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arXiv:1604.06785 · IEEE Transactions on Communications

Rank-Deficient Solutions for Optimal Signaling Over Wiretap MIMO Channels

Sergey L. Loyka, Charalambos D. Charalambous

Published Apr 29, 2016• 10 citations
arXivHTMLPDF

Abstract

Capacity-achieving signaling strategies for the Gaussian wiretap multiple-input multple-output (MIMO) channel are investigated without the degradedness assumption. In addition to known solutions, a number of new rank-deficient solutions for the optimal transmit covariance matrix are obtained. The case of a weak eavesdropper is considered in detail, and the optimal covariance is established in an explicit, closed form with no extra assumptions. This provides lower and upper bounds to the secrecy capacity in the general case with a bounded gap, which are tight for a weak eavesdropper or/and low SNR. Closed-form solutions are also obtained for isotropic and omnidirectional eavesdroppers, based on which lower and upper bounds to the secrecy capacity are established in the general case. Sufficient and necessary conditions for the optimality of three popular transmission techniques, namely, the zero-forcing (ZF), the standard water-filling over the channel eigenmodes, and the isotropic signaling (IS), are established for the MIMO wiretap channel. These solutions are appealing due to their lower complexity. In particular, no wiretap codes are needed for the ZF transmission, and no precoding or feedback is needed for the isotropic signaling.

