An Overview of Physical Layer Security With Finite-Alphabet Signaling <s> A. Transmit Signal Design With Full CSI <s> The capacity of the Gaussian wiretap channel model is analyzed when there are multiple antennas at the sender, intended receiver and eavesdropper. The associated channel matrices are fixed and known to all the terminals. A computable characterization of the secrecy capacity is established as the saddle point solution to a minimax problem. The converse is based on a Sato-type argument used in other broadcast settings, and the coding theorem is based on Gaussian wiretap codebooks. At high signal-to-noise ratio (SNR), the secrecy capacity is shown to be attained by simultaneously diagonalizing the channel matrices via the generalized singular value decomposition, and independently coding across the resulting parallel channels. The associated capacity is expressed in terms of the corresponding generalized singular values. It is shown that a semi-blind "masked" multi-input multi-output (MIMO) transmission strategy that sends information along directions in which there is gain to the intended receiver, and synthetic noise along directions in which there is not, can be arbitrarily far from capacity in this regime. Necessary and sufficient conditions for the secrecy capacity to be zero are provided, which simplify in the limit of many antennas when the entries of the channel matrices are independent and identically distributed. The resulting scaling laws establish that to prevent secure communication, the eavesdropper needs three times as many antennas as the sender and intended receiver have jointly, and that the optimum division of antennas between sender and intended receiver is in the ratio of 2:1. <s> BIB001 </s> An Overview of Physical Layer Security With Finite-Alphabet Signaling <s> A. Transmit Signal Design With Full CSI <s> This work investigates the effect of finite-alphabet source input on the secrecy rate of a multi-antenna wiretap system. Existing works have characterized maximum achievable secrecy rate or secrecy capacity for single and multiple antenna systems based on Gaussian source signals and secrecy code. Despite the impracticality of Gaussian sources, the compact closed-form expression of mutual information between linear channel Gaussian input and corresponding output has led to broad application of Gaussian input assumption in physical secrecy analysis. For practical considerations, we study the effect of finite discrete-constellation on the achievable secrecy rate of multiple-antenna wire-tap channels. Our proposed precoding scheme converts the multi-antenna system into a bank of parallel channels. Based on this precoding strategy, we propose a decentralized power allocation algorithm based on dual decomposition for maximizing the achievable secrecy rate. In addition, we analyze the achievable secrecy rate for finite-alphabet inputs in low and high SNR cases. Our results demonstrate substantial difference in secrecy rate between systems given finite-alphabet inputs and systems with Gaussian inputs. <s> BIB002 </s> An Overview of Physical Layer Security With Finite-Alphabet Signaling <s> A. Transmit Signal Design With Full CSI <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> BIB003 </s> An Overview of Physical Layer Security With Finite-Alphabet Signaling <s> A. Transmit Signal Design With Full CSI <s> The wiretap channel is a setting where one aims to provide information-theoretic privacy of communicated data based solely on the assumption that the channel from sender to adversary is "noisier" than the channel from sender to receiver. It has developed in the Information and Coding IC it has optimal rate; and both the encryption and decryption algorithms are proven to be polynomial-time. <s> BIB004 </s> An Overview of Physical Layer Security With Finite-Alphabet Signaling <s> A. Transmit Signal Design With Full CSI <s> There has recently been significant interest in applying the principles of information-theoretical security and signal processing to secure physical layer systems. Although the community has made progress in understanding how the physical layer can support confidentiality and authentication, it is important to realize that there are many important issues that must be addressed if physical layer security is ever to be adopted by real and practical security systems. In this article, I briefly review several different flavors of physical layer security (at least for wireless systems), and then identify aspects (a.k.a. weaknesses) where the foundation for physical layer security needs to be strengthened. I then highlight that the opportunities for applying physical layer security to real systems will be quite rich if the community can overcome these challenges. In the course of the article, I note new directions for the community to investigate, with the objective of keeping physical layer security research targeted at having a practical impact on real systems. <s> BIB005 </s> An Overview of Physical Layer Security With Finite-Alphabet Signaling <s> A. Transmit Signal Design With Full CSI <s> Conventionally, interference and noise are treated as catastrophic elements in wireless communications. However, it has been shown recently that exploiting known interference constructively can contribute to signal detection ability at the receiving end. This paper exploits this concept to