Secrecy capacity region of a class of two-user Gaussian MIMO BC with degraded message sets

We consider a two-user broadcast channel (BC) with two-degraded message sets where a common message M₂ is intended for both receivers and a private message M₁ is intended for receiver one. There is an eavesdropper whose channel is stochastically degraded with respect to both receivers and both messages must be kept confidential from the eavesdropper. However, we do not assume any form of degradedness between the two receivers. We first characterize the capacity region of this class of discrete memoryless BC. Next, we consider the two-user Gaussian MIMO BC with two-degraded message sets and an eavesdropper. We characterize the capacity region for the case where the eavesdropper is only required to be stochastically degraded with respect to receiver one. We make use of the channel enhancement technique as well as an extremal entropy inequality of Liu et al. that was derived using the generalized Costa´s entropy power inequality (EPI).