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.

翻译：K接收者窃听信道是一种信道模型，其中发射机向K个目标接收机广播K个独立消息，同时要求这些消息对窃听者保密。利用脏纸编码和随机编码方法，K接收者多输入多输出（MIMO）窃听信道的容量区域已被刻画。然而，为了评估该容量区域，可能需要枚举K的阶乘种编码顺序，这使得问题变得难以处理。此外，即使容量区域已知，实现该容量区域的最优信号设计仍属未知。本文确定了在容量区域上每个点均可实现的一种最优编码顺序，从而将编码复杂度降低K的阶乘倍。我们证明K接收者MIMO窃听信道的最优解码顺序与无保密约束的MIMO广播信道相同。具体而言，在加权和速率（WSR）最大化问题中，权重降序排列决定了最优编码顺序。进而，为达到保密容量区域边界，我们构建了WSR最大化问题，并采用块序列最大化方法求解该非凸问题，从而得到每个消息对应的输入协方差矩阵。数值结果验证了编码顺序的最优性，并展示了所提信号设计的有效性。