CCapacity Atlas
gaussian-wiretap-channel

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

Channel and question

Input
\(X\in\mathbb R\) with average power \(P\).
Output
\(Y=X+N_1\) and \(Z=X+N_2\).
Law
Gaussian noise variances satisfy \(\sigma_1^2<\sigma_2^2\).
Quantity
Secrecy capacity \(C_s\), measured in secret bits per channel use.

Criterion. Reliable communication with vanishing leakage.

  • The eavesdropper channel is degraded.
  • Weak or strong secrecy gives the same asymptotic value under standard formulations.

Current status

\[C_s=\frac12\log_2\!\left(1+\frac{P}{\sigma_1^2}\right)-\frac12\log_2\!\left(1+\frac{P}{\sigma_2^2}\right)\]
ResultRelationMethodYear
Lower\(C_s\ge C(\sigma_1^2)-C(\sigma_2^2)\)Gaussian stochastic encoding and binning.1978
Upper\(C_s\le C(\sigma_1^2)-C(\sigma_2^2)\)Degraded secrecy converse and Gaussian extremality.1978

Lean formalization

Canonical statementNone

Version 1 · Lean. A future continuous secrecy layer should share definitions with the discrete wiretap channel.

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. S. K. Leung-Yan-Cheong and Martin E. Hellman (1978). The Gaussian Wire-Tap Channel. IEEE Transactions on Information Theory. DOI 10.1109/TIT.1978.1055917.

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Related problems

Degraded discrete memoryless wiretap channel

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