  The K-receiver wiretap channel is a channel model where a transmitter
broadcasts K independent messages to K intended receivers while keeping them
secret from an eavesdropper. The capacity region of the K-receiver
multiple-input multiple-output (MIMO) wiretap channel has been characterized by
using dirty-paper coding and stochastic encoding. However, K factorial encoding
orders may need to be enumerated to evaluate the capacity region, which makes
the problem intractable. In addition, even though the capacity region is known,
the optimal signaling to achieve the capacity region is unknown. In this paper,
we determine one optimal encoding order to achieve every point on the capacity
region, and thus reduce the encoding complexity K factorial times. We prove
that the optimal decoding order for the K-receiver MIMO wiretap channel is the
same as that for the MIMO broadcast channel without secrecy. To be specific,
the descending weight ordering in the weighted sum-rate (WSR) maximization
problem determines the optimal encoding order. Next, to reach the border of the
secrecy capacity region, we form a WSR maximization problem and apply the block
successive maximization method to solve this nonconvex problem and find the
input covariance matrices corresponding to each message. Numerical results are
used to verify the optimality of the encoding order and to demonstrate the
efficacy of the proposed signaling design.