DOI
10.1109/tcomm.2016.2560173

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Capacity-achieving signaling strategies for the Gaussian wiretap multiple-input multple-output (MIMO) channel are investigated without the degradedness assumption. In addition to known solutions, a number of new rank-deficient solutions for the optimal transmit covariance matrix are obtained. The case of a weak eavesdropper is considered in detail, and the optimal covariance is established in an explicit, closed form with no extra assumptions. This provides lower and upper bounds to the secrecy capacity in the general case with a bounded gap, which are tight for a weak eavesdropper or/and low SNR. Closed-form solutions are also obtained for isotropic and omnidirectional eavesdroppers, based on which lower and upper bounds to the secrecy capacity are established in the general case. Sufficient and necessary conditions for the optimality of three popular transmission techniques, namely, the zero-forcing (ZF), the standard water-filling over the channel eigenmodes, and the isotropic signaling (IS), are established for the MIMO wiretap channel. These solutions are appealing due to their lower complexity. In particular, no wiretap codes are needed for the ZF transmission, and no precoding or feedback is needed for the isotropic signaling.
Capacity-achieving signaling strategies for the Gaussian wiretap MIMO channel are investigated without the degradedness assumption. In addition to known solutions, a number of new rank-deficient solutions for the optimal transmit covariance matrix are obtained. The case of weak eavesdropper is considered in details and the optimal covariance is established in an explicit, closed-form with no extra assumptions. The conditions for optimality of zero-forcing signaling are established, and the standard water-filling is shown to be optimal under those conditions. No wiretap codes are needed in this case. The case of identical right singular vectors for the required and eavesdropper channels is studied and the optimal covariance is established in an explicit closed form. As a by-product of this analysis, we establish a generalization of celebrated Hadamard determinantal inequality using information-theoretic tools.
Secrecy rate maximization in Gaussian MIMO wiretap channels is considered. While the optimality of Gaussian signaling and a general expression for the secrecy capacity have been well established, closed-form solutions for the optimal transmit covariance matrix are known for some special cases only, while the general case remains an open problem. This chapter reviews known closed-form solutions and presents a numerical algorithm for the general case with guaranteed convergence to the global optimum. The known solutions include full-rank and rank-1 cases (which, when combined, provide a complete solution for the case of two transmit antennas), the case of identical right singular vectors for the eavesdropper and legitimate channels, and the cases of weak, isotropic, and omnidirectional eavesdroppers, which also provide lower and upper bounds to the general case. Necessary optimality conditions and a tight upper bound for the rank of the optimal covariance matrix in the general case are discussed. Sufficient and necessary conditions for the optimality of three popular signaling strategies over MIMO channels, namely, isotropic and zero-forcing signaling as well as water-filling over the legitimate channel eigenmodes, are presented. The chapter closes with a detailed description of a numerical globally convergent algorithm to solve the general case, and gives some illustrative examples. Introduction Due to their high spectral efficiency, wireless MIMO (multiple input, multiple output) systems are widely adopted by academia and industry. The broadcast nature of wireless channels stimulated significant interest in their security aspects and the Gaussian MIMO wiretap channel (WTC) has emerged as a popular model to study information theoretic secrecy aspects of wireless systems [1]. A number of results have been obtained for this model, including the proof of optimality of Gaussian signaling [1–4], which is far from trivial and significantly more involved than that of the regular (no wiretap) MIMO channel. Once the functional form of the optimal input is established, the only unknown is its covariance matrix since the mean is always zero. This latter part has not been solved yet in the general case; only a number of special cases have been settled.
Optimal signaling over the Gaussian multiple-input multiple-output wire-tap channel is studied under the total transmit power constraint. A closed-form solution for an optimal transmit covariance matrix is obtained when the channel is strictly degraded. In combination with the rank-1 solution, this provides the complete characterization of the optimal covariance for the case of two transmit antennas. The cases of weak eavesdropper and high SNR are considered. It is shown that the optimal covariance does not converge to a scaled identity in the high-SNR regime. Necessary optimality conditions and a tight upper bound on the rank of an optimal covariance matrix are established for the general case, along with a lower bound to the secrecy capacity, which is tight in a number of scenarios.
Optimal signalling over the wire-tap MIMO Gaussian channel is studied under the total transmit power constraint. The recent results are extended in several directions, including a rank-deficient solution for the optimal covariance, lower and upper capacity bounds for the general case, and characterization of optimality of the isotropic signaling. An isotropic eavesdropper model is studied, which provides (tight) upper and lower capacity bounds for the non-isotropic case and also serves as the worst-case scenario. The optimal signaling for this model is obtained in an explicit form and its properties are studied, including the high and low-SNR behavior, the conditions for the eavesdropper to be negligible and the capacity saturation effect.
We discuss the architecture of wireless multiple-input multiple-output (MIMO) channels with additive white Gaussian noise. The transmission optimization problem over covariance matrix for achieving MIMO channel capacity is reviewed under the assumption that the channel state information (CSI) is available from both the transmitter and the receiver. We also discuss several transmission strategies such as beamforming and water-filling algorithm. The information-theoretic approach to physical-layer security for wireless communications system is discussed. Physical-layer security which is different from cryptographic approach security is achieved by exploiting the physical-layer properties of wireless channel. The principal results and concepts in this field such as equivocation rate and secrecy capacity are reviewed. The wiretap channel model is considered. We mainly discussed the Gaussian MIMO wiretap channel and the methods for obtaining its secrecy capacity. The optimization problem for the secrecy capacity of Gaussian MIMO wiretap channel under total transmit power constraint is discussed. CVX, which is a modeling toolbox for solving convex optimization problems, is reviewed. The precision variable of CVX is also considered. The performances of CVX and Monte Carlo optimization are compared. Differential Evolution algorithm is used to compute the secrecy capacity of Gaussian MIMO wiretap channel. The case of weak eavesdropper is considered and the closed form solution based on the necessary KKT conditions is validated. The algorithm for general cases based on the linear approximation and reformulation of the original optimization problem is also investigated and validated. It has been verified that the lower bound and the upper bound of secrecy capacity can be obtained by such algorithm.
We consider a Gaussian multiple-input multiple-output (MIMO) wiretap channel model, where there exists a transmitter, a legitimate receiver and an eavesdropper, each node equipped with multiple antennas. We study the problem of finding the optimal input covariance matrix that achieves secrecy capacity subject to a power constraint, which leads to a non-convex optimization problem that is in general difficult to solve. Existing results for this problem address the case in which the transmitter and the legitimate receiver have two antennas each and the eavesdropper has one antenna. For the general cases, it has been shown that the optimal input covariance matrix has low rank when the difference between the Grams of the eavesdropper and the legitimate receiver channel matrices is indefinite or semi-definite, while it may have low rank or full rank when the difference is positive definite. In this paper, the aforementioned non-convex optimization problem is investigated. In particular, for the multiple-input single-output (MISO) wiretap channel, the optimal input covariance matrix is obtained in closed form. For general cases, we derive the necessary conditions for the optimal input covariance matrix consisting of a set of equations. For the case in which the transmitter has two antennas, the derived necessary conditions can result in a closed form solution; For the case in which the difference between the Grams is indefinite and has all negative eigenvalues except one positive eigenvalue, the optimal input covariance matrix has rank one and can be obtained in closed form; For other cases, the solution is proved to be a fixed point of a mapping from a convex set to itself and an iterative procedure is provided to search for it. Numerical results are presented to illustrate the proposed theoretical findings.