An Overview of Physical Layer Security With Finite-Alphabet Signaling <s> F. Performance-Complexity Trade-Off in Transmit Signal Design <s> In this paper, we investigate the secrecy rate of finite-alphabet communications over multiple-input-multiple-output-multiple-antenna eavesdropper (MIMOME) channels. Traditional precoding designs based on Gaussian input assumption may lead to substantial secrecy rate loss when the Gaussian input is replaced by practical finite-alphabet input. To address this issue, we investigate linear precoding designs to directly maximize the secrecy rate for MIMOME systems under the constraint of finite-alphabet input. By exploiting the theory of Karush-Kuhn-Tucker (KKT) analysis and matrix calculus, we first present necessary conditions of the optimal precoding design when instantaneous channel-state information (CSI) of the eavesdropper is known at the transmitter. In this light, an iterative algorithm for finding the optimal precoding matrix is developed, utilizing a gradient decent method with backtracking line search. Moreover, we find that the beamforming design in MIMONE systems, which is a secrecy-capacity-achieving approach for Gaussian signaling, no longer provides the maximum secrecy rate for finite-alphabet input data. This case is substantially different from the Gaussian input case. In addition, we derive the closed-form results on the precoding matrix, which maximizes the secrecy rate in the low signal-to-noise ratio (SNR) region, and reveal the optimal precoding structure in the high-SNR region. A novel jamming signal generation method that draws on the CSI of the eavesdropper to additionally increase the secrecy rate is further proposed. The precoding design with only statistical CSI of the eavesdropper available at the transmitter is also considered. Numerical results show that the proposed designs provide significant gains over recent precoding designs through a power control policy and the precoding design with the Gaussian input assumption in various scenarios. <s> BIB001 </s> An Overview of Physical Layer Security With Finite-Alphabet Signaling <s> F. Performance-Complexity Trade-Off in Transmit Signal Design <s> This paper investigates the linear precoder design that maximizes the average mutual information of multiple-input multiple-output fading channels with statistical channel state information known at the transmitter. It formulates the design from the standpoint of finite-alphabet inputs, which leads to a problem that is very important in practice but extremely difficult in theory: First, the average mutual information lacks closed-form expression and involves prohibitive computational burden. Second, the optimization over the precoder is nonconcave and thus easily gets stuck in local maxima. To address these issues, this study first derives lower and upper bounds for the average mutual information, in which the computational complexity is reduced by several orders of magnitude compared to calculating the average mutual information directly. It proves that maximizing the bounds is asymptotically optimal and shows that, with a constant shift, the lower bound actually offers a very accurate approximation to the average mutual information for various fading channels. This paper further proposes utilizing the lower bound as a low-complexity and accurate alternative for developing a two-step algorithm to find a near global optimal precoder. Numerical examples demonstrate the convergence and efficacy of the proposed algorithm. Compared to its conventional counterparts, the proposed linear precoding method provides significant performance gain over existing precoding algorithms. The gain becomes more substantial when the spatial correlation of MIMO channels increases. <s> BIB002 </s> An Overview of Physical Layer Security With Finite-Alphabet Signaling <s> F. Performance-Complexity Trade-Off in Transmit Signal Design <s> With the growth of wireless networks, security has become a fundamental issue in wireless communications due to the broadcast nature of these networks. In this work, we consider MIMO wiretap channels in a fast fading environment, for which the overall performance is characterized by the ergodic MIMO secrecy rate. Unfortunately, the direct solution to finding ergodic secrecy rates is prohibitive due to the expectations in the rates expressions in this setting. To overcome this difficulty, we invoke the large-system assumption, which allows a deterministic approximation to the ergodic mutual information. Leveraging results from random matrix theory, we are able to characterize the achievable ergodic secrecy rates. Based on this characterization, we address the problem of covariance optimization at the transmitter. Our numerical results demonstrate a good match between the large-system approximation and the actual simulated secrecy rates, as well as some interesting features of the precoder optimization. <s> BIB003 </s> An Overview of Physical Layer Security With Finite-Alphabet Signaling <s> F. Performance-Complexity Trade-Off in Transmit Signal Design <s> Security issues in wireless networks have become a subject of growing interest in recent years due to the broadcast nature of wireless channels. In this paper, we investigate secure communication over a multi-antenna wiretap channel in the presence of randomly distributed eavesdroppers. In the fast fading environment, the overall