The Henchman Problem: Measuring Secrecy by the Minimum Distortion in a List

TL;DR

This paper introduces a new secrecy measure based on rate-distortion theory, analyzing the minimum distortion over lists of reconstructions and characterizing the tradeoffs involving secret key, list rate, and eavesdropper distortion.

Contribution

It proposes a novel secrecy measure using list-based distortion and characterizes the optimal tradeoffs in the Shannon cipher system with side information and a henchman.

Findings

Characterization of the secrecy-distortion tradeoff in list-based eavesdropper models.

Solution to lossy compression of codewords from a random codebook.

Analysis of the legitimate receiver's distortion with allowed distortion.

Abstract

We introduce a new measure of information-theoretic secrecy based on rate-distortion theory and study it in the context of the Shannon cipher system. Whereas rate-distortion theory is traditionally concerned with a single reconstruction sequence, in this work we suppose that an eavesdropper produces a list of $2^{nR_{\sf L}}$ reconstruction sequences and measure secrecy by the minimum distortion over the entire list. We show that this setting is equivalent to one in which an eavesdropper must reconstruct a single sequence, but also receives side information about the source sequence and public message from a rate-limited henchman (a helper for an adversary). We characterize the optimal tradeoff of secret key rate, list rate, and eavesdropper distortion. The solution hinges on a problem of independent interest: lossy compression of a codeword drawn uniformly from a random codebook. We also characterize the solution to the lossy communication version of the problem in which distortion is allowed at the legitimate receiver. The analysis in both settings is greatly aided by a recent technique for proving source coding results with the use of a likelihood encoder.