On private broadcasting over independent parallel channels

We study private broadcasting of two messages to two groups of users over reversely degraded parallel channels. Group 1 has two users, both interested in a common message whereas group 2 has only one user. The message for each group of users needs to be kept confidential from the other group. We characterize the capacity region for a special degradation structure of the channels and establish the optimality of a superposition codebook where codewords of a secure product-codebook form the cloud centers and codewords of a secure multicast codebook form the satellite codewords. An extension to Gaussian channels with a sum-power constraint is also obtained, where the optimality of Gaussian codebooks is established using an extremal inequality.