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general-finite-wiretap-channel

General finite wiretap channel

The unconstrained finite wiretap channel has a single-auxiliary secrecy-capacity characterization without a degradedness assumption.

Wiretap Finite alphabet Discrete memoryless Secrecy Secrecy capacity Exact Single-letter characterization

Channel and question

Input
Symbols in a finite alphabet \(\mathcal X\), encoded using private randomness.
Output
Legitimate output \(Y\) and eavesdropper output \(Z\) from two DMC marginals with common input.
Law
Arbitrary finite transition laws \(W_Y(y\mid x)\) and \(W_Z(z\mid x)\).
Quantity
Strong-secrecy capacity \(C_s\), measured in secret bits per channel use.

Criterion. Vanishing legitimate average error and strong information-theoretic secrecy.

  • Messages are uniform and the stochastic encoder randomness is private.
  • Legitimate average decoding error vanishes.
  • Strong secrecy means unnormalized leakage \(I(M;Z^n)\) tends to zero.

Current status

\[C_s=\max_{V-X-(Y,Z)}\bigl[I(V;Y)-I(V;Z)\bigr]\]

Conditions. Finite alphabets and no input-cost constraint.

ResultRelationMethodYear
Exact\(C_s=\max_{V-X-(Y,Z)}[I(V;Y)-I(V;Z)]\)Random binning and an auxiliary-random-variable converse.1978

Lean formalization

Canonical statementStatement

Version 1 · Lean. Strong and weak secrecy predicates and capacities are separate; the statement uses the unconstrained one-auxiliary value.

Substantial proofs0 linked

No external Lean proof is registered. Proofs longer than roughly 50 lines or requiring problem-specific infrastructure should live in a dedicated repository and link back to this statement version.

References

  1. Imre Csiszár and János Körner (1978). Broadcast Channels with Confidential Messages. IEEE Transactions on Information Theory. DOI 10.1109/TIT.1978.1055892.
  2. Sreejith Sreekumar, Alexander Bunin, Ziv Goldfeld, Haim H. Permuter, and Shlomo Shamai (2020). The Secrecy Capacity of Cost-Constrained Wiretap Channels. arXiv preprint.

Discussion

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Related problems

Degraded Gaussian wiretap channel

A Gaussian receiver has a lower noise variance than the eavesdropper, yielding a closed-form secrecy capacity.

Wiretap Continuous alphabet Gaussian Degraded Secrecy Power constraint Secrecy capacity Exact
Solved \(C_s=\frac12\log_2\!\left(1+\frac{P}{\sigma_1^2}\right)-\frac12\log_2\!\left(1+\frac{P}{\sigma_2^2}\right)\)
Degraded discrete memoryless wiretap channel

A sender communicates reliably to a legitimate receiver while hiding the message from a degraded eavesdropper.

Wiretap Finite alphabet Discrete memoryless Degraded Secrecy Secrecy capacity Exact Single-letter characterization
Solved \(C_s=\max_{P_X}\bigl[I(X;Y)-I(X;Z)\bigr]\)
Binary Z-channel Z-channel

One binary symbol is transmitted perfectly while the other can flip in only one direction.

Point-to-point Binary Finite alphabet Discrete memoryless Asymmetric Capacity Exact
Solved \(C_Z(p)=\log_2\!\left(1+(1-p)p^{p/(1-p)}\right)\)
Binary erasure channel BEC

Each transmitted bit is received correctly or replaced by a visible erasure symbol.

Point-to-point Binary Finite alphabet Discrete memoryless Symmetric Erasure Capacity Exact
Solved \(C_{\mathrm{BEC}}(\varepsilon)=1-\varepsilon\)