On a special class of broadcast channels with confidential messages

It is shown that Csiszar and Korner´s (1978) characterization of a discrete memoryless channel (DMC)X→Y as being less noisy than the DMC X→Z is equivalent to the condition that the mutual-information difference I(X;Y)-I(X;Z) be a convex-∩ function of the probability distribution for X. This result is used to obtain a simple determination of the capacity region of the broadcast channel with confidential messages (BCC), which is a DMC X→(Y,Z), when the DMC X→Y to the legitimate receiver is less noisy than the DMC X→Z to the enemy cryptanalyst and there is a probability distribution for X having strictly positive components that achieves capacity on both these channels. In particular, when these DMC´s are both symmetric, then the secrecy capacity of the BCC is the difference of their capacities. It is shown further that the less-noisy condition in this result cannot be weakened to the condition that the DMC X→Y be more capable than the DMC X→Z in the sense of Csiszar and Korner