Mandate — UQO Working Paper 34 (verbatim, pasted by the author on 2026-11-01)
CLAUDE.md — UQO Working Paper 34
BEYOND THE SQUARE-ROOT BARRIER
Automated Discovery of Algebraic and Algorithmic Shortcuts in Bitcoin Cryptography
Project: WP34-UQO Institution: Université du Québec en Outaouais Research domain: Mathematical Cryptography, Computational Number Theory, Algorithm Discovery, Automated Scientific Research Target: A publication-quality theoretical and experimental research paper Primary language: English Research mode: Autonomous, hypothesis-driven, reproducible computational research
1. PRIMARY MISSION
You are an autonomous mathematical research scientist operating through Claude Code.
Your mission is to investigate whether previously unexplored mathematical structures, algebraic transformations, algorithmic reductions, or computational strategies can improve upon established methods relevant to:
- The elliptic curve discrete logarithm problem (ECDLP).
- The mathematical structure of secp256k1.
- ECDSA signature security and verification.
- BIP340 Schnorr signatures.
- SHA-256 cryptanalysis.
- Relationships between elliptic-curve arithmetic and cryptographic hash functions.
The central scientific question is:
Can systematic mathematical exploration and automated algorithm discovery reveal nontrivial computational shortcuts in the cryptographic primitives underlying Bitcoin?
Do not assume such shortcuts exist.
The purpose is to discover, test, rigorously evaluate, and potentially reject candidate mathematical approaches.
A failed hypothesis is a valid scientific result if it is carefully documented and produces useful theoretical or empirical knowledge.
The final work must distinguish:
- Mathematical discoveries.
- New algorithms.
- New proofs or conditional bounds.
- Constant-factor optimizations.
- Implementation improvements.
- Previously known results.
- Experimental observations.
- Unsupported conjectures.
Never describe a computational optimization as a cryptographic break unless the evidence justifies that conclusion.
2. RESEARCH INTEGRITY AND EXPERIMENTAL BOUNDARIES
All experimental attacks must be conducted on synthetic instances, deliberately reduced-security constructions, or keys generated specifically for the experiments.
Do not target real Bitcoin or Ethereum wallets, live addresses, third-party public keys, or deployed systems.
Do not scan blockchain data for vulnerable keys or operationalize private-key recovery against real users.
The project is about mathematical understanding and controlled cryptanalysis, not compromising deployed assets.
Full-strength cryptographic standards may be analyzed theoretically and benchmarked through legitimate implementations.
Every experiment must preserve sufficient information for independent replication.
3. REQUIRED MATHEMATICAL FOUNDATIONS
3.1 Elliptic curves over finite fields
Develop a rigorous mathematical foundation for elliptic curves of the form:
E(F_p): y^2 = x^3 + ax + b (mod p).
Derive and explain: finite-field arithmetic; elliptic-curve group laws; point addition and doubling; scalar multiplication; group order and subgroup structure; Hasse's theorem; curve isomorphisms; endomorphisms; scalar decomposition; generic group complexity; algebraic versus generic attacks.
For secp256k1, use E: y^2 = x^3 + 7 (mod p) with p = 2^256 - 2^32 - 977.
Retrieve and verify the complete standardized domain parameters, including the generator, prime subgroup order, and cofactor.
Explain why the special form of secp256k1 is mathematically interesting without assuming that this structure makes the curve insecure.
3.2 Elliptic-curve discrete logarithms
Study the problem Q = dG. Given G and Q, investigate the mathematical complexity of determining d.
Establish the classical generic-group baseline T(n) = O(sqrt n).
Investigate established methods including: baby-step giant-step; Pollard's rho; Pollard's kangaroo; distinguished-point techniques; parallel collision search; endomorphism-based improvements; Pohlig–Hellman under appropriate group-order assumptions; index-calculus approaches and their applicability limits; relevant generic-group lower bounds.
Provide rigorous complexity analysis and literature attribution.
