbf{c} \rangle$ and flattened linear constraints(still public and easily computed by verifier): + \begin{equation*} + \begin{aligned} + \mathbf{w}_L &= \mathbf{W}_L^T \cdot (z \cdot \mathbf{z}^q) & + \mathbf{w}_R &= \mathbf{W}_R^T \cdot (z \cdot \mathbf{z}^q) \\ + \mathbf{w}_O &= \mathbf{W}_O^T \cdot (z \cdot \mathbf{z}^q) & + \mathbf{w}_V &= \mathbf{W}_V^T \cdot (z \cdot \mathbf{z}^q) + \end{aligned} + \end{equation*} + Again doing some linear algebra witchcraft we could separate $\mathbf{a}_L, \mathbf{a}_O$ to be on the left side of the inner-product and $\mathbf{a}_R$ to be on the right: + \begin{equation*} + \begin{aligned} + w_c + \langle \mathbf{w}_V, \mathbf{v} \rangle + \delta(y, z) =\\ + \langle \textcolor{teal}{\mathbf{a}_L + \mathbf{y}^{-n} \circ \mathbf{w}_R}, \textcolor{orange}{\mathbf{y}^n \circ \mathbf{a}_R + \mathbf{w}_L} \rangle + \langle \textcolor{purple}{\mathbf{a}_O}, \textcolor{blue}{ -\mathbf{y}^n + \mathbf{w}_O} \rangle + \end{aligned} + \end{equation*} + Where $\delta(y, z) = \langle \mathbf{y}^{-n} \circ \mathbf{w}_R, \mathbf{w}_L \rangle$ -- easily computable by $\mathcal{V}$. + + Here we have a sum of 2 separate inner-products, we could express it as second-degree coefficient of the following polynomial: + \begin{equation*} + \begin{aligned} + \langle \textcolor{teal}{\mathbf{a}}x + \textcolor{purple}{\mathbf{c}}x^2, \textcolor{blue}{\mathbf{d}} + \textcolor{orange}{\mathbf{b}}x \rangle = s_1x + s_2x^2 + s_3x^3 =\\ + x \cdot \langle \textcolor{teal}{\mathbf{a}}, \textcolor{blue}{\mathbf{d}} \rangle + x^2 \cdot (\langle \textcolor{teal}{\mathbf{a}}, \textcolor{orange}{\mathbf{b}} \rangle + \langle \textcolor{purple}{\mathbf{c}}, \textcolor{blue}{\mathbf{d}} \rangle) + x^3 \cdot \langle \textcolor{purple}{\mathbf{c}}, \textcolor{orange}{\mathbf{b}} \rangle + \end{aligned} + \end{equation*} +\end{frame} + +\begin{frame}{Arithmetic circuits: compiling into inner-product} + \vspace{-1em} + \begin{equation*} + \begin{aligned} + &\textcolor{teal}{\mathbf{a}} \gets \textcolor{teal}{\mathbf{a}_L + \mathbf{y}^{-n} \circ \mathbf{w}_R} &&\textcolor{orange}{\mathbf{b}} \gets \textcolor{orange}{\mathbf{y}^n \circ \mathbf{a}_R + \mathbf{w}_L} \\ + &\textcolor{purple}{\mathbf{c}} \gets \textcolor{purple}{\mathbf{a}_O} &&\textcolor{blue}{\mathbf{d}} \gets \textcolor{blue}{-\mathbf{y}^n + \mathbf{w}_O} + \end{aligned} + \end{equation*} + Desired sum of inner products is the second-degree coefficient $s_2$: + $$ w_c + \langle \mathbf{w}_V, \mathbf{v} \rangle + \delta(y, z) = s_2$$ + To obtain final polynomials $\mathbf{l}(x), \mathbf{r}(x)$ we must firstly blind $\mathbf{a}_L, \mathbf{a}_R$: + \begin{equation*} + \begin{aligned} + \mathbf{a}_L \gets \mathbf{a}_L + \mathbf{s}_Lx^2 && \mathbf{a}_R \gets \mathbf{a}_R + \mathbf{s}_Rx^2 + \end{aligned} + \end{equation*} + And finally compute polynomials $\mathbf{l}(x), \mathbf{r}(x)$ as follows: + \begin{equation*} + \begin{aligned} + {\mathbf{l}}(x) &= \mathbf{s}_L \cdot x^3 + \mathbf{a}_O \cdot x^2 + (\mathbf{a}_L + \mathbf{y}^{-n} \circ \mathbf{w}_R) \cdot x \\ + {\mathbf{r}}(x) &= \mathbf{y}^n \circ \mathbf{s}_R \cdot x^3 + (\mathbf{y}^n \circ \mathbf{a}_R + \mathbf{w}_L) \cdot x - \mathbf{y}^n + \mathbf{w}_O \\ + t(x) &= \langle \mathbf{l}(x), \mathbf{r}(x) \rangle = \sum_{i=0}^6 t_i x_i + \end{aligned} + \end{equation*} + Where $t_2 = w_c + \langle \mathbf{w}_V, \mathbf{v} \rangle + \delta(y, z)$ -- desired sum of inner-products. +\end{frame} + +\begin{frame}{Arithmetic circuits: witness commitments} + Here we could again apply modified \textbf{zk-mul} to prove that $t_2$ is a valid sum of inner-products: \begin{itemize} - \item Proof size: $2\log_2 n + O(1)$ group elements ($n$ = number of constraints) - \item No trusted setup - \item Efficient for small/medium circuits - \item Verification time linear in $n$ + \item $\mathsf{Setup}$: returns vectors of independent generators $\mathbf{G,H} \in \mathbb{G}^n$. + \item Prover $\mathcal{P}$ choses blinding factors $\alpha, \beta, \gamma \in \mathbb{F}_p, \mathbf{s}_L, \mathbf{s}_R \in \mathbb{F}_p^n$ and sends the following commitments to $\mathcal{V}$: + \begin{equation*} + \begin{aligned} + A_I &= \langle \mathbf{a}_L, \mathbf{G} \rangle + \langle \mathbf{a}_R, \mathbf{H} \rangle + [\alpha]B\\ + A_O &= \langle \mathbf{a}_O, \mathbf{G} \rangle + [\gamma]B\\ + S &= \langle \mathbf{s}_L, \mathbf{G} \rangle + \langle \mathbf{s}_R, \mathbf{H} \rangle + [\beta]B + \end{aligned} + \end{equation*} + \item Verifier samples challenges $y,z \xleftarrow{R} \mathbb{F}_p$ and sends them to $\mathcal{P}$. \end{itemize} \end{frame} -\section{Security, efficiency, and open questions} - -\begin{frame}{Security and efficiency} +\begin{frame}{Arithmetic circuits: product commitments} \begin{itemize} - \item \textbf{Security:} Discrete log assumption, no trusted setup - \item \textbf{Efficiency:} Logarithmic proof size, fast prover for small $n$ - \item \textbf{Limitations:} Verification time linear in $n$ + \item Using challenges $y,z$ prover forms polynomials $\mathbf{l}(x), \mathbf{r}(x), t(x)$: + \begin{equation*} + \begin{aligned} + \mathbf{l}(x) &= \mathbf{s}_L \cdot x^3 + \mathbf{a}_O \cdot x^2 + (\mathbf{a}_L + \mathbf{y}^{-n} \circ \mathbf{w}_R) \cdot x \\ + \mathbf{r}(x) &= \mathbf{y}^n \circ \mathbf{s}_R \cdot x^3 + (\mathbf{y}^n \circ \mathbf{a}_R + \mathbf{w}_L) \cdot x - \mathbf{y}^n + \mathbf{w}_O \\ + t(x) &= \langle \mathbf{l}(x), \mathbf{r}(x) \rangle = t_1 x + t_2 x^2 + t_3 x^3 + t_4 x^4 + t_5 x^5 + t_6 x^6 + \end{aligned} + \end{equation*} + $\mathcal{P}$ choses random blinding factors $\tau_1, \tau_3, \tau_4, \tau_5, \tau_6 \in \mathbb{F}_p$ and sends to $\mathcal{V}$ commitments to its coefficients: + \begin{equation*} + \begin{aligned} + T_1 &= [t_1]G + [\tau_1]B & + T_3 &= [t_3]G + [\tau_3]B & + T_4 &= [t_4]G + [\tau_4]B\\ + T_5 &= [t_5]G + [\tau_5]B & + T_6 &= [t_6]G + [\tau_6]B & + \end{aligned} + \end{equation*} + \textbf{Note:} Prover does not send separate commitment to $t_2$ as the verifier could derive it from $\mathbf{V}$ and the circuit public parameters: + \begin{align*} + t_2 &= w_c + \langle \mathbf{w}_V, \mathbf{v} \rangle + \delta(y, z) \\ + T_2 &= \langle \mathbf{w}_V, \mathbf{V} \rangle + [\delta(y, z) + w_c]G + \end{align*} \end{itemize} \end{frame} -\begin{frame}{Open questions and extensions} +\begin{frame}{Arithmetic circuits: evaluating polynomials} \begin{itemize} - \item Can we get sublinear verification? - \item Can we combine with SNARKs for better efficiency? - \item Applications: privacy, blockchains, verifiable computation - \item Extensions: batch proofs, recursive proofs, Halo/Nova + \item Verifier samples and sends to $\mathcal{P}$ random evaluation point $u \xleftarrow{R} \mathbb{F}_p$. + \item