ecdlp-breakthrough

Research and experimental implementations of novel approaches to the Elliptic Curve Discrete Logarithm Problem (ECDLP).
Explorations include KRAKEN-4D, ISLE, S-UNIT-LOG, TWIST-WALK, and constrained-search frameworks.

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ECDLP Breakthrough — Research Repository

Short: Research on constrained-search approaches to the ECDLP — instance-aware filters + statistical validation + constrained random-walks.
Research only. No exploit code. No third-party wallet targeting. Contributions welcome under the ethical rules below.

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Table of Contents

• Abstract
• Background & Motivation
• Formal Problem Statement
• Core Idea & Novelty
• Mathematical Formulation
• Filter Taxonomy
• Filter Soundness & Validation
• Cost Model & Break-Even Condition
• Experimental Design & Reproducibility
• Ledger Schema
• Run Examples (Conceptual)
• Research Directions: KRAKEN-4D, ISLE, S-UNIT-LOG, TWIST-WALK
• Open Problems & Research Agenda
• Ethics, Legal & Safety Statement
• Contributing
• License & Citation

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Abstract

We investigate whether instance-specific side information and empirically observed fingerprints of key-generation can be used to safely restrict the scalar search space for ECDLP instances.
Our approach combines sound residue/structural filters, statistical validation harnesses, and constrained-search strategies (conceptual Pollard-style adaptations).
The emphasis is on reproducible research and strict filter validation — not operational exploitation.

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Background & Motivation

Classical ECDLP attacks (Pollard’s rho, baby-step/giant-step) assume uniformly random private scalars and operate with (O(\sqrt{n})) complexity.
In practice, some real-world instances may carry auxiliary information (metadata, keygen quirks, statistical fingerprints).
We ask: can such information be transformed into necessary conditions that safely reduce the effective search space?

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Formal Problem Statement

Let:

• (\mathbb{E}/\mathbb{F}_p) be an elliptic curve (e.g., secp256k1) with generator (G) of order (n),
• (d \in \mathbb{Z}_n) be the unknown private scalar,
• (Q = dG) be the observed public point.

Goal: identify a filter family (F={f_1,\dots,f_m}), each (f_i:\mathbb{Z}_n\to{\text{true},\text{false}}), such that the surviving scalar set

$$ \mathcal{S}_F = {, d' \in \mathbb{Z}_n ;\mid; \forall f_i \in F,; f_i(d') = \text{true} ,} $$

satisfies (|\mathcal{S}_F| \ll n) and (critically) (d \in \mathcal{S}_F).

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Core Idea & Novelty

• Instance-aware shaping: using metadata and fingerprints to prune the space.
• Soundness-first: filters must prove 100% retention on diverse corpora before use.
• Constrained random-walks: Pollard-style walks restricted to (\mathcal{S}_F).
• Cost model: explicit break-even condition ensures filters save more than they cost.

This is not an algebraic break of secure curves; it’s a disciplined search-space shaping paradigm with rigorous validation and provenance.

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Mathematical Formulation

Residue filter (example):

$$ f_{\mathrm{res}}^{(m,R)}(d) := d \bmod m \in R $$

Effective cost per surviving EC multiply:
Let (T_{\mathrm{EC}}) be the average time for one EC scalar multiply, (T_f) the time to evaluate all filters per candidate scalar, and (\rho) the fraction rejected by filters. Then:

$$ C_{\mathrm{eff}} = \frac{T_f + (1-\rho),T_{\mathrm{EC}}}{1-\rho} $$

Break-even condition: deploy filters only if

$$ C_{\mathrm{eff}} < T_{\mathrm{EC}} \quad\Longleftrightarrow\quad T_f < \rho , T_{\mathrm{EC}} $$

Filter soundness requirement (operational): for a validation corpus (\mathcal{K}),

$$ \forall d' \in \mathcal{K}, \quad d' \in \mathcal{S}_F $$

In practice we demand 100% retention on a diverse, documented test corpus before filters are used.

