Synonym-channel uniformity · P2

A flat, unreadable channel

If every public symbol shows up equally often no matter the secret, counting symbols reveals nothing.

~6 min overview · ~5 min formalContinue: Observer bit budget

What “flat” means

Frequency attacks are classical cryptanalysis: skewed letter counts become a compass toward the secret. Uniformity is an information-theoretic structural denial, not “a larger key.” Conditional on the public challenge, every public response symbol is designed to be equiprobable.

A flat unreadable channel: six equal-height bearing bars with no frequency compass — Uniform marginals · no lean → no leak
Read the six equal-height bearing bars as equiprobable marginals under the challenge: no bar leans, so frequency analysis finds no compass. A crossed-out frequency cue marks the denied attack. Takeaway: uniform channel statistics keep the public transcript flat so lean cannot become leak.

In the reference model with six bearings, that reads: P(W=w | C) = 1/6 for each public symbol W given challenge C. This is axiom synonym-channel uniformity (P2): a load-bearing pillar, not a visual preference (notation).

Conditional on the challenge, every public response symbol is equiprobable.ValidatedAxiom P2 (synonym-channel uniformity) defeats frequency analysis on the transcript.

Why skew breaks secrecy

Toy counterexample: if one zone is chosen more often when the secret has a certain property, the histogram leans. That lean is a compass. An observer who tallies public symbols starts learning something about the hidden choice, even without breaking any hard math problem.

Engineering must preserve balance under the model. This page states the requirement conceptually, not a manufacturing recipe. Deployment skew (biased randomness, broken zone selection, leaked maps) is a model-scope failure mode, not a “try harder later” hardness clock.

Uniformity vs zero mutual information

Uniform public symbols are necessary structure for emptiness claims. Axiom P2 (flat marginals) and axiom P3 (zero mutual information with the secret) are related but not identical. Flatness denies frequency analysis; zero $I(X;O\mid C)$ is the perfect-secrecy pillar for observers; see Perfect secrecy (Shannon). How much an archive can still “measure” is the observer budget story at What observers can measure.

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

Channel language

Prefer channel for the public transcript path an observer can record. The empty-channel story (capture is not knowledge) lives at Channel Zero: When the channel reveals nothing. Classical attack families that assume skewed ciphertext miss this model; see Why classical attacks fail.

Parameter notes

The reference architecture often uses |B|=6 (six zones / six bearings). That cardinality is a reference parameter, not sacred. What matters for the uniformity claim is equiprobability given the challenge, whatever the alphabet size under the axioms. Full axiom tables and bounds live at Axioms, theorems, and bounds.

Model scope: Claims.

Formal mathematics and notation

Formal uniformity claim

Conditional on the public challenge C, every public response symbol B is designed to be equiprobable. Frequency analysis finds no compass. Combined with Channel Zero emptiness:

Flat public-symbol marginal

P(W=w\mid C)=1/|B|

Read as: Given challenge C, each public bearing symbol is equally likely.

Arguments

$W$: : public bearing witness emitted on the channel
$w$: : one bearing value an observer can count
$B$: : bearing alphabet or public response set
$C$: : public challenge conditioning the marginal

Operators

$P$: : probability of the observable bearing
$\mid$: : conditioned on the challenge
$1/|B|$: : uniform probability across the bearing set

Significance: This is the local P2 condition: counting bearings should not reveal a preferred secret direction.

Computational contrast: A computational channel can leak biased residue and rely on key-recovery cost; a flat channel first removes the frequency compass.

References:Synonym-channel uniformity

Combined empty-channel target

I(X;O\mid C)=0

Read as: Given challenge C, observable transcript O reveals no mutual information about hidden state X.

Arguments

$X$: : hidden state: private map, secret choice, key material, or local ceremony state
$O$: : observable public transcript or archive
$C$: : public challenge or context conditioning the observation

Operators

$I$: : mutual information between hidden state and observation
$\mid$: : conditioning on the public challenge
$= 0$: : zero effective leakage under the stated model

Significance: Uniform marginals support the stronger Channel Zero claim that the transcript carries no effective information about the secret.

Computational contrast: The claim is about zero transcript leakage, not the work factor for decoding a biased or reusable credential.

References:Channel Zero Information-theoretic authorization

Flat marginals are axiom P2; emptiness is P3 (notation). Scope: Claims.