Prover evaluates polynomials at $u$: + \begin{equation*} + \begin{aligned} + \mathbf{l}_u &= \mathbf{l}(u) \\ + \mathbf{r}_u &= \mathbf{r}(u) \\ + t_u &= \langle \mathbf{l}_u, \mathbf{r}_u \rangle = t(u) \\ + \tau_u &= \tau_1 \cdot u + \langle \mathbf{w}_V, \mathbf{r} \rangle u^2 + \tau_3 \cdot u^3 + \tau_4 \cdot u^4 + \tau_5 \cdot u^5 + \tau_6 \cdot u^6\\ + \alpha_u &= \alpha u + \gamma u^2 + \beta u^3 + \end{aligned} + \end{equation*} + and sends $(\mathbf{l}_u, \mathbf{r}_u, t_u, \alpha_u, \tau_u)$ to $\mathcal{V}$. \end{itemize} \end{frame} -\begin{frame}{Summary} +\begin{frame}{Arithmetic circuits: verification} \begin{itemize} - \item Bulletproofs: efficient, no trusted setup, versatile - \item IPA polynomial commitment: core primitive - \item Range proofs and circuit proofs: practical applications - \item Still active area of research! + \item Verifier performs checks: + \begin{equation*} + \begin{aligned} + &[u]A_I + [u^2]A_O + [u^3]S - \langle \mathbf{1}, \mathbf{H} \rangle + \\ + &u \cdot (\langle \mathbf{y}^{-n} \circ \mathbf{w}_L, \mathbf{G} \rangle + \langle \mathbf{y}^{-n} \circ \mathbf{w}_R, \mathbf{H} \rangle) + \langle \mathbf{y}^{-n} \circ \mathbf{w}_O, \mathbf{H} \rangle \\ + &\stackrel{\text{?}}{=} \langle \mathbf{l}_u, \mathbf{G} \rangle + \langle \mathbf{r}_u, \mathbf{y}^{-n} \circ \mathbf{H} \rangle + [\alpha_u]B\\ + [t_u]G + [\tau_u]B &\stackrel{\text{?}}{=} [u]T_1 + u^2 \cdot (\langle \mathbf{w}_V, \mathbf{V} \rangle + [\delta(y,z) + w_c]G) +\\ + &[u^3]T_3 + [u^4]T_4 + [u^5]T_5 + [u^6]T_6\\ + t_u &\stackrel{\text{?}}{=} \langle \mathbf{l}_u, \mathbf{r}_u \rangle + \end{aligned} + \end{equation*} \end{itemize} + \begin{block}{Remark} + To provide logarithmic proof instead of sending $\mathbf{l}_u, \mathbf{r}_u$ parties could run \textbf{IPA} on inputs $(\mathbf{G},\mathbf{y}^{-n} \circ \mathbf{H}, P, t_u; \mathbf{l}_u, \mathbf{r}_u)$ where: + \begin{align*} + P =& [u]A_I + [u^2]A_O + [u^3]S - \langle \mathbf{1}, \mathbf{H} \rangle +\\ + &u \cdot (\langle \mathbf{y}^{-n} \circ \mathbf{w}_L, \mathbf{G} \rangle + \langle \mathbf{y}^{-n} \circ \mathbf{w}_R, \mathbf{H} \rangle) + \langle \mathbf{y}^{-n} \circ \mathbf{w}_O, \mathbf{H} \rangle - [\alpha_u]B + \end{align*} + \end{block} \end{frame} -\begin{frame}{Thank you!} - \begin{center} - \Huge Questions? - \end{center} +\begin{frame}{Arithmetic circuits: efficiency \& extensions} + \begin{theorem} + The \textbf{arithmetic circuits protocol} has \textit{perfect completeness, computational extended witness emulation, perfect honest-verifier zero-knowledge} + \end{theorem} + The protocol is efficient as it has logarithmic proof size. + \begin{block}{Remark} + The \textbf{arithmetic circuits protocol} protocol could be slightly modified to provide intermediate random challenges inside the circuit. For example it would allow proving \textit{permutation check}: $\{a,b\} = \{c,d\} \iff (a-x)\cdot(b-x) = (c-x)\cdot(d-x)$ for some random challenge $x$. + \end{block} \end{frame} +\begin{frame}{Questions?