performance of this channel is traditionally characterized by the ergodic secrecy capacity, which, in general, cannot be derived explicitly. Nevertheless, based on the assumption that the numbers of antennas at legitimate terminals and the number of eavesdroppers grow large without bound, we derive a deterministic approximation of an achievable ergodic secrecy rate for arbitrary inputs. In addition, we characterize the secrecy rates for practically relevant separate-decoding scheme at the receiver. We validate the proposed large-system approximation through numerical simulations and observe a good match with the actual secrecy rates. Finally, we also analyze some interesting behavior of the secrecy rates in the given scenario depending on the geometry of the nodes. <s> BIB004 </s> An Overview of Physical Layer Security With Finite-Alphabet Signaling <s> F. Performance-Complexity Trade-Off in Transmit Signal Design <s> We propose a low complexity transmit signal design scheme for achieving information-theoretic secrecy over a MIMO wiretap channel driven by finite-alphabet inputs. We assume that the transmitter has perfect channel state information (CSI) of the main channel and also knows the statistics of the eavesdropper's channel. The proposed transmission scheme relies on jointly optimizing the precoder matrix and the artificial noise so as to maximize the achievable secrecy rates. In order to lower the computational complexity associated with the transmit signal design, we employ a design metric using the cut-off rate instead of the mutual information. We formulate a gradient-descent based optimization algorithm and demonstrate via extensive numerical examples that the proposed signal design scheme can yield an enhanced secrecy performance compared with the existing solutions in spite of its relatively lower computational complexity. The impacts of the modulation order as well as the number of antennas at the transmitter and receiver ends on the achievable secrecy rates are also investigated. <s> BIB005 </s> An Overview of Physical Layer Security With Finite-Alphabet Signaling <s> F. Performance-Complexity Trade-Off in Transmit Signal Design <s> We consider precoder and artificial noise (AN) design for multi-antenna wiretap channels under the finite-alphabet input assumption. We assume that the transmitter has access to the channel coefficients of the legitimate receiver and knows the statistics of the eavesdropper’s channel. Accordingly, we propose a secrecy rate maximization algorithm using a gradient descent-based optimization of the precoder matrix and an exhaustive search over the power levels allocated to the AN. We also propose algorithms to reduce the complexities of direct ergodic secrecy rate maximization by: 1) maximizing a cut-off rate-based approximation for the ergodic secrecy rate, simplifying the mutual information expression, which lacks a closed-form and 2) diagonalizing the channels toward the legitimate receiver and the eavesdropper, which allows for employing a per-group precoding-based technique. Our numerical results reveal that jointly optimizing the precoder and the AN outperforms the existing solutions in the literature, which rely on the precoder optimization only. We also demonstrate that the proposed low complexity alternatives result in a small loss in performance while offering a significant reduction in computational complexity. <s> BIB006
The iterative algorithms proposed in BIB001 and BIB006 for optimizing the precoder matrix are shown to offer promising results in terms of the achievable secrecy rates. However, their implementation complexity may be too high for some practical applications. There are two main reasons for this high computational complexity. First, both the mutual information and its gradient lack closed-form expressions. Therefore, Monte Carlo simulations need to be utilized to evaluate them, which require averaging over a large number of noise realizations in order to maintain a high level of accuracy. More importantly, evaluation of the mutual information and the minimum MSE (MMSE) term involves additions over the modulation signal space, which grows exponentially with the number of transmit antennas, resulting in a prohibitively high computational complexity when the number of transmit antennas is large. Different strategies are proposed for lowering the computational complexity associated with the transmit signal design for secrecy under the finite-alphabet input assumption. Wu et al. BIB001 and Aghdam and Duman BIB005 derive closed-form approximations for the secrecy rate expressions using bounds on the mutual information. While the former employs the bounds given in BIB002 , the latter approximates the mutual information using the cut-off rate expression. Girnyk et al. BIB003 , BIB004 derive asymptotic secrecy rates in a regime where the numbers of antennas at the transmitter and both receivers grow infinitely large with a constant ratio. Maximization of this expression is shown to yield a satisfactory performance even in the cases of small number of antennas. Aghdam and Duman BIB006 and Wu et al. [63] propose per-group precoding strategies under perfect and partial CSIT assumptions, respectively. In these works, the channels are diagonalized, and the precoder matrices are designed by grouping the transmit antennas and designing the precoder matrix for each group, independently.