Information Leakage in Zero-Error Source Coding: A Graph-Theoretic Perspective

TL;DR

This paper analyzes the amount of information leakage in zero-error source coding using graph theory, providing a single-letter characterization of maximum leakage and extending it to more general adversarial models.

Contribution

It introduces a single-letter formula for maximum leakage in zero-error source coding and proposes optimal coding schemes, extending the analysis to complex adversarial scenarios.

Findings

Optimal normalized leakage equals the fixed-length source coding rate.

Deterministic coding schemes can achieve both optimal leakage and compression.

Extended bounds for leakage under multiple guesses and distortion are derived.

Abstract

We study the information leakage to a guessing adversary in zero-error source coding. The source coding problem is defined by a confusion graph capturing the distinguishability between source symbols. The information leakage is measured by the ratio of the adversary's successful guessing probability after and before eavesdropping the codeword, maximized over all possible source distributions. Such measurement under the basic adversarial model where the adversary makes a single guess and allows no distortion between its estimator and the true sequence is known as the maximum min-entropy leakage or the maximal leakage in the literature. We develop a single-letter characterization of the optimal normalized leakage under the basic adversarial model, together with an optimum-achieving scalar stochastic mapping scheme. An interesting observation is that the optimal normalized leakage is equal to the optimal compression rate with fixed-length source codes, both of which can be simultaneously achieved by some deterministic coding schemes. We then extend the leakage measurement to generalized adversarial models where the adversary makes multiple guesses and allows certain level of distortion, for which we derive single-letter lower and upper bounds.