On a special class of broadcast channels with confidential messages

It is shown that Csiszár and Körner's 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 Csiszár and Körner. © 1997 IEEE.