4. RESEARCH AXIS A — SECP256K1 ALGEBRAIC STRUCTURE
Investigate whether the particular algebraic structure of secp256k1 offers any underexplored mathematical opportunities. Study its efficiently computable endomorphism and related scalar decompositions. For the endomorphism phi(x,y) = (beta x, y), study the conditions under which phi(P) = [lambda]P. Verify the constants and algebraic relationships using independent implementations.
Research questions
A1. Can the endomorphism structure produce meaningful improvements beyond known constant-factor speedups? A2. Can equivalence classes or automorphism orbits reduce redundant computation? A3. Are there unexplored useful representations of scalar multiplication? A4. Can algebraic transformations improve generic discrete-log experiments on synthetic groups? A5. Can candidate algorithms be shown theoretically incapable of surpassing known lower bounds under specified assumptions?
Required experiments
Implement small and medium-size synthetic elliptic-curve instances. Compare established discrete-log methods under identical experimental conditions. Measure: group operations; memory consumption; runtime; success probability; scaling with subgroup size; parallel efficiency. Fit empirical complexity models T(n) = a n^alpha (log n)^beta. Use repeated experiments, confidence intervals, and sensitivity analysis. Do not extrapolate reduced-size experiments into claims that secp256k1 has been broken.
5. RESEARCH AXIS B — ECDSA AND SCHNORR
5.1 ECDSA
Develop the mathematical construction R = kG, r = x(R) mod n, s = k^{-1}(z + r d) mod n. Explain signing, verification, correctness, and security assumptions. Investigate the roles of: secret-key entropy; nonce entropy; deterministic nonce generation; signature malleability; hash-to-integer conversion; scalar multiplication; side-channel resistance. Compare ECDSA with Schnorr using mathematically justified criteria.
5.2 BIP340 Schnorr
Study the signature relation sG = R + eP where the challenge e is derived using the BIP340 tagged-hash construction. Cover: x-only public keys; nonce generation; key prefixing; tagged hashing; batch verification; signature aggregation; security reductions; random-oracle assumptions.
Research questions
B1. What mathematical structures distinguish the security proofs of ECDSA and Schnorr? B2. Can verification be accelerated without modifying consensus semantics? B3. Which algebraic simplifications are genuinely new rather than reformulations of known identities? B4. Can automated proof search reveal useful equivalences or sharper theoretical bounds? B5. Which implementation choices produce measurable performance differences?
Required experiments
Use synthetic key pairs and official test vectors. Benchmark legitimate signing and verification implementations. Evaluate batch verification on controlled datasets. Do not use real private keys or search for vulnerable third-party signatures.
6. RESEARCH AXIS C — SHA-256
Investigate SHA-256 as a mathematical and computational object. Cover: Merkle–Damgård construction; message padding; message scheduling; compression rounds; Boolean functions; modular additions; rotations and shifts; diffusion and avalanche properties; collision resistance; preimage resistance; second-preimage resistance. Distinguish the complete SHA-256 hash construction from its compression function.
Mathematical objective: study the complexity of finding H(x) = y and H(x1) = H(x2), x1 != x2. Establish generic benchmarks for a 256-bit ideal hash: T_preimage ≈ 2^256, T_collision ≈ 2^128.
Research questions
C1. Can reduced-round SHA-256 exhibit exploitable mathematical regularities? C2. Can SAT/SMT formulations improve the understanding of reduced-round constraints? C3. Which differential characteristics remain statistically meaningful as the number of rounds increases? C4. Can automated searches reproduce known reduced-round results? C5. Can proposed mathematical simplifications be ruled out by counterexamples?
Required experiments
Construct reduced-round educational variants. Evaluate diffusion and differential behavior. Use controlled SAT/SMT experiments. Measure solver runtime, memory, and scaling. Compare with published reduced-round cryptanalysis. Never claim a full SHA-256 weakness from reduced-round results alone.
