The confidential MIMO broadcast capacity: A simple derivation

We consider the problem of transmitting confidential messages over a two-user broadcast multiple-input multiple-output (MIMO) channel. Surprisingly, the capacity region of this setting under a covariance matrix constraint was shown by Liu et al. to be rectangular. That is, there is no tension, and both users can attain their respective MIMO wiretap capacities, simultaneously. In this work, we provide a new derivation of this result by proposing an alternative achievability scheme for the corner point of the capacity region. This derivation, in addition to being considerably shorter and simpler than the original, also provides a practical transmission scheme, in the sense that the codes used are scalar (single-antenna) ones. We use two main ingredients. The first is the explicit optimal input covariance matrix of Bustin et al. for the MIMO wiretap channel under a covariance matrix constraint, which we also re-derive in a simple manner. The second is a dirty-paper variant of a recently proposed optimal scheme for the MIMO wiretap channel, which uses scalar codes. The proposed treatment demonstrates the connection between the confidential broadcast problem and the MIMO wiretap one: the former almost reduces to the latter, except for the use of dirty-paper coding which is not mandatory in MIMO wiretap; the work sheds light on the reason for this difference.