The Secrecy Capacity Region of the Degraded Vector Gaussian Broadcast Channel

The information-theoretic single user secure communication problem was first characterized by Wyner in [1]. Wyner considered a scenario in which a wire-tapper receives the transmitted signal over a degraded channel with respect to the legitimate receiver’s channel. He measured the level of ignorance at the eavesdropper by its equivocation and characterized the capacity-equivocation region. Wyner’s work was then extended to the general broadcast channel with confidential messages by Csiszar et al. [2]. They considered transmitting confidential information to the legitimate receiver while transmitting common information to both the legitimate receiver and the wire-tapper. They established a capacityequivocation region of this channel. The secrecy capacity for the Gaussian wire-tap channel was characterized by Leung-Yan-Cheong in [3].

The Gaussian MIMO wire-tap channel has recently been considered by Khisti et al. in [4], [5]. Finding the optimal 1 Financial support provided by Nortel and the corresponding matching funds by the Natural Sciences and Engineering Research Council of Canada (NSERC), and Ontario Centers of Excellence (OCE) are gratefully acknowledged.

distribution, which maximizes the secrecy capacity for this channel is a nonconvex problem. Khisti et al., however, followed an indirect approach to evaluate the secrecy capacity of Csiszar et al. They used a genie-aided upper bound and characterized the secrecy capacity as the saddle-value of a min-max problem to show that Gaussian distribution is optimal. Motivated by the broadcast nature of the wireless communication systems, we considered the secure broadcast channel in [6]. In this work, we characterized the secrecy capacity region of the degraded broadcast channel and showed that the secret superposition coding is optimal.

The capacity region of the conventional Gaussian MIMO broadcast channel is studied in [7] by Weingarten et al. The notion of an enhanced broadcast channel is introduced in this work and is used jointly with entropy power inequality to characterize the capacity region of the degraded vector Gaussian broadcast channel. They showed that the superposition of Gaussian codes is optimal for the degraded vector Gaussian broadcast channel and that dirty-paper coding is optimal for the nondegraded case.

In this paper, we aim to characterize the secrecy capacity region of a secure degraded vector Gaussian MIMO broadcast channel. Our achievability scheme is a combination of the superposition of Gaussian codes and randomization within the layers. To prove the converse, we use the notion of enhanced channel and show that the secret superposition of Gaussian codes is optimal. We have extended the results of this paper to the general Gaussian MIMO broadcast channel in [8] and showed that secret dirty paper coding of Gaussian codes is optimal.

We acknowledge two other independent and concurrent works of [9], [10] where the authors considered the secrecy capacity region of the Gaussian MIMO broadcast channel.

The rest of the paper is organized as follows. In section II we introduce some preliminaries. In section III, we establish the secrecy capacity region of the Gaussian vector broadcast channel. In Section V, we conclude the paper.

Consider a Secure Gaussian Multiple-Input Multiple-Output Broadcast Channel (SGMBC) as depicted in Fig. 1. In this Encoder Decoder1 Decoder2 Eavesdropper

Fig. 1. Secure Gaussian MIMO Broadcast Channel confidential setting, the transmitter wishes to send two independent messages (W 1 , W 2 ) to the respective receivers in n uses of the channel and prevent the eavesdropper from having any information about the messages. At a specific time, the signals received by the destinations and the eavesdropper are given by

• x is a real input vector of size t × 1 under an input covariance constraint. We require that E[x T x] S for a positive semi-definite matrix S 0. Here,≺, , ≻, and represent partial ordering between symmetric matrices where B

A means that (B -A) is a positive semidefinite matrix.

• y 1 , y 2 , and z are real output vectors which are received by the destinations and the eavesdropper respectively. These are vectors of size r 1 × 1, r 2 × 1, and r 3 × 1, respectively. • H 1 , H 2 , and H 3 are fixed, real gain matrices which model the channel gains between the transmitter and the receivers. These are matrices of size r 1 × t, r 2 × t, and r 3 × t respectively. The channel state information is assumed to be known perfectly at the transmitter and at all receivers.

• n 1 , n 2 and n 3 are real Gaussian random vectors with zero means

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