Tight Exponential Strong Converse for Source Coding Problem with Encoded Side Information

TL;DR

This work derives the tight exponential strong converse exponent for the Wyner-Ahlswede-Körner (WAK) network with encoded side information by employing a change-of-measure approach. The exponent $G(R_1,R_2|P_{XY})$ is shown to equal a variational function $F(R_1,R_2|P_{XY})$ that incorporates an auxiliary random variable and a soft Markov constraint, treating the constraint as an integral part of the exponent. A numerical study on a doubly symmetric binary source illustrates the positive contribution of the soft Markov term, and in the no-side-information limit the result recovers the known tight single-user exponent while yielding improvements over Oohama's bound. The analysis also extends to privacy amplification, linking the strong converse exponent to cryptographic security against bounded-storage eavesdroppers. Overall, the paper advances a tight, methodical understanding of exponential decay in distributed source coding with encoded side information and offers tools potentially applicable to broader network settings.

Abstract

The source coding problem with encoded side information is considered. A lower bound on the strong converse exponent has been derived by Oohama, but its tightness has not been clarified. In this paper, we derive a tight strong converse exponent. For the special case such that the side-information does not exists, we demonstrate that our tight exponent of the WAK problem reduces to the known tight expression of that special case while Oohama's lower bound is strictly loose. The converse part is proved by a judicious use of the change-of-measure argument, which was introduced by Gu-Effros and further developed by Tyagi-Watanabe. Interestingly, the soft Markov constraint, which was introduced by Oohama as a proof technique, is naturally incorporated into the characterization of the exponent. A technical innovation of this paper is recognizing that the soft Markov constraint is a part of the exponent, rather than a penalty term that should be vanished. In fact, via numerical experiment, we provide evidence that the soft Markov constraint is strictly positive. The achievability part is derived by a careful analysis of the type argument; however, unlike the conventional analysis for the achievable rate region, we need to derive the soft Markov constraint in the analysis of the correct probability. Furthermore, we present an application of our derivation of strong converse exponent to the privacy amplification.

Figures (5)

• Figure 1: WAK network
• Figure 2: Graphs of \ref{['eq:exp_bound']} for $(p, R_2) = (0.1,1-h(0.2))$. The vertical axis shows the value of \ref{['eq:exp_bound']} and the horizontal axis shows the value of $R_1$. The solid line indicates the value without the Markov constraint for $P_{\tilde{U}\tilde{X}\tilde{Y}}$, and the dashed line indicate the value with the constraint.
• Figure 3: Network with non-encoded side information
• Figure 4: Single-user network
• Figure 5: Privacy amplification for bounded storage eavesdropper

Theorems & Definitions (13)

• Theorem 1
• Theorem 2
• Corollary 3
• Proposition 4
• Remark
• Corollary 5
• Corollary 6
• Proposition 7
• Lemma 8
• Lemma 9