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: <formula formulatype="inline"><tex Notation="TeX">$U$</tex></formula>, for rate splitting and <formula formulatype="inline"> <tex Notation="TeX">$V$</tex></formula>, 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, <formula formulatype="inline"><tex Notation="TeX">$V=X$</tex></formula> is optimal and the boundary of the rate-equivocation region is achieved by varying rate splitting <formula formulatype="inline"> <tex Notation="TeX">$U$</tex></formula> alone. Conversely, we show under a mild condition that if the wiretap channel is not more capable, then <formula formulatype="inline"><tex Notation="TeX">$V=X$</tex> </formula> is strictly suboptimal. Next, we focus on the class of cyclic shift symmetric wiretap channels. We show that optimal rate splitting <formula formulatype="inline"><tex Notation="TeX">$U$</tex> </formula> that achieves the boundary of the rate-equivocation region is uniform with cardinality <formula formulatype="inline"><tex Notation="TeX">$\vert{\cal X}\vert$</tex></formula> and the prefix channel between optimal <formula formulatype="inline"><tex Notation="TeX">$U$</tex></formula> and <formula formulatype="inline"><tex Notation="TeX">$V$</tex> </formula> 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 <formula formulatype="inline"><tex Notation="TeX">$U=\phi$</tex></formula> 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.