Wiretap channels: Roles of rate splitting and channel prefixing

Csiszár and Körner´s characterization of the rate-equivocation region of a general wiretap channel involves two auxiliary random variables: U, which represents rate splitting and V, which represents channel prefixing. For some channels, one or both of these auxiliary random variables are unnecessary, simplifying the expression and evaluation of the rate-equivocation region. In this paper, we provide new conditions under which channel prefixing or rate splitting does not improve the rate-equivocation region. In particular, we show that when the main channel is more capable than the eavesdropping channel, channel prefixing is unnecessary; the entire rate-equivocation region can be achieved by rate splitting alone. Conversely, we show under a mild assumption that if the main receiver is not more capable, then channel prefixing is strictly necessary. Moreover, we show that if the main channel is more capable but not less noisy, then rate splitting is strictly necessary. Next, we focus on the set of cyclic shift symmetric channels. We prove that for these channels, if in addition I(X; Y) - I(X;Z) is maximized at the uniform distribution, then rate splitting is unnecessary. Our results apply to BSC-BEC and BEC-BSC wiretap channels. We identify the conditions on the parameters of the BSC and BEC under which channel prefixing and/or rate splitting are unnecessary.