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Electronics Letters

Comments on ‘Precoding for secrecy rate maximisation in cognitive MIMO wiretap channels’

Mahdi Khojastehnia, Sergey L. Loyka

Published Jun 5, 2020• 5 citations
Publisher

Abstract

Several gaps and errors in Fang et al. (2015) are identified. Their corrections as well as new results are reported. After these corrections, the successive convex approximation algorithm in Fang et al. (2015) is shown to generate an increasing, converging sequence of secrecy rates, which, for a degraded channel, converges to its secrecy capacity. The failure of the popular convex solver CVX to maximise secrecy rate is illustrated. Alternative algorithms are pointed out.

DOI
10.1049/el.2020.1684

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Several gaps and errors in Fang et al. (2015) are identified. Their corrections as well as new results are reported. After these corrections, the successive convex approximation algorithm in Fang et al. (2015) is shown to generate an increasing, converging sequence of secrecy rates, which, for a degraded channel, converges to its secrecy capacity. The failure of the popular convex solver CVX to maximise secrecy rate is illustrated. Alternative algorithms are pointed out.
Several gaps and errors in “Precoding and Artificial Noise Design for Cognitive MIMOME Wiretap Channels” by Fang et al. are identified and corrected. While accommodating these corrections, a rigours proof is given that the successive convex approximation algorithm of Fang et al. for secrecy rate maximization (SRM) does generate an increasing and bounded sequence of true secrecy rates and hence converges. It is further shown that its convergence point is a KKT point of the original SRM problem and, if the original problem is convex, this convergence point is globally-optimal, which is not necessarily the case in general. An interlacing property of the sequences of the true and approximate secrecy rates is established.
Several gaps and errors in (B. Fanget al.,“AN-aided secrecy precoding for SWIPT in cognitive MIMO broadcast channels,”IEEE Commun. Lett.,vol. 19, no. 9, pp. 1632–1635, Sep. 2015) are identified. Their corrections as well as new results are reported. After these corrections in the analysis, the successive convex approximation algorithm for secrecy rate maximization with artificial noise is shown to generate an increasing, converging sequence of secrecy rates. For a degraded channel, (i) this sequence converges to the maximum secrecy rate of the channel, and (ii) it is optimal to use no artificial noise. For a reversely-degraded channel, the maximum achievable secrecy rate is zero.
In this paper, we study the precoding problem for physical layer security in a general multiple-input multiple-output (MIMO) wiretap channel. Since the resultant secrecy capacity maximization (SCM) problem is nonconvex in general, we solve it by employing a successive convex approximation method, where the nonconvex part of the formulated problem is approximated by its first-order Taylor expansion. Thus, the SCM problem can be iteratively solved through convex programming of its convexified version. Finally, an iterative precoding algorithm with provable convergence is presented. Numerical simulations are also provided to verify the proposed algorithm.
We investigate the fading cognitive multiple-access wiretap channel (CMAC-WT), in which two secondary-user transmitters (STs) send secure messages to a secondary-user receiver (SR) in the presence of an eavesdropper and subject to interference threshold constraints at multiple primary-user receivers (PRs). We design linear precoders to maximize the average secrecy sum rate for a multiple-input-multiple-output (MIMO) fading CMAC-WT under finite-alphabet inputs and statistical channel state information at STs. For this nondeterministic polynomial-time NP-hard problem, we utilize an accurate approximation of the average secrecy sum rate to reduce the computational complexity and then present a two-layer algorithm by embedding the convex-concave procedure into an outer-approximation framework. The idea behind this algorithm is to reformulate the approximated average secrecy sum rate as a difference of convex functions and then generate a sequence of simpler relaxed sets to approach the nonconvex feasible set. Subsequently, we maximize the approximated average secrecy sum rate over the sequence of relaxed sets by using the convex-concave procedure. Numerical results indicate that our proposed precoding algorithm is superior to the conventional Gaussian precoding method in the medium and high signal-to-noise ratio (SNR) regimes.
This paper presents novel numerical approaches to finding the secrecy capacity of the multiple-input multiple-output (MIMO) wiretap channel subject to multiple linear transmit covariance constraints, including sum power constraint, per antenna power constraints and interference power constraint. An analytical solution to this problem is not known and existing numerical solutions suffer from slow convergence rate and/or high per-iteration complexity. Deriving computationally efficient solutions to the secrecy capacity problem is challenging since the secrecy rate is expressed as a difference of convex functions (DC) of the transmit covariance matrix, for which its convexity is only known for some special cases. In this paper we propose two low-complexity methods to compute the secrecy capacity along with a convex reformulation for degraded channels. In the first method we capitalize on the accelerated DC algorithm which requires solving a sequence of convex subproblems, for which we propose an efficient iterative algorithm where each iteration admits a closed-form solution. In the second method, we rely on the concave-convex equivalent reformulation of the secrecy capacity problem which allows us to derive the so-called partial best response algorithm to obtain an optimal solution. Notably, each iteration of the second method can also be done in closed form. The simulation results demonstrate a faster convergence rate of our methods compared to other known solutions. We carry out extensive numerical experiments to evaluate the impact of various parameters on the achieved secrecy capacity.