} + \begin{figure} + \centering + \includegraphics[width=0.44\textwidth]{images/lecture_17/circuit.png} + \label{fig:np-completeness} + \end{figure} +\end{frame} \end{document} \ No newline at end of file diff --git a/presentations/images/lecture_17/circuit.png b/presentations/images/lecture_17/circuit.png new file mode 100644 index 0000000000000000000000000000000000000000..f331e5bffbde8e921f741ede41ca81bf38a1bba4 GIT binary patch literal 93392 zcmce8^;^_i_qGxeN{L9ffOL0C$Iva^IdrFh5>f(^LrJIfP(yc1NetcHo$qiw$LISO zyw|mV;$qj@YuzjE^_egw1t~OC0@P>Eo}tM|i>o|)1_yZd4E_i53)o)}`fli-JtKQ2 zBmP0nLw_#~DM@{DGI%5nzvzuJ7M{}AP-5{=0vt?Qx`gRn`{xW{)L7+`@_`kdaG?a) zgod~rZYGF0C9GK#Sy|G`cp~MWWI@T^5AJ%y_!v{PwYAq5r>UFZs!RP_LHC1e^ZlAB 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.../{lecture_17 => lecture_14}/ipa_meme.jpg | Bin ...c3bd2c20f632edf509ff6c41010dbd29-pjlq.jpeg | Bin .../{lecture_17 => lecture_14}/meme.jpg | Bin 10 files changed, 7 insertions(+), 7 deletions(-) rename presentations/images/{lecture_17 => lecture_14}/circuit.png (100%) rename presentations/images/{lecture_17 => lecture_14}/compressed.png (100%) rename presentations/images/{lecture_17 => lecture_14}/ipa.png (100%) rename presentations/images/{lecture_17 => lecture_14}/ipa_meme.jpg (100%) rename presentations/images/{lecture_17 => lecture_14}/main-qimg-c3bd2c20f632edf509ff6c41010dbd29-pjlq.jpeg (100%) rename presentations/images/{lecture_17 => lecture_14}/meme.jpg (100%) diff --git a/presentations/14-bulletproofs-intro.pdf b/presentations/14-bulletproofs-intro.pdf index 20bca390abb3b5bbcc1d68c862700130bf153c86..9452ed8319102356cf9a71f78eaafaf5f3725f43 100644 GIT binary patch delta 148 zcmbPpPi@XUwT2eP7N!>F7M2#)7Pc1l7LF~PXL~d(3``BowGE8a4Gh#Zx%7SWQ(O{D zQWZ2@tc(ndObrbUEeyd*wtwy6Ea!7}a&&TXaWrF7M2#)7Pc1l7LF~PXL~fvP0dUVv<-~Z4Gh#Zx%7SWQ(O{D zQWZ2@tc(ndObrbUEeyd*wtwy6Ea!7JGcq$ZcQJ5wvNW|YbT&3Hb~QC~Hg~dgH8V6a Ta5FcuQ?MbVWc&KroVmgPEvqMP diff --git a/presentations/14-bulletproofs-intro.tex b/presentations/14-bulletproofs-intro.tex index f806313..0ff08ee 100644 --- a/presentations/14-bulletproofs-intro.tex +++ b/presentations/14-bulletproofs-intro.tex @@ -28,7 +28,7 @@ \section{Introduction} \begin{frame}{Bulletproofs: just some basic linear algebra} \begin{figure}[h!] \centering - \includegraphics[width=0.7\textwidth]{images/lecture_17/meme.jpg} + \includegraphics[width=0.7\textwidth]{images/lecture_14/meme.jpg} \end{figure} \end{frame} @@ -229,7 +229,7 @@ \section{Inner-product argument} \begin{frame}{Compression step: illustration} \begin{figure}[h!] \centering - \includegraphics[width=0.8\textwidth]{images/lecture_17/compressed.png} + \includegraphics[width=0.8\textwidth]{images/lecture_14/compressed.png} \label{fig:compression} \end{figure} \end{frame} @@ -320,7 +320,7 @@ \section{Inner-product argument} \begin{frame}{Inner-product argument: illustration} \begin{figure}[h!] \centering - \includegraphics[width=0.8\textwidth]{images/lecture_17/ipa.png} + \includegraphics[width=0.8\textwidth]{images/lecture_14/ipa.png} \label{fig:ipa} \end{figure} \end{frame} @@ -340,7 +340,7 @@ \section{Inner-product argument} \begin{frame}{What's next?} \begin{figure}[h!] \centering - \includegraphics[width=0.5\textwidth]{images/lecture_17/ipa_meme.jpg} + \includegraphics[width=0.5\textwidth]{images/lecture_14/ipa_meme.jpg} \end{figure} \end{frame} diff --git a/presentations/14-bulletproofs.pdf b/presentations/14-bulletproofs.pdf index 89f65a2da8084710c498c0bb910324dcbaf57a2d..ef568ff3b4b5ff1053d470108d4f66ce3a619e03 100644 GIT binary patch delta 209962 zcmb5VbCf0BlQ!CA+qSFAUAAr8wtdP*mu=g&ZQJa!U0?s^oo{CDoteAVT{|Q4*%4<& z?))nvDLGppbq?tIfc;>hzfwa;kd#2q@&l@l4O$^;z7Bvp 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