Rate-equivocation region of cyclic shift symmetric wiretap channels

In this paper, we study cyclic shift symmetric wiretap channels in which the channels between Alice and Bob and Alice and Eve are both cyclic shift symmetric. We characterize the rate-equivocation region by determining the optimal selection of rate splitting U and channel prefixing V for these channels. 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. We find the cardinality bound on the necessary auxiliary variable and formulate the problem as a constrained optimization problem. We determine the optimality conditions for the binary-input cyclic shift symmetric wiretap channels. We find the optimum by inspecting each point of the I(X;Y) - I(X; Z) function and ruling out the sub-optimal candidates that satisfy the optimality conditions. In particular, we address BSC-BEC and BEC-BSC wiretap channels. By using the optimality conditions, we determine the optimal selections of U and V for the rate-equivocation regions of these channels.