Formal cryptography & information theory

Math

If you want equations and bounds, start here. Fundamentals stay metaphorical; this section turns claims into falsifiable statements under stated axioms.

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~4 min overview · ~10 min formalContinue: Perfect secrecy

Overview

Ordinary web crypto bets that hard math stays hard. Channel Zero authorization asks a different question: does the public transcript even carry secret-relevant information? This section turns that question into Shannon entropy, mutual information, unicity, and named axioms, without dumping formulas in the hero.

Formal mathematics

The mathematical case

Authorization transcripts can be analyzed with entropy, mutual information, and unicity: a security class distinct from hardness clocks. The Math door is not a product tour. It is the ordered path through Shannon perfect secrecy, uniform channels, observer budgets, anti-unicity, axioms, quantum myths, and paradigm comparison.

Formal flavor uses notation such as I(·;·), H(·), and u (notation table). Absolute-sounding lines stay model-scoped; see Claims before treating any equation as a guarantee.

Empty-channel authorization target

I(X;O\mid C)=0

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

Arguments

$X$: : hidden state: private map, key material, and local ceremony state
$O$: : observable public transcript recorded by an observer
$C$: : public challenge or context fixed before 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: This is the section's central authorization claim: surface activity may exist, but the transcript is information-neutral about the secret.

Computational contrast: The claim asks whether secret-relevant transcript information exists, not whether a guessed key is computationally expensive to recover.

References:Channel Zero Information-theoretic authorization

Mutual-information identity

I(A;B)=H(A)-H(A\mid B)

Read as: Information that B gives about A is the uncertainty in A before observation minus the uncertainty left after observing B.

Arguments

$A$: : the variable whose uncertainty is being measured
$B$: : the observation or side information being compared

Operators

$I$: : mutual information
$H$: : entropy or remaining uncertainty
$\mid$: : conditional uncertainty after B is known

Significance: This gives the reader the algebra behind effective leakage: if conditional uncertainty does not drop, the observation taught nothing about the secret.

Computational contrast: A hardness proof can tolerate leaked information if recovery is slow; an information-theoretic claim measures the leak itself.

References:Mutual information

Classical unicity sketch

u \approx \frac{H(K)}{D}

Read as: Unicity distance is roughly keyspace entropy divided by the redundancy available per observed symbol.

Arguments

$u$: : unicity distance: how much ordinary ciphertext collapses ambiguity
$K$: : keyspace whose entropy is being consumed by observations
$D$: : redundancy per symbol that helps rule out wrong keys

Operators

$H$: : entropy of the keyspace
$\approx$: : classical approximation, not a deployment guarantee
$/$: : redundancy consumes ambiguity as observations accumulate

Significance: It previews why ordinary archives become dangerous when redundancy exists, and why anti-unicity pages require empty effective information.

Computational contrast: Computational-bargain systems hope the archive stays hard to solve; anti-unicity asks whether additional transcripts collapse the secret at all.

References:Unicity distance

Technology Math hub: hardness clock versus entropy and mutual-information bounds with seven formal steps — Technology · Math · bounds not hardness clocks
Read left hardness clock versus right entropy and mutual-information panel, then follow the seven-step formal path across the center. The plate frames Math as bounds and model assumptions, not a stronger padlock. Takeaway: Technology Math proves emptiness claims with Shannon-style tools, not with hardness clocks alone.

Two roads

This section is organized by a single contrast. One road bets that hard math stays hard: a computational bargain with a lifespan clock. The other asks whether the public channel even carries secret-relevant information: information-theoretic authorization, security that does not rely on computers being too slow.

Computational hardness versus information-theoretic authorization
Dimension	Hardness clock	Empty channel
Core bet	Problems stay computationally hard	Transcript reveals nothing usable about the secret
Expires when	Machines, algorithms, or archives catch up	Model assumptions fail, not a faster CPU alone
Formal flavor	Complexity / hardness games	Entropy, $I=0$ , unicity under axioms

What's covered

Seven formal children in locked order. Read them this way so later pages can assume the vocabulary is already established.

01
Perfect secrecy (Shannon)
Define the gold standard: posterior equals prior. Mutual information with the secret is zero, even for an unbounded adversary.
02
A flat, unreadable channel
Why every public symbol must be equiprobable given the challenge: flat histograms deny frequency compasses.
03
What observers can measure
Surface bits fill notebooks; effective bits are mutual information with the secret. Capture is not knowledge.
04
Why more data doesn't help
Classical unicity collapses ciphers with redundancy. Authorization transcripts aim at anti-unicity under the model.
05
Axioms, theorems, and bounds
The load-bearing pillars and parameterized reference bounds that make the formal claims falsifiable.
06
Quantum myths vs real bounds
What quantum search needs, and why empty effective information changes the story.
07
Compare security systems
Side-by-side paradigms: hardness clocks versus empty-channel authorization claims.

Prefer a guided reading path? Where to start includes the Math track.

Papers & Claims

Formal pages route to axiom-backed statements and public abstracts, not to product SKUs.

Out of scope

Product SKUs, Gate stage marketing, and patent drawings do not belong here. Engineering stack detail for implementers lives at /technology/stack. Where it ships in the estate is a Products and Security story: Math shows why the claim class is different.