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IEEE Globecom Workshops (GC Wkshps)

On beamforming solution for secrecy capacity of MIMO wiretap channels

Jiangyuan Li, Athina P. Petropulu

Published Dec 1, 2011• 12 citations
Publisher

Abstract

A Gaussian multiple-input multiple-output (MIMO) wiretap channel model is considered, where there exists a transmitter, a legitimate receiver and an eavesdropper each equipped with multiple antennas. The problem of finding the optimal input covariance matrix that achieves secrecy capacity subject to a power constraint is studied. In particular, it is shown that when the difference between the Grams of legitimate and eavesdropper channel matrices has all negative eigenvalues except one positive eigenvalue, beamforming is optimal. For that case, the secrecy capacity is obtained.

DOI
10.1109/glocomw.2011.6162584

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A Gaussian multiple-input multiple-output (MIMO) wiretap channel model is considered, where there exists a transmitter, a legitimate receiver and an eavesdropper each equipped with multiple antennas. The problem of finding the optimal input covariance matrix that achieves secrecy capacity subject to a power constraint is studied. In particular, it is shown that when the difference between the Grams of legitimate and eavesdropper channel matrices has all negative eigenvalues except one positive eigenvalue, beamforming is optimal. For that case, the secrecy capacity is obtained.
A Gaussian multiple-input multiple-output (MIMO) wiretap channel model is considered, where there exists a transmitter, a legitimate receiver and an eavesdropper each equipped with multiple antennas. The optimality of beamforming for secrecy capacity subject to sum power constraint is studied, and two sufficient conditions for beamforming to be globally optimal are given. The first sufficient condition states that when the difference between the Gram matrices of legitimate and eavesdropper channel matrices has exactly one positive eigenvalue, then beamforming is globally optimal. An alternative sufficient condition, which involves convex optimization, is also provided. For the case in which beamforming is globally optimal, the secrecy capacity is obtained. Otherwise, an upper bound of the difference between the secrecy capacity and beamforming secrecy rate is provided.
We consider a Gaussian multiple-input multiple-output (MIMO) wiretap channel model, where there exists a transmitter, a legitimate receiver and an eavesdropper, each node equipped with multiple antennas. We study the problem of finding the optimal input covariance matrix that achieves secrecy capacity subject to a power constraint, which leads to a non-convex optimization problem that is in general difficult to solve. Existing results for this problem address the case in which the transmitter and the legitimate receiver have two antennas each and the eavesdropper has one antenna. For the general cases, it has been shown that the optimal input covariance matrix has low rank when the difference between the Grams of the eavesdropper and the legitimate receiver channel matrices is indefinite or semi-definite, while it may have low rank or full rank when the difference is positive definite. In this paper, the aforementioned non-convex optimization problem is investigated. In particular, for the multiple-input single-output (MISO) wiretap channel, the optimal input covariance matrix is obtained in closed form. For general cases, we derive the necessary conditions for the optimal input covariance matrix consisting of a set of equations. For the case in which the transmitter has two antennas, the derived necessary conditions can result in a closed form solution; For the case in which the difference between the Grams is indefinite and has all negative eigenvalues except one positive eigenvalue, the optimal input covariance matrix has rank one and can be obtained in closed form; For other cases, the solution is proved to be a fixed point of a mapping from a convex set to itself and an iterative procedure is provided to search for it. Numerical results are presented to illustrate the proposed theoretical findings.
A Gaussian multiple-input multiple-output (MIMO) wiretap channel model is considered. The input is a two-antenna transmitter, while the outputs are the legitimate receiver and an eavesdropper, both equipped with multiple antennas. All channels are assumed to be known. The problem of obtaining the optimal input covariance matrix that achieves secrecy capacity subject to a sum power constraint is addressed, and a closed-form expression for the secrecy capacity is obtained. The sufficient and necessary condition for beamforming to be optimal is also given.
We consider a multiple-input multiple-output (MIMO) Gaussian wiretap channel model, where there exists a transmitter, a legitimate receiver and an eavesdropper, each node equipped with multiple antennas. We study the problem of finding the optimal input covariance that achieves secrecy capacity subject to a power constraint, which in general leads to a difficult non-convex optimization problem. For the Gaussian multiple-input single-output (MISO) wiretap channel where there exists a transmitter equipped with multiple antennas, a legitimate receiver and an eavesdropper each equipped with one antenna, the optimal input covariance is obtained in closed form. For the general case, we derive the necessary conditions for the optimal input covariance in the form of a set of equations. We show that for MISO case, the derived conditions guarantee the optimal input covariance. Numerical results are presented to illustrate the proposed theoretical findings.
This paper considers a multiple-input multiple-output (MIMO) Gaussian wiretap channel, where there exists a transmitter, a legitimate receiver and an eavesdropper, each equipped with multiple antennas. In particular, we study the rank of the optimal input covariance matrix that achieves the secrecy capacity of the multiple antenna MIMO Gaussian wiretap channel under an average power constraint. The rank and other properties of the optimal solution are derived based on certain relationships between the channel matrices for the legitimate receiver and eavesdropper. Such properties are useful steps towards characterizing the general solution to the MIMO wiretap problem with an average power constraint.