A Framework of Distributed Source Encryption using Mutual Information Security Criterion and the Strong Converse Theorem

TL;DR

This work analyzes distributed secure source coding with a common-key cryptosystem under the standard mutual-information secrecy criterion. It leverages the information-spectrum method and a variant of the Birkhoff-von Neumann theorem to derive explicit inner and outer bounds on the reliable-and-secure rate region and to establish a strong converse. The main contributions include an explicit inner bound via the intersection of source- and key-based rate regions, a matching outer bound in key cases, and a strong-converse result that holds generally for the sum-rate bound and under entropy-compatibility conditions $H(X_i)\le H(K_i)$. The results clarify fundamental limits of secure distributed source coding with correlated keys and connect to PEC/Shannon-cipher-type frameworks, with potential impact on practical encryption-assisted distributed compression systems.

Abstract

We reinvestigate the general distributed secure source coding based on the common key cryptosystem proposed by Oohama and Santoso (ITW 2021). They proposed a framework of distributed source encryption and derived the necessary and sufficient conditions to have reliable and secure transmission. However, the bounds of the rate region, which specifies both necessary and sufficient conditions to have reliable and secure transmission under the proposed cryptosystem, were derived based on a self-tailored non-standard} security criterion. In this paper we adopt the standard security criterion, i.e., standard mutual information. We successfully establish the bounds of the rate region based on this security criterion. Information spectrum method and a variant of Birkhoff-von Neumann theorem play an important role in deriving the result.

Figures (5)

• Figure 1: Distributed source coding without encryption.
• Figure 2: Distributed source coding with encryption.
• Figure 3: The coding scheme $(\varphi_1^{(n)},\varphi_2^{(n)},\widetilde{\psi}^{(n)})$ internally connected with $(\Phi_1^{(n)},\Phi_2^{(n)},{\Psi}^{(n)})$.
• Figure 4: Shapes of the four sets ${\cal V}_{(\underline{H}_1,\underline{H}_2)}, {\cal V}_{2\gamma_0}(\widetilde{R}_1,\widetilde{R}_2), {\cal V}_{2\gamma}(\widetilde{R}_1,\widetilde{R}_2),$ and ${\cal V}_{(2\gamma,3\gamma)} (\widetilde{R}_1,\widetilde{R}_2)$ related to the case of $\underline{I}>0$.
• Figure 5: The set $\widetilde{\cal V}_{(3\gamma,4\gamma), (\underline{H}_1,\underline{H}_2)}$ related to the case of $\underline{I}=0$.

Theorems & Definitions (22)

• Lemma 1
• Definition 1: Reliable and Secure Rate Pair
• Definition 2: Reliable and Secure Rate Region
• Theorem 1
• Theorem 2
• Theorem 3
• Corollary 1
• Theorem 4
• Proposition 1
• Theorem 5