Wiretap Channels: Implications of the More Capable Condition and Cyclic Shift Symmetry

TL;DR

This paper simplifies the characterization of the rate-equivocation region in certain wiretap channels by identifying conditions under which auxiliary variables are optimal, focusing on more capable and cyclic shift symmetric channels.

Contribution

It provides explicit solutions and optimal strategies for the rate-equivocation region in classes of wiretap channels, including more capable and cyclic shift symmetric types.

Findings

V=X is optimal when the main channel is more capable.

Explicit optimal U and V are determined via cyclic shifts in symmetric channels.

U=ϕ is optimal for certain cyclic shift symmetric channels.

Abstract

Characterization of the rate-equivocation region of a general wiretap channel involves two auxiliary random variables: U, for rate splitting and V, for channel prefixing. Evaluation of regions involving auxiliary random variables is generally difficult. In this paper, we explore specific classes of wiretap channels for which the expression and evaluation of the rate-equivocation region are simpler. In particular, we show that when the main channel is more capable than the eavesdropping channel, V=X is optimal and the boundary of the rate-equivocation region can be achieved by varying U alone. Conversely, we show under a mild condition that if the main receiver is not more capable, then V=X is strictly suboptimal. Next, we focus on the class of cyclic shift symmetric wiretap channels. We explicitly determine the optimal selections of rate splitting U and channel prefixing V that achieve the boundary of the rate-equivocation region. We show that optimal U and V are determined via cyclic shifts of the solution of an auxiliary optimization problem that involves only one auxiliary random variable. In addition, we provide a sufficient condition for cyclic shift symmetric wiretap channels to have U=\phi as an optimal selection. Finally, we apply our results to the binary-input cyclic shift symmetric wiretap channels. We solve the corresponding constrained optimization problem by inspecting each point of the I(X;Y)-I(X;Z) function. We thoroughly characterize the rate-equivocation regions of the BSC-BEC and BEC-BSC wiretap channels. In particular, we find that U=\phi is optimal and the boundary of the rate-equivocation region is achieved by varying V alone for the BSC-BEC wiretap channel.