Successive Refinement of Shannon Cipher System Under Maximal Leakage

TL;DR

The paper addresses the secrecy-reliability tradeoff in a multiterminal Shannon cipher system under the maximal leakage metric for discrete memoryless sources with distortion constraints. It develops inner and outer leakage-region characterizations for both joint excess-distortion probability and expected distortion through a type-based achievability and a guessing-based converse, and then extends the converse to the expected-distortion setting via a causal-disclosure framework. Key findings include explicit region bounds and conditions under which the inner and outer bounds coincide, revealing that maximal leakage can be tightly controlled in a successive refinement setup with partial secrecy. The results provide insights for designing multi-user secret communications with keys, showing when successive refinability preserves secrecy performance and highlighting the role of key rates and rate-distortion constraints in shaping leakage.

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.

Figures (4)

• Figure 1: Successive refinement of Shannon cipher system with an eavesdropper.
• Figure 2: Illustration of the boundaries of maximal leakage region $(\Lambda_1,\Lambda_2)$ with respect to the JEP constraint $\alpha$ for $P=\mathrm{Bern}(0.3)$, $D_1=0.2$, $D_2=0.1$, $r_1=0.06$ and $r_2=0.1$.
• Figure 3: Illustration of $\Lambda_1$ with respect to $\alpha$ and $r_1$ for $P=\mathrm{Bern}(0.3)$ and $D_1=0.2$. The slice of $\Lambda_1$ is determined by a fixed key rate $r_1=0.06$.
• Figure 4: Illustration of the causal disclosure setting for Eve.

Theorems & Definitions (24)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Theorem 1
• Remark 1
• Remark 2
• Remark 3
• Corollary 2
• Remark 4