Research
Formal security
Four pillars stated as axioms, the theorems they imply, and the parameterized bounds used on the public site. This page does not publish gap analysis or residual-risk inventories (those belong in the Deployment Security Package under NDA).
Given the public challenge, the observable transcript carries zero Shannon mutual information about the secret.ValidatedAxiom P3 / perfect-secrecy pillar: structural absence of secret-relevant information in O, not merely computational hardness.Holds under the reference architecture
A transcript from one freshness epoch is not accepted in another.ValidatedAxiom P4 binds each proof to tau/nonce; burn-before-validate spends the nonce before twin validation.Holds under the reference architecture
Axioms (four pillars)
P1
Path-complete geometric projection
The public challenge enumerates every selector path any secret in the family could take. Completeness is required so that choosing one walk over another does not itself leak information.
P2
Uniform synonym marginals
Conditional on the public challenge, every public symbol is equiprobable. Frequency analysis finds no direction in the transcript.
P3
Zero mutual information
Given the challenge, the observable transcript carries zero Shannon mutual information about the secret index, the private synonym, and the selector path. The information was never present, not merely expensive to extract.
P4
Session-bound no-replay
Every proof event is bound to freshness (tau / nonce). A transcript from session τ_a is not accepted in τ_b ≠ τ_a.
Theorems
Synonym-channel uniformity
Under the private synonym morphism, each observable symbol is uniform over the response alphabet given the challenge leaf.
Perfect secrecy (observer)
I(secret ; observation | challenge) = 0. Equivalent in spirit to Shannon perfect secrecy on the transcript channel.
Replay resistance
Verify(O_τa, τ_b) accepts with probability 0 for τ_a ≠ τ_b when freshness binding and nonce burn are enforced.
Parameterized bounds
| Parameter | Value | Note |
|---|---|---|
| H_surface / character | log₂ |B| = log₂ 6 ≈ 2.585 bits | Observable bearing entropy only. |
| I_effective | 0 bits | Mutual information with the secret under the axioms. |
| Ring sizes (reference) | 26 + 26 + 10 = 62 symbols | Disjoint uppercase, lowercase, digit rings. |
| Zone / bearing cardinality | |B| = 6 | Six zones, six bearings, 6! color–bearing bijections in the full construction. |
Absolute mechanism claims on product pages assume the reference architecture.