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

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. In this paper, we explore specific classes of wiretap channels for which the evaluation of the rate-equivocation region is simpler. We show that if the wiretap channel is more capable, V=X is optimal and the boundary of the rate-equivocation region is achieved by varying rate splitting U alone. Conversely, we show under a mild condition that if the wiretap channel is not more capable, then V=X is strictly suboptimal. Next, we focus on the class of cyclic shift symmetric wiretap channels. We show that optimal rate splitting U that achieves the boundary of the rate-equivocation region is uniform with cardinality |X| and the prefix channel between optimal U and V is expressed as cyclic shifts of the solution of an auxiliary optimization problem over a single variable. We provide a special class of cyclic shift symmetric wiretap channels for which U=φ is optimal. We apply our results to the binary-input cyclic shift symmetric wiretap channels and thoroughly characterize the rate-equivocation regions of the BSC-BEC and BEC-BSC wiretap channels.