  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.
