Ergodic Secrecy Capacity Region of the Fading Broadcast Channel

We consider the fading broadcast channel from a secrecy point of view. In this channel, each user views the other user as an eavesdropper, and wants to keep its information as secret from the other user as possible. First, we consider a more general channel model which consists of L independent sub-channels, where in each sub-channel, one of the users´ channel is less noisy with respect to the other user. Since the user which has the less noisy observation can be different in each sub-channel, the overall channel is not less noisy for any one of the users. We establish the secrecy capacity region of this channel for the case where the transmitter sends a common message to both users and an individual confidential message to each user. This channel model encompasses the sub-class of channels, where in each sub-channel, one of the users´ observation is degraded with respect to the other user. The parallel Gaussian broadcast channel belongs to this sub-class. In the Gaussian case, we identify the optimum input distribution, which is Gaussian, and the optimum power allocation corresponding to each point on the boundary of the secrecy capacity region. Finally, noting that the fading Gaussian broadcast channel is equivalent to a parallel Gaussian broadcast channel from an ergodic capacity perspective, we explicitly evaluate the ergodic secrecy capacity region of the fading broadcast channel.