MIMO Gaussian broadcast channels with common, private and confidential messages

The two-user multiple-input multiple-output (MIMO) Gaussian broadcast channel (BC) with common, private and confidential messages is considered. The transmitter sends a common message to both users, a confidential message to User 1 and a private (non-confidential) message to User 2. The secrecy-capacity region is characterized by showing that certain inner and outer bounds coincide and that the boundary points are achieved by Gaussian inputs. The proof relies on factorization of upper concave envelopes and a variant of dirty-paper coding (DPC). The entire region is exhausted by using DPC to cancel out the signal of the non-confidential message at Receiver 1, making DPC against the signal of the confidential message unnecessary. The secrecy-capacity results are visualized using a numerical example.