Researcher: Cody Michael Allison — the.wandering.god@gmail.com ORCID: https://orcid.org/0009-0007-7239-6760 Disclosure posture: White hat. SHA-1 is already publicly broken (SHAttered, Stevens et al. 2017); nothing here re-breaks it. Nature of this document: a white paper on one exact algebraic fact — not a research paper, and it keeps no history of superseded results.
In the Cayley–Dickson algebra T₃₂ over GF(2), the trace-Laplacian
Δ(w) = w · 𝟏₃₂ (𝟏₃₂ = 0xFFFFFFFF, the all-ones vector)
is zero iff w is nilpotent (w² = 0), iff w lies on the zero-divisor
nodal line — machine-verified exhaustively at low dimension and over 20 000 random
elements at dim 32. This is exact: it holds for an element or it does not, no
statistics. (Note: 𝟏₃₂ is not a global annihilator — for involutory w,
including the round constants and e₀ itself, w · 𝟏₃₂ = 𝟏₃₂ ≠ 0. It annihilates
exactly the nilpotents. An earlier write-up's "global annihilator" lemma is
retracted; the theorem stands.)
Feed SHA-1's own constants through it. The five initialization constants (the fractional parts of √2, √3, √5, √7, √11) all land on the boundary; the four round constants land as far from it as the 32-dimensional space allows.
IV constant value Δ(w) spectral distance on the locus?
H₀ frac√2 0x67452301 0x00000000 0 ✓ nilpotent
H₁ frac√3 0xEFCDAB89 0x00000000 0 ✓ nilpotent
H₂ frac√5 0x98BADCFE 0x00000000 0 ✓ nilpotent
H₃ frac√7 0x10325476 0x00000000 0 ✓ nilpotent
H₄ frac√11 0xC3D2E1F0 0x00000000 0 ✓ nilpotent
round const value spectral distance on the locus?
K₀ 0x5A827999 32 ✗ maximally far
K₁ 0x6ED9EBA1 32 ✗ maximally far
K₂ 0x8F1BBCDC 32 ✗ maximally far
K₃ 0xCA62C1D6 32 ✗ maximally far
A clean bimodal split, with nothing in between: the constants that name the
hash's initial state collide onto the zero-divisor locus; the constants that do
its arithmetic sit at the opposite extreme. This is the name collision. It is
exact and it is reproducible in seconds — python3 hypercomplex_laplacian.py.
Reproduce the table directly:
from hypercomplex_laplacian import trace_laplacian
for w in (0x67452301, 0xEFCDAB89, 0x98BADCFE, 0x10325476, 0xC3D2E1F0):
assert trace_laplacian(w)['spectral_dist'] == 0 # every IV nilpotent
for w in (0x5A827999, 0x6ED9EBA1, 0x8F1BBCDC, 0xCA62C1D6):
assert trace_laplacian(w)['spectral_dist'] == 32 # every round const maximalThe five IVs are not just individually nilpotent — they span a null subalgebra
of T₃₂/GF(2): every self-product Hᵢ² and every cross-product Hᵢ·Hⱼ annihilates.
Two design choices make this happen and are visible in the constants themselves:
H₀ ⊕ H₂ = 0xFFFFFFFF = 𝟏₃₂
H₁ ⊕ H₃ = 0xFFFFFFFF = 𝟏₃₂
so H₂ = H₀ + 𝟏₃₂ and H₃ = H₁ + 𝟏₃₂ in GF(2). Then, because H₀ is itself on the
nodal line (H₀·𝟏₃₂ = Δ(H₀) = 0, since H₀ is nilpotent — not because 𝟏₃₂
annihilates everything),
H₀·H₂ = H₀·(H₀ + 𝟏₃₂) = H₀² + H₀·𝟏₃₂ = 0 + 0 = 0
and the remaining cross-products are machine-verified. ("Null subalgebra," not "ideal": T₃₂/GF(2) is non-associative, so classical ideal theory doesn't transfer — what's exact is that all internal products vanish.) The consequence, stated plainly: SHA-1 begins its compression function with its entire initial state already on the zero-divisor boundary of the ambient algebra, before a single message bit is read. In these coordinates a collision is a zero-divisor event — two messages whose differential lands on a locus that is describable analytically rather than only by search. That is a structural reading of the known SHAttered result, offered as mathematical context, not a new attack.
UDEO — the Unified Dimensional Entropy Oracle — is the name for this instrument: the trace-Laplacian that reads whether a constant sits on the T₃₂ boundary. The SHA-1 name collision is what it was built to measure, and what it measures exactly.
- It IS: an exact algebraic theorem about Cayley–Dickson algebras over GF(2), and an exact, reproducible observation that SHA-1's five IV constants are nilpotent in T₃₂ while its four round constants are maximally far from the locus — with the IVs spanning a null subalgebra (all internal products vanish).
- It is NOT: a working recovery of any RSA or ECC private key, a signature-forgery capability, an attack on SHA-2 or SHA-3, or a re-break of SHA-1. No currently deployed cryptographic system is shown to be broken by anything in this repository.
- Open, and labelled open: reaching the zero-divisor locus analytically (in polynomial time) rather than by search. Nothing here closes that gap. The value of the name collision is as mathematical structure, not as an exploit.
The correct disclosure vehicle for this content is academic publication — the GF(2) theorem plus the SHA-1 retrospective — not a CVE against unbroken primitives.
| File | What it contains |
|---|---|
hypercomplex_laplacian.py |
The T₃₂/GF(2) trace-Laplacian theorem and the SHA-1 name-collision table. The instrument. |
UDEO_Cryptographic_Vulnerability.md |
The white paper: the tower, the boundary, the T₃₂ embedding of SHA-1, the IV nilpotency theorem. |
sha1_zero_divisor_engine.py |
The name-collision result in the standard one-claim engine format. |
sha1_chladni_figure.py / .png |
The nodal lines of the T₃₂ Laplacian as a Chladni figure — the locus, visualized. |
stix_bundle_udeo.json |
STIX 2.1 bundle at the honest scope (algebraic theorem + SHA-1 retrospective; no exploit provided). |
DISCLOSURE_CHECKLIST.md |
Disclosure process record. |
Gemini_Generated_Image_Breaking_Enigma.png is thematic cover art (Enigma /
Turing) — decorative, not a figure.
White-hat research, provided for defensive and scholarly purposes. See LICENSE.
Researcher: Cody Michael Allison — the.wandering.god@gmail.com