The popular technique of secure signaling over Gaussian MIMO wiretap channels, which makes use of artificial noise (AN) to increase secrecy rates, is considered. First, we briefly review the current state of affairs in this area and then provide new analytical results and insights on the usefulness of AN (jamming) to boost secrecy rates. The settings considered here go beyond the total transmit power constraint and include a number of additional constraints, such as interference (cognitive radio) and energy-harvesting constraints, for which the feasible set is not isotropic anymore so that the standard tools of the analysis cannot be used. By closely examining optimal precoding for the information-bearing and AN signals, we identify a number of cases where it is optimal to transmit no artificial noise at all (so that all the transmit power goes to the information-bearing signal). These cases include a fixed (no fading) MIMO WTC with single eavesdropper (for which we give a direct matrix-theoretic proof using novel matrix inequalities), multi-eavesdropper (com-pound) degraded and reversely-degraded channels, and multi-eavesdropper channels where there exists a dominant eavesdropper (for which we give a precise definition) or when the eavesdroppers collude. To improve secrecy rates by using AN, one has to look elsewhere.
The popular technique of secure signaling over Gaussian MIMO wiretap channels, which makes use of artificial noise (AN) to increase secrecy rates, is considered. First, we briefly review the current state of affairs in this area and then provide new analytical results and insights on the usefulness of AN (jamming) to boost secrecy rates. The settings considered here go beyond the total transmit power constraint and include a number of additional constraints, such as interference (cognitive radio) and energy-harvesting constraints, for which the feasible set is not isotropic anymore so that the standard tools of the analysis cannot be used. By closely examining optimal precoding for the information-bearing and AN signals, we identify a number of cases where it is optimal to transmit no artificial noise at all (so that all the transmit power goes to the information-bearing signal). These cases include a fixed (no fading) MIMO WTC with single eavesdropper (for which we give a direct matrix-theoretic proof using novel matrix inequalities), multi-eavesdropper (com-pound) degraded and reversely-degraded channels, and multi-eavesdropper channels where there exists a dominant eavesdropper (for which we give a precise definition) or when the eavesdroppers collude. To improve secrecy rates by using AN, one has to look elsewhere.
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.
This paper considers a Gaussian multi-input multi-output (MIMO) wiretap channel with a legitimate transmitter, a legitimate receiver (Bob), an eavesdropper (Eve), and a cooperative jammer. All nodes may be equipped with multiple antennas. Traditionally, the jammer transmits Gaussian noise (GN) to enhance the security. However, using this approach, the jamming signal interferes not only with Eve but also with Bob. In this paper, besides the GN strategy, we assume that the jammer can also choose to use the encoded jammer (EJ) strategy, i.e., instead of GN, it transmits a codeword from an appropriate codebook. In certain conditions, the EJ scheme enables Bob to decode the jamming codeword and thus cancel the interference, while Eve remains unable to do so even if it knows all the codebooks. We first derive an inner bound on the system’s secrecy rate under the strong secrecy metric, and then consider the maximization this bound through precoder design in a computationally efficient manner. In the single-input multi-output (SIMO) case, we prove that although non-convex, the power control problems can be optimally solved for both GN and EJ schemes. In the MIMO case, we propose to solve the problems using the matrix simultaneous diagonalization (SD) technique, which requires quite a low computational complexity. Simulation results show that by introducing a cooperative jammer with coding capability, and allowing it to switch between the GN and EJ schemes, a dramatic increase in the secrecy rate can be achieved. In addition, the proposed algorithms can significantly outperform the current state of the art benchmarks in terms of both secrecy rate and computation time.
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.
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.
Cooperative jamming approach has been commonly used in the field of physical layer security to achieve a good secrecy rate of wireless networks. In this approach, the Gaussian noise signal sent by a friendly jammer can disrupt the reception of the eavesdropper, however, it may also interfere with the legitimate receiver. In this paper, we propose a novel approach, namely, information jamming, to go beyond the secrecy rate achieved by the traditional cooperative jamming approach in a multiple-input single-output (MISO) wiretap channel. We propose that a multi-antenna friendly jammer transmits the source message rather than Gaussian noise signal for simultaneously enhancing the signal strength at the legitimate receiver and canceling the received signal at the eavesdropper. In particular, we aim to maximize the achievable secrecy rate by optimizing the information jammer's beamforming vector. Though the optimization problem is non-convex, we successfully solve it optimally via a two-stage based numerical approach. To gain more useful analytical results, we also propose two suboptimal jamming schemes with low computation, by targeting at improving the legitimate reception and avoiding the eavesdropping, respectively. Finally, numerical results show that the proposed information jamming approach significantly helps go beyond the secrecy rate of the existing cooperative jamming approach, especially when the transmit power of the jammer is large enough.
We consider the multiple-input multiple-output (MIMO) wiretap channel under a minimum receiver-side power constraint in addition to the usual maximum transmitter-side power constraint. This problem is motivated by energy harvesting communications with wireless energy transfer, where an added goal is to deliver a minimum amount of energy to a receiver in addition to delivering secure data to another receiver. In this paper, we characterize the exact secrecy capacity of the MIMO wiretap channel under transmitter and receiver-side power constraints. We first show that solving this problem is equivalent to solving the secrecy capacity of a wiretap channel with a double-sided correlation matrix constraint on the channel input. We show the converse by extending the channel enhancement technique to our case. We present two achievable schemes that achieve the secrecy capacity: the first achievable scheme uses a Gaussian codebook with a fixed mean, and the second achievable scheme uses artificial noise (or cooperative jamming) together with a Gaussian codebook. The role of the mean or the artificial noise is to enable energy transfer without sacrificing from the secure rate. This is the first instance of a channel model where either the use of a mean signal or use of channel prefixing via artificial noise is strictly necessary in the MIMO wiretap channel.