Successive Refinement of Shannon Cipher System Under Maximal Leakage

TL;DR

This paper extends the Shannon cipher system to a successive refinement setting under maximal leakage secrecy, characterizing optimal leakage regions and proposing coding schemes for discrete sources with distortion constraints.

Contribution

It generalizes the threat model to multiterminal settings and provides the first characterization of maximal leakage regions under joint and expected distortion constraints.

Findings

Optimal normalized maximal leakage regions characterized for both JEP and expected distortion.

Type-based coding scheme achieves the optimal leakage bounds.

Normalized maximal leakage regions are identical under certain conditions.

Abstract

We study the successive refinement setting of Shannon cipher system (SCS) under the maximal leakage secrecy metric for discrete memoryless sources under bounded distortion measures. Specifically, we generalize the threat model for the point-to-point rate-distortion setting of Issa, Wagner and Kamath (T-IT 2020) to the multiterminal successive refinement setting. Under mild conditions that correspond to partial secrecy, we characterize the asymptotically optimal normalized maximal leakage region for both the joint excess-distortion probability (JEP) and the expected distortion reliability constraints. Under JEP, in the achievability part, we propose a type-based coding scheme, analyze the reliability guarantee for JEP and bound the leakage of the information source through compressed messages. In the converse part, by analyzing a guessing scheme of the eavesdropper, we prove the optimality of our achievability result. Under expected distortion, the achievability part is established similarly to the JEP counterpart. The converse proof proceeds by generalizing the corresponding results for the rate-distortion setting of SCS by Schieler and Cuff (T-IT 2014) to the successive refinement setting. Somewhat surprisingly, the normalized maximal leakage regions under both JEP and expected distortion constraints are identical under certain conditions, although JEP appears to be a stronger reliability constraint.