Fast secrecy rate optimization for MIMO wiretap channels in the presence of a multiple-antenna eavesdropper

Highlights

• We develop a fast secrecy rate optimization algorithm for MIMO Wiretap Channels.

• We also extend the proposed algorithm to tackle the transmission power minimization problem under the secrecy constraint.

• The proposed method attains the same performance as the competing algorithms but is computationally more efficient.

Abstract

We consider the secrecy rate maximization for multiple-input-multiple-output (MIMO) wiretap channels in the presence of a multiple-antenna eavesdropper. We propose an efficient algorithm based on minorization-maximization (MM) technique. The proposed algorithm constructs a simple surrogate function and has a closed-form solution at each iteration. In addition, we prove that the proposed algorithm has guaranteed convergence of the sequence of objective values. Moreover, an acceleration scheme, called squared iterative method (SQUAREM), is used to enhance the convergence rate of the proposed algorithm. Then, we extend the proposed algorithm to handle the transmission power minimization problem. Results show that the proposed algorithm have the same secrecy rate performance as the state-of-art algorithm but is much more computationally efficient.

Introduction

Wireless communication technologies have progressed rapidly due to the open and shared nature. However, the broadcast characteristics of wireless communications make it difficult to prevent transmitted signals from unintended recipients on the communication link. Therefore, information security has become an essential task for a wireless communication system due to the proliferation of privacy-sensitive wireless services (see, e.g., Shiu et al. [1], Liu et al. [2], Poor and Schaefer [3], Wang et al. [4], and the references therein).

Traditionally, information security relies on cryptography mechanisms implemented in media access control (MAC) and the upper-layer communication protocol [5], [6]. However, due to the emergence of new encryption cracking algorithms and the drastically increasing computing power of wireless devices, traditional encryption by secret keys is challenged [7], [8], [9]. As an alternative, physical layer security can also guarantee confidentiality of the messages transmitted from one transmitter to the legitimate receiver. Remarkably, the theoretical foundation for physical-layer security has been established in the seminal works including [10], [11], [12], [13].

The concept of “wiretap channel”, proposed by Wyner in [11], is the one of the basic physical layer models that capture the nature of communication security. Moreover, the notion of secrecy rate was introduced as a popular metric to quantify the achievable maximum information rate from the source to the destination. Subsequent works extended the work of Wyner to non-degraded discrete broadcast channels [12] and applied them to the basic Gaussian channel [13].

Substantial research efforts have been dedicated to the secrecy rate analysis of multiple-antenna wiretap channels from an information-theoretic perspective (see, e.g., Khisti and Wornell [14], Li and Ma [15], Liu and Shamai [16], Weingarten et al. [17], Li and Petropulu [18], Oggier and Hassibi [19], Fakoorian and Swindlehurst [20], Khisti and Wornell [21], Huang and Swindlehurst [22] and the references therein). In [14], the authors derived a closed-form optimal transmit covariance matrix for multiple-input-single-output (MISO) secrecy channels with multiple-antenna eavesdroppers. In [15], the secrecy rate optimization for MISO secrecy channels with multiple multiple-antenna eavesdroppers was recast as a semidefinite programming (SDP) problem. However, the proposed design therein cannot be extended to a multiple-input-multiple-output (MIMO) Gaussian wiretap channel because of the non-convexity associated with this channel. The authors in [16] characterized the MIMO secrecy capacity using an information-theoretic approach motivated by Weingarten in [17]. In [18], a fixed-point based secrecy rate maximization algorithm was proposed for the MIMO wiretap channel to compute the optimal input covariances under the average power constraint. However, the convergence of the proposed algorithm is not guaranteed. In [19], Khisti and Wornell obtained a closed-form expression of the secrecy capacity for MIMO wiretap channels in the case of high signal-to-noise-ratio (SNR). To gain more insights, the authors in [20] revealed the relationship between the rank of the optimal transmit covariance matrix and the channel conditions at the legitimate receiver and the eavesdropper. A semi-closed-form solution is proposed for the case of full-rank covariance matrix. Moreover, when the solution is not a full rank matrix, there exists an equivalent channel with a full rank solution to achieve the same secrecy capacity. However, finding the equivalent channel is intractable. To further enhance the secrecy rates, relays and cooperative jamming have been introduced as a promising technique in wireless networks. They can protect the legitimate users against being intercepted by the eavesdroppers [23], [24], [25], [26], [27]. Moreover, transmitting artificial noise along with the useful data signals is also proposed to improve the physical layer security [28], [29], [30], [31], [32].

In this paper, we consider the secrecy rate maximization problem for MIMO Gaussian wiretap channels, where a multiple-antenna eavesdropper might overhear the communication. In [33], the authors developed an iterative algorithm based on first-order Taylor series expansion to tackle the non-convex problem. The convergence of the developed algorithm is guaranteed. However, at each iteration, the algorithm in [33] needs to solve a determinant maximization (MAXDET) problem, whose computational complexity is very high. Therefore, it is desirable to devise more efficient methods. To this purpose, we propose a simple surrogate function to minorize the non-convex objective function. With the linear minorizer, we can obtain a closed-form optimal solution at each iteration. Moreover, we employ the squared iterative methods (SQUAREM) to speed up the convergence rate of the proposed algorithm. Finally, we adapt the proposed algorithm to deal with the the transmission power minimization problem under the secrecy rate constraint.

The rest of this paper is organized as follows. In Section 2, we establish the signal model for the MIMO wiretap channels and formulate the secrecy rate maximization problem. Section 3 introduces the principle of MM and develops the proposed algorithm based on MM. Section 4 presents an accelerated scheme to speed up the convergence rate of the proposed algorithm. In Section 5, we extend the proposed algorithm to tackle the transmission power minimization problem under the secrecy rate constraint. In Section 6, we present several numerical examples to verify the proposed algorithms. Finally, conclusions are drawn in Section 7.

Notations: We use bold lowercase and capital letters to denote vectors and matrices, respectively. The transpose, the conjugate, and the conjugate transpose are denoted by ()^T, ()*, and ()^H, respectively. $\mathbb{C},$ ${\mathbb{C}}^n,$ and ${\mathbb{C}}^{m\times n}$ represent the set of complex numbers, vectors, and matrices with complex-valued entries, respectively. The symbols Tr( · ), ‖ · ‖_F, and | · | indicate the trace, the Frobenius norm, and the determinant of a matrix, respectively. $\mathbb{E}\left\{\right\}$ stands for the statistical expectation for random variables. $\mathbf{x}\sim \mathcal{CN}(\mathbf{0},\mathit{\Sigma })$ denotes that x is a complex Gaussian random vector with zero mean and covariance matrix Σ. Re(x) means the real part of a complex-valued number x. I_M denotes the M × M identity matrix. Finally, if we say that D≻B (D⪰B), we mean that $\mathbf{D}-\mathbf{B}$ is positive definite (semi-definite).