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

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.

💡 Research Summary

The paper revisits the distributed secure source coding framework introduced by Oohama and Santoso (ITW 2021), which is based on a common‑key cryptosystem. The original work derived necessary and sufficient conditions for reliable and secure transmission, but the rate region was characterized using a non‑standard, self‑tailored security metric. In this study the authors replace that metric with the standard mutual‑information leakage criterion, i.e., the mutual information (I(C_{1}C_{2};X_{1}X_{2})) between the pair of ciphertexts and the original sources.

The system model consists of two correlated discrete memoryless sources ((X_{1},X_{2})) and two correlated keys ((K_{1},K_{2})). Each source‑key pair is processed at a separate terminal by an encryption function (\Phi^{(n)}{i}) producing ciphertext (C^{(n)}{i}). Ciphertexts travel over public noiseless links, while the keys are delivered over private links. A decoder (\Psi^{(n)}) at the central node attempts to recover ((X_{1}^{n},X_{2}^{n})).

Reliability is measured by the decoding error probability (p^{(n)}{e}) and security by the leakage (\Delta^{(n)}{\text{MI}}=I(C^{(n)}{1}C^{(n)}{2};X_{1}^{n}X_{2}^{n})). A rate pair ((R_{1},R_{2})) is called ((\epsilon,\delta))-reliable‑and‑secure if, for any (\gamma>0), there exists a block length (n) large enough such that (\frac{1}{n}\log|C^{(n)}{i}|\le R{i}+\gamma), (p^{(n)}{e}\le\epsilon) and (\Delta^{(n)}{\text{MI}}\le\delta).

The authors define two fundamental regions. The source‑coding region (\mathcal{R}{\text{sw}}(p{X_{1}X_{2}})) is the Slepian‑Wolf region: (R_{1}\ge H(X_{1}|X_{2})), (R_{2}\ge H(X_{2}|X_{1})), and (R_{1}+R_{2}\ge H(X_{1}X_{2})). The key region (\mathcal{R}{\text{key}}(p{K_{1}K_{2}})) imposes (R_{1}\le H(K_{1})), (R_{2}\le H(K_{2})), and (R_{1}+R_{2}\le H(K_{1}K_{2})). Their intersection, after allowing a “padding” of the source rates by any feasible key rates, yields the inner bound (\mathcal{R}(p_{X},p_{K})).

Theorem 1 shows that for every ((\epsilon,\delta)) the inner bound is contained in the true reliable‑and‑secure region, which in turn is contained in the outer bound derived later. Consequently, (\mathcal{R}(p_{X},p_{K})) is a provable achievable region.

Property 2 characterizes when the region is non‑empty: it requires (H(X_{i})\le H(K_{i}\mid K_{3-i})) for each terminal and (H(X_{1}X_{2})\le H(K_{1}K_{2})). These conditions state that the keys must collectively carry at least as much entropy as the sources they protect. When the keys are sufficiently “rich” (i.e., each key entropy exceeds its corresponding source entropy and the joint key entropy exceeds the joint source entropy), the inner bound collapses to the classic Slepian‑Wolf region, meaning the encryption does not impose extra rate penalties.

The paper’s most significant contribution is the strong converse theorem for the distributed encryption problem. Using two technical tools—(i) the information‑spectrum method, which handles arbitrary (non‑i.i.d.) source and key distributions by analyzing limsup/liminf of normalized information densities, and (ii) a variant of the Birkhoff‑von Neumann theorem (Lemma 1) that guarantees a doubly‑stochastic structure for the mapping from plaintext‑key pairs to ciphertexts—the authors prove that any rate pair outside (\mathcal{R}(p_{X},p_{K})) forces the error probability to converge to one, regardless of how small (\epsilon) and (\delta) are chosen. The strong converse holds under an additional “information‑spectrum regularity” assumption on the ciphertext outputs; without this assumption only a weak converse can be guaranteed.

Furthermore, the authors introduce an optimistic definition of achievability, where the reliability and security constraints need only be satisfied along a subsequence of block lengths ({k_{n}}). This notion provides flexibility for practical system design, allowing engineers to select block lengths that meet performance targets without requiring uniform guarantees for all (n).

In summary, the paper accomplishes three major advances: (1) it aligns the security analysis of distributed source encryption with the standard mutual‑information criterion, (2) it derives tight inner and outer bounds for the reliable‑and‑secure rate region using information‑spectrum techniques and a novel matrix‑theoretic lemma, and (3) it establishes a strong converse, thereby delineating the exact boundary beyond which secure and reliable communication is impossible. These results furnish a rigorous theoretical foundation for designing distributed encryption schemes in networks where correlated sources share correlated keys, and they clarify the precise trade‑off between compression rates, key resources, and security guarantees.