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We study the optimality of linear precoding for the two-receivermultiple-input multiple-output (MIMO) Gaussian broadcast channel (BC) withconfidential messages. Secret dirty-paper coding (SDPC) is optimal under aninput covariance constraint, but there is no computable secrecy capacityexpression for the general MIMO case under an average power constraint. Inprinciple, for this case, the secrecy capacity region could be found through anexhaustive search over the set of all possible matrix power constraints.Clearly, this search, coupled with the complexity of dirty-paper encoding anddecoding, motivates the consideration of low complexity linear precoding as analternative. We prove that for a two-user MIMO Gaussian BC under an inputcovariance constraint, linear precoding is optimal and achieves the samesecrecy rate region as S-DPC if the input covariance constraint satisfies aspecific condition, and we characterize the corresponding optimal linearprecoders. We then use this result to derive a closed-form sub-optimalalgorithm based on linear precoding for an average power constraint. Numericalresults indicate that the secrecy rate region achieved by this algorithm isclose to that obtained by the optimal S-DPC approach with a search over allsuitable input covariance matrices.

On the Optimality of Linear Precoding for Secrecy in the MIMO Broadcast Channel