As multimedia applications become more popular, wireless communication systems are expected to reliably provide increased data rates. Multiple Input Multiple Output (MIMO) technologies can meet this demand without using additional bandwidth or transmit power. MIMO is part of modern wireless communication standards. Another critical aspect of communications is to secure the confidentiality of data transmission. Cryptography accomplishes this at the upper layers of the protocol stack. At the physical layer, data travels unencrypted and can be secured by using the channel characteristics to ``hide'' data transmission from potential eavesdroppers. We consider a Gaussian MIMO wiretap channel and are looking for the maximal rate at which data can be transmitted both reliably and securely to the intended receiver: the secrecy capacity. This quantity is difficult to find analytically and is known precisely in only a few cases. This thesis proposes several numerical optimization methods, both stochastic and deterministic, to evaluate the secrecy capacity and to find the optimal transmit covariance matrix. The stochastic approaches are based on Monte-Carlo and on Differential Evolution (a genetic algorithm). The deterministic approaches are based on successive linear approximation. The accuracy of the results obtained with these methods is, in general, better than the one offered by popular numerical optimization tools such as CVX or YALMIP.
As multimedia applications become more popular, wireless communication systems are expected to reliably provide increased data rates. Multiple Input Multiple Output (MIMO) technologies can meet this demand without using additional bandwidth or transmit power. MIMO is part of modern wireless communication standards. Another critical aspect of communications is to secure the confidentiality of data transmission. Cryptography accomplishes this at the upper layers of the protocol stack. At the physical layer, data travels unencrypted and can be secured by using the channel characteristics to ``hide'' data transmission from potential eavesdroppers. We consider a Gaussian MIMO wiretap channel and are looking for the maximal rate at which data can be transmitted both reliably and securely to the intended receiver: the secrecy capacity. This quantity is difficult to find analytically and is known precisely in only a few cases. This thesis proposes several numerical optimization methods, both stochastic and deterministic, to evaluate the secrecy capacity and to find the optimal transmit covariance matrix. The stochastic approaches are based on Monte-Carlo and on Differential Evolution (a genetic algorithm). The deterministic approaches are based on successive linear approximation. The accuracy of the results obtained with these methods is, in general, better than the one offered by popular numerical optimization tools such as CVX or YALMIP.
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.
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.
In this modern era, wireless networks continue to expand worldwide and play a pronounced role in daily life. However, wireless communication has become vulnerable to intelligent invaders in the network. Hence, the security aspects of wireless communication have been the center of modern-era research. Physical Layer Security (PLS) schemes have been proven to be very effective against potential eavesdroppers thanks to their low-complexity nature compared to encryption techniques. Therefore, this thesis is dedicated to the physical layer security of multiple-input multiple-output (MIMO) systems. More specifically, we provide efficient optimization techniques to maximize the achievable secrecy rate of MIMO Wiretap channels (WTC). First, we present novel numerical approaches to finding the achievable secrecy rate of the MIMO WTC subject to multiple power constraints. There is a challenge in deriving computationally efficient solutions to the secrecy rate maximization problem because the secrecy rate is expressed as a difference of convex functions (DC) of the transmit covariance matrix. However, to deal with it, we provided local optimization methods, such as the accelerated DC algorithm (ADCA) and a numerical method named partial best response algorithm (PBRA) based on the concave-convex equivalent reformulation of the secrecy capacity problem. Along with it, we provide a convex reformulation for degraded channels. Interestingly, our experiments have shown that the local optimization methods achieve global optimality for the secrecy rate maximization problem under power constraints. In this regard, we provide analytical proof with Karush-Kuhn-Tucker (KKT) conditions to validate our observation. Motivated by our finding, we propose an accelerated gradient projection algorithm with adaptive momentum parameters (AGPnc) that directly solves the secrecy rate maximization problem rather than the equivalent convex-concave form. We then extend the scope of our research of physical layer security enhancement by considering the Intelligent reflecting surface (IRS)-aided MIMO WTC system, as the IRS has achieved enormous attention recently for enhancement in spectral efficiency. In this context, we provided a numerical solution for the problem of maximizing the achievable secrecy rate of the considered system by jointly optimizing the input covariance matrix and the IRS phase shifts. The proposed is an iterative method based on block successive maximization (BSM) where each iteration is done in closed form. We then extend our research further toward maximizing the secrecy energy efficiency (SEE) of the (IRS)-aided MIMO WTC system. In relation to the SEE maximization problem, we provide a low-complexity based local optimization method. The challenge of this considered problem is the coupling between design variables in both the objective and the constraint. The main idea of the proposed method is to apply a penalty dual decomposition based alternating gradient projection (PDDAPG) method to obtain an efficient solution. Thanks to the low complexity nature of proposed methods, these methods can scale favourably with large-scale wireless systems, which has been an inherent issue of the existing studies. In this way, the contributions of this thesis will significantly impact future research in the field of physical layer security, especially toward the future generation of wireless communication.
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.
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.
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.