MIMO Broadcast Channel with an Unknown Eavesdropper: Secrecy Degrees of Freedom

We study a multi-antenna broadcast channel with two legitimate receivers and an external eavesdropper. We assume that the channel matrix of the eavesdropper is unknown to the legitimate terminals but satisfies a maximum rank constraint. As our main result we characterize the associated secrecy degrees of freedom for the broadcast channel with common and private messages. We show that a direct extension of the single-user wiretap codebook does not achieve the secrecy degrees of freedom. Our proposed optimal scheme involves decomposing the signal space into a common subspace, which can be observed by both receivers, and private subspaces which can be observed by only one of the receivers, and carefully transmitting a subset of messages in each subspace. We also consider the case when each user´s private message must additionally remain confidential from the other legitimate receiver and characterize the s.d.o.f. region in this case.