Distributions Attaining Secret Key at a Rate of the Conditional Mutual Information

TL;DR

This paper characterizes when the secret key rate equals the conditional mutual information across different communication scenarios, providing necessary conditions and operational interpretations for such distributions.

Contribution

It offers a complete solution for secret key distillation under no-communication and introduces conditions for achieving the key rate with one- or two-way communication.

Findings

Distributions with key rate equal to conditional mutual information are characterized.

Secret key rate equals $H(S|Z)$ with shared randomness, where $S$ is Gács-Körner Common Information.

Two-way communication can be necessary to achieve the optimal key rate in certain distributions.

Abstract

In this paper we consider the problem of extracting secret key from an eavesdropped source $p_{XYZ}$ at a rate given by the conditional mutual information. We investigate this question under three different scenarios: (i) Alice ($X$) and Bob ($Y$) are unable to communicate but share common randomness with the eavesdropper Eve ($Z$), (ii) Alice and Bob are allowed one-way public communication, and (iii) Alice and Bob are allowed two-way public communication. Distributions having a key rate of the conditional mutual information are precisely those in which a "helping" Eve offers Alice and Bob no greater advantage for obtaining secret key than a fully adversarial one. For each of the above scenarios, strong necessary conditions are derived on the structure of distributions attaining a secret key rate of $I(X:Y|Z)$. In obtaining our results, we completely solve the problem of secret key distillation under scenario (i) and identify $H(S|Z)$ to be the optimal key rate using shared randomness, where $S$ is the G\'acs-K\"orner Common Information. We thus provide an operational interpretation of the conditional G\'acs-K\"orner Common Information. Additionally, we introduce simple example distributions in which the rate $I(X:Y|Z)$ is achievable if and only if two-way communication is allowed.