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Filter Taxonomy

• Modular / residue filters: (d \bmod m \in R). Fast and deterministic.
• Public-key congruence filters: constraints derived from small-modulus behavior of (Q) or (x(Q)) projections (only when provable).
• Statistical / heuristic filters: empirical fingerprints (e.g., non-uniform bits from a particular keygen path). Must be validated before use.
• Metadata-derived constraints: conservative transformations of on-chain or timing metadata into scalar constraints.

Note: Heuristic filters are permitted only after passing the validation harness and being recorded in the ledger.

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Cost Model & Break-Even Condition

Measure baseline (T_{\mathrm{EC}}) (the average time for one EC scalar multiply),
filter cost (T_f) (time to evaluate all filters per candidate),
and reject rate (\rho) (fraction of scalars rejected by the filters).

A filter family is beneficial only if:

$$ T_f < \rho , T_{\mathrm{EC}} $$

In other words, the cost of applying the filters must be lower than the savings from the fraction of expensive EC operations they prevent.

Run reproducible A/B benchmarks on the same shard and hardware to verify this condition empirically.

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Experimental Design & Reproducibility

Every run must record:

• Commit hash (code provenance)
• Environment snapshot (uname -a, compiler, CUDA, driver versions, nvidia-smi)
• Dataset checksum (md5/sha256)
• RNG seed (if applicable)
• Ledger row (see schema below)

Prefer Docker/OCI for reproducibility.

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Ledger Schema

CSV header:

timestamp,commit\_hash,node\_id,shard\_start,shard\_end,seed,filter\_flags,iterations,keys\_per\_sec,rejects\_per\_sec,disk\_io\_MBps,git\_branch,notes

Example:

2025-09-18T12:00:00Z,abc1234,my-node,0x00000000,0x0000FFFF,42,MOD661|MOD499,1e7,298000000,120000000,108,main,"baseline GTX1070"

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Run Examples (Conceptual)

Validation:

python3 filters/validate.py --filter=MOD661,MOD499 --keys=test_keys.csv --report=report.json

• Microbench: run baseline vs. filtered run and compare effective keys/sec using the ledgered metrics.

• Heatmap:

  python3 heatmap/render.py --ledger ledger/latest.csv --out heatmap.png

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Research Directions: KRAKEN-4D, ISLE, S-UNIT-LOG, TWIST-WALK

In addition to constrained-search filters, this repo explores:

• KRAKEN-4D: 4D GLV/GLS decomposition applied to Pollard’s Kangaroo.
• ISLE: Isogeny-class exploration of secp256k1 for alternative curve embeddings.
• S-UNIT-LOG: Algebraic number theory (S-units, CM fields) to derive scalar constraints.
• TWIST-WALK: Quadratic twist random walks for dual-space mixing.

All tested first on toy curves (48–56 bits) before scaling.

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Open Problems & Research Agenda

• Provable necessary conditions from observables.
• Mixing/collision analysis of constrained random walks.
• Optimal filter subset selection under a soundness constraint.
• Public benchmark corpus for validation (owned/synthetic keys only).

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Ethics, Legal & Safety Statement

This repository is research-only.

• No exploit code.
• Not intended to compromise real wallets.
• Use only with keys you control or on testnets.
• Unauthorized key-recovery attempts are illegal and unethical.

If you discover a practical reduction, disclose responsibly via academic or security channels.

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Contributing

Rules for PRs:

1. No exploit code.
2. Filters must ship with a validation harness + test corpus (100% retention).
3. Document provenance (commit hash, env, ledger).
4. Add ETHICS.md if touching sensitive ground.

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License & Citation

MIT (suggested).

Citation (example):

Ghostaxe & collaborators, "ECDLP Breakthrough — Constrained-Search Research", GitHub repo, 2025.

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Final Checklist Before Runs

• Filters validated 100%? ✅/❌
• Commit hash logged? ✅/❌
• Env snapshot stored? ✅/❌
• Dataset checksummed? ✅/❌
• Experiment restricted to test keys/testnet? ✅/❌

If any ❌ → stop.

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Authors & Credits

• SteelGhost17 (Lead Researcher) — founder & primary developer.
• Open collaboration: PRs and discussions welcome.