design artificial noise (AN) beamformers constructive to the intended receiver (IR) yet keeping AN disruptive to possible eavesdroppers (Eves). The scenario considered here is a multiple-input single-output wiretap channel with multiple Eves. This paper starts from AN design without any knowledge of Eve’s CSI, builds with solutions with statistical CSI up to full CSI. Both perfect and imperfect channel information have been considered, in particular, with different extent of Eves’ channel responses. The main objective is to improve the receive signal-to-interference and noise ratio at IR through exploitation of AN power in an attempt to minimize the total transmit power, while hindering detection at the Eves. Numerical simulations demonstrate that the proposed constructive AN precoding approach yields superior performance over conventional AN schemes in terms of transmit power. Critically, they show that, while the statistical constraints of conventional approaches may lead to instantaneous IR outages and security breaches from the Eves, the instantaneous constraints of our approach guarantee both IR performance and secrecy at every symbol period. <s> BIB006
The most optimistic assumption regarding the transmitter's CSI knowledge is availability of perfect instantaneous realizations of the channel matrices H B and H E . This assumption is justifiable in the scenarios where Eve is an authorized user in the network, however, she should be kept ignorant about the confidential messages transmitted from Alice to Bob. In this case, the transmitter can take advantage of the full CSI knowledge to design a precoder, which results in the largest quality difference between the signals received at the legitimate receiver and the eavesdropper. With the aid of the perfect CSI corresponding to both channels, a generalized singular value decomposition (GSVD) precoding solution is proposed in BIB002 using which the MIMO multi-antenna eavesdropper (MIMOME) channel is converted into a bank of parallel channels, and a power allocation strategy is formulated to maximize the achievable secrecy rate. The necessary conditions for optimum transmit precoding is derived in BIB003 , and it is demonstrated that the GSVD precoding BIB002 is suboptimal. Alternatively, a gradient descent optimization is proposed in which the precoder matrices are updated with steps proportional to the gradient of the instantaneous secrecy rate. Furthermore, it is shown that transmission along the null-space of the eavesdropper's channel is the optimal secure transmission strategy at the high-SNR regime. This is because, with a precoder matrix along the null-space of the eavesdropper's channel, her achievable rate is suppressed to zero, and when the SNR is sufficiently high, the rate at the legitimate receiver approaches the maximum value that can be achieved with the use of specific finite-alphabet inputs, which guarantees that the secrecy rate is maximized. However, it should be noted that, when the number of transmit antennas is less than or equal to the number of antennas at the eavesdropper, it is not possible to suppress the Eve's information rate by transmitting along the null-space of H E . Therefore, to maximize the secrecy rates, only part of the available power should be allocated to data transmission. Given N t > N B , Wu et al. BIB003 suggest to use the excess power for transmission of an artificial noise signal along the null-space of H B so as to prevent the secrecy rates from dropping to zero. Fig. 5 compares the achievable secrecy rates by the schemes proposed in BIB002 and BIB003 for a wiretap channel with N t = 2, N B = 1 and N E = 1, and fixed channels given by 1 The precoding approach proposed in BIB003 outperforms the GSVD-based precoding BIB002 . This improved performance is attained at the price of increased computational complexity. Secrecy rates with QPSK inputs over fixed channels given in BIB005 and BIB004 . Furthermore, comparing the results given in Fig. 5 with those of Fig. 3 for QPSK inputs over the Gaussian wiretap channel reveals the advantages offered by fading and CSI-based precoding approaches. An important conclusion that can be drawn from the results in BIB002 and BIB003 is that, as in the case of non-fading channels, the structure of the optimal precoders under the finite-alphabet input assumption is different from that of the optimal precoders for Gaussian inputs. For example, over MISO multi-antenna eavesdropper (MISOME) channels, the optimal transmission strategy is beamforming along the eigenvector corresponding to the largest generalized eigenvalue of (H b , H e ) BIB001 . However, these precoders (which correspond to signaling with rank one covariance), when applied to finitealphabet inputs, undergo a considerable loss with respect to the precoders specifically optimized for this case (see Fig. 7 in BIB003 ). As an alternative method to design a suboptimal precoder, one may also replace the secrecy capacity with practical metrics (as opposed to those ensuring secrecy in an informationtheoretic sense) such as SINR-based metrics. For instance, Khandaker et al. BIB006 design artificial noise beamformers, which serve as constructive interference to the legitimate receiver (improving Bob's SINR) while disrupting the reception at the eavesdropper (degrading Eve's SINR). In Table II , we summarize the precoder design approaches with full CSI.