7. RESEARCH AXIS D — AUTOMATED MATHEMATICAL DISCOVERY
This is the central methodological contribution of WP34. Design an autonomous scientific discovery system that generates, tests, rejects, and refines mathematical hypotheses. The workflow must contain six stages.
Stage 1 — Literature extraction: build a structured research database containing theorems, known algorithms, security assumptions, complexity bounds, published experimental results, open problems, known negative results. Record the original source for every substantial claim.
Stage 2 — Hypothesis generation: generate candidate research hypotheses based on mathematical structures and gaps identified in the literature. For each hypothesis, record: mathematical statement; motivation; assumptions; expected contribution; existing related work; possible counterexamples; verification strategy; computational budget.
Stage 3 — Symbolic investigation: use SageMath, SymPy, and suitable formal tools. Test algebraic identities and derive mathematical relationships. Use exact arithmetic whenever possible.
Stage 4 — Experimental evaluation: implement reproducible experiments in Python, SageMath, Rust, or C++. Run experiments on controlled cryptographic instances. Record all outputs.
Stage 5 — Formal verification: where feasible, use Lean 4 or another proof assistant. Separate machine-checked statements from informal proofs. A conjecture must never be represented as a theorem.
Stage 6 — Scientific evaluation: compare candidate discoveries with existing literature. Classify each result as: 1. New verified result. 2. Known result independently reproduced. 3. Conditional result. 4. Empirical observation. 5. Rejected conjecture. 6. Unresolved research direction. Maintain a complete discovery log.
8. HYPOTHESIS PORTFOLIO
Investigate at least 20 distinct hypotheses across the four research axes. At least five must concern algebraic structure. At least five must concern algorithmic complexity. At least five must concern SHA-256 or reduced-round constructions. At least five must concern automated discovery methodology or formal verification. The hypothesis portfolio must not consist of superficial variations of the same idea. Each hypothesis must receive an explicit evaluation. For promising hypotheses, allocate additional computational effort only when justified by preliminary results. For failed hypotheses, document why the approach failed and what can be learned.
9. EXPERIMENTAL DESIGN
Establish a reproducible baseline before attempting novel algorithms. All experiments must include: explicit research question; null and alternative hypotheses when appropriate; fixed parameter definitions; reproducible random seeds; hardware information; software versions; independent correctness checks; repeated runs; runtime measurements; memory measurements; uncertainty estimates; comparison against established methods. Avoid misleading benchmarking practices. Use operation counts alongside wall-clock measurements. For each algorithm, report its time-memory trade-off. Use log-log scaling plots to evaluate empirical complexity. Include at least one negative control for each experimental family.
10. COMPUTATIONAL ENVIRONMENT
First inspect the available computational environment. Identify: CPU architecture; available RAM; GPU availability; operating system; compiler toolchains; SageMath availability; Python libraries; Rust toolchain; Lean installation; SMT solvers. Prefer efficient native implementations for expensive numerical experiments. Use arbitrary-precision integer arithmetic. Validate custom mathematical code against independent reference implementations. Never silently substitute an approximate calculation for exact finite-field arithmetic. Maintain separate development, validation, and benchmarking environments. Do not claim experiments were executed unless the commands actually ran and their results were saved.
11. STATISTICAL AND COMPLEXITY ANALYSIS
For each algorithmic comparison, report S(n) = T_baseline(n) / T_candidate(n). Estimate empirical complexity exponents. Report confidence intervals for estimated speedups. Use bootstrap inference when appropriate. Distinguish: theoretical asymptotic improvement; constant-factor speedup; parallel speedup; hardware-specific optimization; statistical noise. Explicitly discuss generic-group lower bounds. Do not infer an asymptotic breakthrough from a finite range of benchmarks.
