Analysis of nonce defects ECDSA secp256k1

Stratification Analysis of Nonce Defects in ECDSA on secp256k1

ECDSA signatures on the secp256k1 curve are analyzed as phase corpora, where nonce generation defects manifest as persistent families (defect-family), rather than isolated failures. Transitioning to (u_r, u_z) coordinates enables the detection of stratified patterns through toric geometry and kNN search. The study confirms the transferability of repeated-r patterns in 58 out of 58 reconstruction tests with full signature validation, and no cross-address collisions in external corpora.

Standard ECDSA uses a nonce k to generate point R = k·G, from which r = x(R) mod n. The s component is computed as s = k^{-1} · (z + r · d) mod n. Verification introduces w = s^{-1} mod n, u_z = z · w mod n, u_r = r · w mod n, with recovery of R' = u_z · G + u_r · Q, where x(R') mod n = r.

These (u_r, u_z) coordinates project the signature onto the torus Z_n × Z_n, revealing geometric metrics of defective modes.

Theoretical Foundations and the Role of the Nonce

The nonce k is critical: its repetition (repeated-r) indicates a systemic generator defect, not a random failure. In the phase representation PhasePoint = (u_r, u_z, verification_x, verification_y_raw, branch_y), PhaseCorpus aggregates points from a single address context.

Key ECDSA relationships:

Q = d · G
R = k · G, r = x(R) mod n
s = k^{-1} · (z + r · d) mod n

w = s^{-1} mod n
u_z = z · w mod n, u_r = r · w mod n
R' = u_z · G + u_r · Q
x(R') mod n = r

Repetition of r occurs with repeated k, making u_r, u_z markers of a defect-family. The method combines ECDSA algebra with persistent homology and permutation significance tests.

| Notation | Description | Role in Analysis |

|-------------|----------|---------------|

| G | Base point | Generation of R and Q |

| n | Subgroup order | Computation modulus |

| d | Private key | Signer's secret |

| k | Nonce | Source of defects |

| (r,s) | Signature | Observable data |

| (u_r, u_z) | Phase coordinates | Pattern detection |

Experimental Design and Materials

Object — secp256k1 signature corpora as sets of PhasePoint. Subject — defect-family signatures: repeated-r, geometry of (u_r, u_z), transferability.

Hypothesis: defects leave stratified traces on the torus, with repeated-r being the surface of a deeper mode.

Methodology:

• Transition to phase coordinates.
• kNN detector and corpus resultants.
• Synthetic replay on controlled data.
• Publication-safety audit.

Corpus: 30 address contexts, 6257 signatures. Repeated-r in 1 context, cross-address r collisions — 0. Property sweep confirmed transferability for 4 scenarios, reconstruction — 58/58 with ECDSA validation. State entanglement — 0 detections. Audit identified 498 issues (30 critical), blocking publication of sensitive data.

Comparison with commercial tools:

| Characteristic | Commercial (CertiK, Hacken) | ECDSA-Stratification-Suite |

|-----------------|--------------------------------|-----------------------------|

| Task | Contract vulnerabilities | Defect-family in nonce |

| Apparatus | Fuzzing, verification | Toric geometry, kNN |

| Output | Vulnerability reports | Mode classification |

| Audience | Developers, exchanges | Laboratories, standards bodies |

Results and Proven Assertions

Proven transfer of defect-family from a real donor to 58 address targets with cryptographic validation. Spontaneous transfer absent in external corpora.

Limitations: mathematical transferability proven, but publication excludes operational steps, raw r/s/z, recovery scripts. Aggregates sanitized for reproducibility.

Code snippets for synthetic replay (safe):

import numpy as np
# Example PhasePoint generation (sanitized)
def phase_point(r, s, z, Q):
    n = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141  # secp256k1
    w = pow(s, -1, n)
    u_z = (z * w) % n
    u_r = (r * w) % n
    return (u_r, u_z)

Permutation significance checks on synthetic data confirm robustness.

Reproducibility and Publication Safety

The project defines interpretation boundaries: scientific rigor does not lower the exploitation threshold. Synthetic-only control and audit ensure a safe pipeline.

Scientific novelty: integration of ECDSA algebra, toric geometry of Z_n × Z_n, corpus analysis, and publication-safety.

Key Takeaways:

• Repeated-r is an indicator of a defect-family, transferable in 58/58 tests.
• Phase coordinates (u_r, u_z) reveal stratification on the torus.
• No cross-address r collisions in external corpora.
• Publication-safety blocks 30 critical risks.
• Method applicable for classifying nonce generators without disclosing vulnerabilities.

— Editorial Team