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

## 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 Gács-Körner Common Information. We thus provide an operational interpretation of the conditional Gács-Körner 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.

## Keywords

Information-theoretic security Public key agreement Gács-Körner Common Information## References

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