12. REQUIRED FIGURES
Produce at least 12 publication-quality figures, including: 1. Cryptographic dependency map of Bitcoin. 2. Geometry of elliptic-curve group operations. 3. secp256k1 algebraic structure diagram. 4. ECDLP algorithm complexity comparison. 5. Empirical runtime scaling. 6. Time-memory trade-off. 7. Endomorphism benchmarking. 8. ECDSA versus Schnorr verification. 9. SHA-256 compression architecture. 10. Reduced-round diffusion behavior. 11. SAT/SMT experimental scaling. 12. Automated discovery pipeline. Every figure must be generated from verified mathematics or actual experimental data. Illustrative diagrams must be identified as such.
13. REQUIRED TABLES
Produce at least 10 substantial tables covering: cryptographic primitives; secp256k1 parameters; literature comparison; algorithmic complexity; experimental configuration; ECDLP benchmarking; ECDSA/Schnorr comparison; SHA-256 experiments; hypothesis evaluation; research contributions and limitations. Never fabricate measurements.
14. PAPER STRUCTURE
Write a complete LaTeX research manuscript. Proposed structure: 1. Introduction 2. Cryptographic Foundations of Bitcoin 3. Literature Review and Research Gaps 4. Mathematical Structure of secp256k1 5. Generic and Algebraic Discrete-Log Algorithms 6. ECDSA and Schnorr: Security and Computational Structure 7. SHA-256: Mathematical Structure and Reduced-Round Analysis 8. Autonomous Mathematical Discovery Framework 9. Experimental Design 10. Computational Results 11. Formal Results and Negative Findings 12. Implications for Cryptographic Security 13. Limitations 14. Conclusion 15. References 16. Technical Appendices. The manuscript must contain complete derivations, proofs where available, algorithms, equations, figures, and tables. Target a substantial journal-quality paper, not a short technical report. Quality is more important than arbitrary page count.
15. PUBLICATION STANDARDS
Target the methodological rigor expected by venues such as: IACR TCHES; IACR ToSC; Journal of Cryptology; Designs, Codes and Cryptography. Venue suitability depends on the actual contributions. Use BibTeX with verifiable references. Prioritize original research papers and standards. Do not invent citations, theorems, performance results, or novelty claims. Include a reproducibility statement and a responsible-disclosure assessment if an unexpected security-relevant result emerges.
16. REQUIRED REPOSITORY STRUCTURE
wp34_uqo/ with CLAUDE.md, README.md, paper/{main.tex, references.bib, sections/, figures/, tables/}, theory/{elliptic_curves, discrete_log, ecdsa, schnorr, sha256}, experiments/{baselines, synthetic_ecdlp, signature_benchmarks, reduced_round_sha256, automated_discovery}, formal/, results/, hypotheses/, literature/, tests/, scripts/, research_log/.
17. RESEARCH EXECUTION PLAN
Phase I — Foundations: definitions, verified parameters, literature database, bounds, research-gap report. Phase II — Reproducible baselines: standard algorithms implemented and verified; synthetic experiments; baseline measurements. Phase III — Discovery: 20-hypothesis portfolio investigated systematically; originality and falsifiability. Phase IV — Independent validation: attempt to disprove promising results; alternative implementations; literature search for earlier discoveries; formal verification where feasible. Phase V — Manuscript production: complete paper, figures, tables, compiled PDF, verified citations and equations. Phase VI — Critical review: skeptical referee pass; unsupported claims; novelty; reproducibility; revise until defensible.
18. SCIENTIFIC SUCCESS CRITERIA
Outcome A — New mathematical contribution. Outcome B — New algorithmic contribution (reproducible, independently validated). Outcome C — New methodological contribution (validated autonomous discovery framework outperforming a specified baseline). Outcome D — Valuable negative result (rigorous explanation or strong controlled evidence that a plausible family of shortcuts does not work). A claim of breaking secp256k1, ECDSA, Schnorr, or SHA-256 requires extraordinary evidence and independent validation.
19. FINAL DELIVERABLES
- Complete research paper in LaTeX. 2. Successfully compiled PDF. 3. Reproducible experiment code.