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arXiv:2007.13187 · IEEE Transactions on Signal Processing

Algorithms for Globally-Optimal Secure Signaling Over Gaussian MIMO Wiretap Channels Under Interference Constraints

Limeng Dong, Sergey L. Loyka, Yong Li

Published Jan 1, 2020• 11 citations
arXivHTMLPDF

Abstract

Multi-user Gaussian MIMO wiretap channel is considered under interference power constraints (IPC), in addition to the total transmit power constraint (TPC). Algorithms for global maximization of its secrecy rate are proposed. Their convergence to the secrecy capacity is rigorously proved and a number of properties are established analytically. Unlike known algorithms, the proposed ones are not limited to the MISO case and are proved to converge to a global rather than local optimum in the general MIMO case, even when the channel is not degraded. In practice, the convergence is fast as only a small to moderate number of Newton steps is required to achieve a high precision level. The interplay of TPC and IPC is shown to result in an unusual property when an optimal point of the max-min problem does not provide an optimal transmit covariance matrix in some (singular) cases. To address this issue, an algorithm is developed to compute an optimal transmit covariance matrix in those singular cases. It is shown that this algorithm also solves the dual (nonconvex) problems of globally minimizing the total transmit power subject to the secrecy and interference constraints; it provides the minimum transmit power and respective signaling strategy needed to achieve the secrecy capacity, hence allowing power savings.

DOI
10.1109/tsp.2020.3010451

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Multi-user Gaussian MIMO wiretap channel is considered under interference power constraints (IPC), in addition to the total transmit power constraint (TPC). Algorithms for global maximization of its secrecy rate are proposed. Their convergence to the secrecy capacity is rigorously proved and a number of properties are established analytically. Unlike known algorithms, the proposed ones are not limited to the MISO case and are proved to converge to a global rather than local optimum in the general MIMO case, even when the channel is not degraded. In practice, the convergence is fast as only a small to moderate number of Newton steps is required to achieve a high precision level. The interplay of TPC and IPC is shown to result in an unusual property when an optimal point of the max-min problem does not provide an optimal transmit covariance matrix in some (singular) cases. To address this issue, an algorithm is developed to compute an optimal transmit covariance matrix in those singular cases. It is shown that this algorithm also solves the dual (nonconvex) problems of globally minimizing the total transmit power subject to the secrecy and interference constraints; it provides the minimum transmit power and respective signaling strategy needed to achieve the secrecy capacity, hence allowing power savings.
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.
Secure signaling over multiple-input multiple-output (MIMO) wiretap channel (WTC) is studied under interference and transmit power constraints. The classical MIMO WTC model is extended to interference-limited scenarios, so that interference to other users does not exceed a given threshold while ensuring simultaneously no information leakage to an eavesdropper. The operational secrecy capacity of the Gaussian MIMO WTC under interference and transmit power constraints is rigorously established in two forms (a non-convex max problem and a convex-concave max-min problem), to which per-antenna power constraints can be added as well. Optimal signaling directions are characterized in the general case, from which (tight) upper bounds to the rank of optimal transmit covariance matrix are derived. A sufficient condition for the optimality of beamforming and a necessary condition for optimal full-rank signaling are given. Closed-form rank-1 and high-rank solutions are obtained in the case of zero interference constraints. Sufficient and necessary conditions for non-zero secrecy capacity are established. The results are extended to multi-user scenarios. A sufficient and necessary condition for the unbounded growth of the secrecy capacity with transmit power is obtained. The interplay between transmit and interference power constraints is studied, and its significant impact on optimal signaling is demonstrated (so that neither constraint can be absorbed into the other one in general, as was sometimes suggested in the literature). Overall, these results provide insights into fundamental information-theoretic limits and optimal signaling strategies for secure communications under interference constraints.
The multiple-input multiple-output (MIMO) Gaussian wire-tap channel (WTC) model is considered in the cognitive radio (CR) setting, where there is a interference power constraint (IPC), in addition to the total transmit power constraint (TPC). Based on the recently established mini-max characterization of its secrecy capacity, a numerical algorithm for global maximization of secrecy rates over such channels is proposed, which is based on the barrier method in combination with the residual-form Newton method and backtracking line search. Unlike the known algorithms, the proposed algorithm is guaranteed to converge to a global (rather than local) optimum. Its efficient performance is illustrated by examples.
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.
A closed-form full-rank solution for an optimal transmit covariance matrix over a strictly degraded MIMO wiretap channel and its secrecy capacity are obtained under an interference constraint, in addition to the transmit power constraint. A number of its properties are pointed out. The regimes when one of the constraints is inactive are studied. The interference constraint affects significantly the optimal covariance and induces its new properties, which cannot be found under the total transmit power constraint only.