An Overview of Physical Layer Security With Finite-Alphabet Signaling <s> C. Secrecy Capacity of Multi-Antenna Wiretap Channels <s> Given two discrete memoryless channels (DMC's) with a common input, it is desired to transmit private messages to receiver 1 rate R_{1} and common messages to both receivers at rate R_{o} , while keeping receiver 2 as ignorant of the private messages as possible. Measuring ignorance by equivocation, a single-letter characterization is given of the achievable triples (R_{1},R_{e},R_{o}) where R_{e} is the equivocation rate. Based on this channel coding result, the related source-channel matching problem is also settled. These results generalize those of Wyner on the wiretap channel and of Korner-Marton on the broadcast Channel. <s> BIB001 </s> An Overview of Physical Layer Security With Finite-Alphabet Signaling <s> C. Secrecy Capacity of Multi-Antenna Wiretap Channels <s> We consider the MIMO wiretap channel, that is a MIMO broadcast channel where the transmitter sends some confidential information to one user which is a legitimate receiver, while the other user is an eavesdropper. Perfect secrecy is achieved when the transmitter and the legitimate receiver can communicate at some positive rate, while insuring that the eavesdropper gets zero bits of information. In this paper, we compute the perfect secrecy capacity of the multiple antenna MIMO broadcast channel, where the number of antennas is arbitrary for both the transmitter and the two receivers. Our technique involves a careful study of a Sato-like upper bound via the solution of a certain algebraic Riccati equation. <s> BIB002 </s> An Overview of Physical Layer Security With Finite-Alphabet Signaling <s> C. Secrecy Capacity of Multi-Antenna Wiretap Channels <s> The secrecy capacity of the multiple-antenna wiretap channel under the average total power constraint was recently characterized, independently, by Khisti and Wornell and Oggier and Hassibi using a Sato-like argument and matrix analysis tools. This paper presents an alternative characterization of the secrecy capacity of the multiple-antenna wiretap channel under a more general matrix constraint on the channel input using a channel-enhancement argument. This characterization is by nature information-theoretic and is directly built on the intuition regarding to the optimal transmission strategy in this communication scenario. <s> BIB003 </s> An Overview of Physical Layer Security With Finite-Alphabet Signaling <s> C. Secrecy Capacity of Multi-Antenna Wiretap Channels <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 3 times as many antennas as the sender and intended receiver have jointly, and that the optimimum division of antennas between sender and intended receiver is in the ratio of 2:1. <s> BIB004 </s> An Overview of Physical Layer Security With Finite-Alphabet Signaling <s> C. Secrecy Capacity of Multi-Antenna Wiretap Channels <s> A Gaussian multiple-input single-output (MISO) wiretap channel model is considered, where there exists a transmitter equipped with multiple antennas, a legitimate receiver and an eavesdropper, each equipped with a single antenna. We study the problem of finding the optimal input covariance that maximizes the ergodic secrecy rate subject to a power constraint, where only statistical information about the eavesdropper channel is available at the transmitter. This is a non-convex optimization problem that is in general difficult to solve. Existing results address the case in which the eavesdropper or/and legitimate channels have independent and identically distributed Gaussian entries with zero mean and unit variance, i.e., the channels have trivial covariances. This paper addresses the general case in which the eavesdropper and legitimate channels have nontrivial covariances. A set of equations describing the optimal input covariance matrix are proposed along with an algorithm to obtain the solution. Based on this framework, it is shown that when full information on the legitimate channel is available to the transmitter, the optimal input covariance has always rank one. It is also shown that when only statistical information on the legitimate channel is available to the transmitter, the legitimate channel has some general non-trivial covariance and the eavesdropper channel has trivial covariance, the optimal input covariance has the same eigenvectors as the legitimate channel covariance. <s> BIB005 </s> An Overview of Physical Layer Security With Finite-Alphabet Signaling <s> C. Secrecy Capacity of Multi-Antenna Wiretap Channels <s> This paper considers transmit optimization in multi-input multi-output (MIMO) wiretap channels, wherein we aim at maximizing the secrecy capacity or rate of an MIMO channel overheard by one or multiple eavesdroppers. Such optimization problems are nonconvex, and appear to be difficult especially in the multi-eavesdropper scenario. In this paper, we propose an alternating optimization (AO) approach to tackle these secrecy optimization problems. We first consider the secrecy capacity maximization (SCM) problem in the single eavesdropper scenario. An AO algorithm is derived through a judicious SCM reformulation. The algorithm conducts some kind of reweighting and water-filling in an alternating fashion, and thus is computationally efficient to implement. We also prove that the AO algorithm is guaranteed to converge to a Karush-Kuhn-Tucker (KKT) point of the SCM problem. Then, we turn our attention to the multiple eavesdropper scenario, where the artificial noise (AN)-aided secrecy rate maximization (SRM) problem is considered. Although the AN-aided SRM problem has a more complex problem structure than the previous SCM, we show that AO can be extended to deal with the former, wherein the problem is handled by solving convex problems in an alternating fashion. Again, the resulting AO method is proven to have KKT point convergence guarantee. For fast implementation, a custom-designed AO algorithm based on smoothing and projected gradient is also derived. The secrecy rate performance and computational efficiency of the proposed algorithms are demonstrated by simulations. <s> BIB006 </s> An Overview of Physical Layer Security With Finite-Alphabet Signaling <s> C. Secrecy Capacity of Multi-Antenna Wiretap Channels <s> 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. <s> BIB007
Multiple-input multiple-output (MIMO) wiretap channel is an extension of the wiretap channel to a scenario where all the nodes, namely, Alice, Bob and Eve, are equipped with multiple antennas. Under the full CSI assumption, secrecy capacity for this setup is determined independently by Khisti and Wornell BIB004 and Oggier and Hassibi BIB002 . The results in BIB001 are employed by Khisti and Wornell so as to develop a genie-aided upper bound on the secrecy capacity in the general case of non-degraded MIMO wiretap channels. While their characterization is through a saddlepoint, Oggier and Hassibi propose an alternative approach through a single optimization process. It is shown that the achievable secrecy rate is maximized when the channel input is Gaussian, and the secrecy capacity of a MIMO wiretap channel is given by where K x = E{xx H }, and H B ∈ C N B ×Nt and H E ∈ C N E ×Nt are the channel matrices corresponding to the legitimate receiver and the eavesdropper, respectively. An alternative characterization of the secrecy capacity of the MIMO wiretap channel is presented by Liu and Shamai BIB003 where they consider a more general matrix constraint on the channel input. We highlight that, while BIB004 - BIB003 prove that the optimal input is Gaussian, the optimal input covariance matrix needs to be determined using numerical optimization approaches (see BIB006 , BIB007 ). A set of equations describing the optimal channel input under partial CSI at the transmitter (CSIT) along with algorithms for obtaining the solution are given in BIB005 .