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International Conference on Communications

Signaling strategy optimization for Gaussian MIMO wiretap channel

Qingjiang Shi, Enbin Song, Guoan Chen

Published Jun 1, 2012• 6 citations
Publisher

Abstract

MIMO wiretap channel consists of a multiple antenna transmitter, a legitimate multiple antenna receiver and a multiple antenna eavesdropper. Signaling strategy optimization for MIMO wiretap channel, including secrecy capacity maximization problem and power minimization problem with a secrecy rate constraint, are studied. We identify when the two problems are globally solvable. Further, iterative algorithms based on concave-convex procedure are proposed for the two problems, which can monotonically converge to a Karush-Kuhn-Tucker (KKT) point of the two problems. Numerical examples validate the effectiveness of the proposed algorithms.

DOI
10.1109/icc.2012.6364133

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MIMO wiretap channel consists of a multiple antenna transmitter, a legitimate multiple antenna receiver and a multiple antenna eavesdropper. Signaling strategy optimization for MIMO wiretap channel, including secrecy capacity maximization problem and power minimization problem with a secrecy rate constraint, are studied. We identify when the two problems are globally solvable. Further, iterative algorithms based on concave-convex procedure are proposed for the two problems, which can monotonically converge to a Karush-Kuhn-Tucker (KKT) point of the two problems. Numerical examples validate the effectiveness of the proposed algorithms.
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.
A Gaussian multiple-input multiple-output (MIMO) wiretap channel model is considered, where there exists a transmitter, a legitimate receiver and an eavesdropper each equipped with multiple antennas. The optimality of beamforming for secrecy capacity subject to sum power constraint is studied, and two sufficient conditions for beamforming to be globally optimal are given. The first sufficient condition states that when the difference between the Gram matrices of legitimate and eavesdropper channel matrices has exactly one positive eigenvalue, then beamforming is globally optimal. An alternative sufficient condition, which involves convex optimization, is also provided. For the case in which beamforming is globally optimal, the secrecy capacity is obtained. Otherwise, an upper bound of the difference between the secrecy capacity and beamforming secrecy rate is provided.
This paper studies multiple-input multiple-output multiple access wiretap channels (MAC-WT) where an eavesdropper tries to tap the communication between multiple legitimate transmitters and a legitimate receiver. In this system, we propose precoder optimization methods at the transmitters in order to maximize the sum secrecy rate performance. Although this problem can be solved by the well-known difference of convex (DC) programming, we present a more efficient algorithm whose computational complexity is much lower than that of the conventional DC approach. By investigating the Karush-Kuhn-Tucker conditions, it is confirmed that the proposed low-complexity algorithm achieves the same performance as the conventional DC method. Our analysis also reveals that the proposed algorithm ensures global optimality for multiple-input single-output MAC-WT cases, while binary power control is optimal for single-input multiple-output scenarios. Simulation results demonstrate the efficacy of the proposed precoding methods.
We consider a multiple-input multiple-output (MIMO) Gaussian 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 that achieves secrecy capacity subject to a power constraint, which in general leads to a difficult non-convex optimization problem. For the Gaussian multiple-input single-output (MISO) wiretap channel where there exists a transmitter equipped with multiple antennas, a legitimate receiver and an eavesdropper each equipped with one antenna, the optimal input covariance is obtained in closed form. For the general case, we derive the necessary conditions for the optimal input covariance in the form of a set of equations. We show that for MISO case, the derived conditions guarantee the optimal input covariance. Numerical results are presented to illustrate the proposed theoretical findings.
Optimal signaling for secrecy rate maximization in Gaussian MIMO wiretap channels is considered. While this channel has attracted a significant attention recently and a number of results have been obtained, including the proof of the optimality of Gaussian signalling, an optimal transmit covariance matrix is known for some special cases only and the general case remains an open problem. An iterative custom-made algorithm to find a globally-optimal transmit covariance matrix in the general case is developed in this paper, with guaranteed convergence to a global optimum. While the original optimization problem is not convex and hence difficult to solve, its minimax reformulation can be solved via the convex optimization tools, which is exploited here. The proposed algorithm is based on the barrier method extended to deal with a minimax problem at hand. Its convergence to a global optimum is proved for the general case (degraded or not) and a bound for the optimality gap is given for each step of the barrier method. The performance of the algorithm is demonstrated via numerical examples. In particular, 20 to 40 Newton steps are already sufficient to solve the sufficient optimality conditions with very high precision (up to the machine precision level), even for large systems. Even fewer steps are required if the secrecy capacity is the only quantity of interest. The algorithm can be significantly simplified for the degraded channel case and can also be adopted to include the per-antenna power constraints (instead or in addition to the total power constraint). It also solves the dual problem of minimizing the total power subject to the secrecy rate constraint.
To achieve perfect secrecy in a multiple-input multiple-output (MIMO) Gaussian wiretap channel (WTC), we need to find its secrecy capacity and optimal signaling, which involves solving a difference of convex functions program known to be non-convex for the non-degraded case. To deal with this, a class of existing solutions have been developed but only local optimality is guaranteed by standard convergence analysis. Interestingly, our extensive numerical experiments have shown that these local optimization methodsindeed achieve global optimality. In this letter, we provide an analytical proof for this observation. To achieve this, we show that the Karush-Kuhn-Tucker (KKT) conditions of the secrecy rate maximization problem admit a unique solution forboth degraded and non-degraded cases. Motivated by this, we also propose a low-complexity algorithm to find a stationary point. Numerical results are presented to verify the theoretical analysis.