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arXiv:1803.02107 · Globecom

Optimal Grassmann Manifold Eavesdropping: A Huge Security Disaster for M-1-2 Wiretap Channels

Dongyang Xu, Pinyi Ren, James A. Ritcey

Published Dec 1, 2017• 5 citations
arXivHTMLPDF

Abstract

We in this paper introduce an advanced eavesdropper that aims to paralyze the artificial-noise-aided secure communications. We consider the M-1-2 Gaussian MISO wiretap channel, which consists of a M-antenna transmitter, a single-antenna receiver, and a two-antenna eavesdropper. This type of eavesdropper, by adopting an optimal Grassmann manifold (OGM) filtering structure, can reduce the maximum achievable secrecy rate (MASR) to be zero by using only two receive antennas, regardless of the number of antennas at the transmitter. Specifically, the eavesdropper exploits linear filters to serially recover the legitimate information symbols and intends to find the optimal filter that minimizes the mean-square error (MSE) in estimating the symbols. During the process, a convex semidefinite programming (SDP) problem with constraints on the filter matrix can be formulated and solved. Interestingly, the resulted optimal filters constitute a complex Grassmann manifold on the matrix space. Based on the filters, a novel expression of MASR is derived and further verified to be zero under the noiseless environment. Besides this, an achievable variable region (AVR) that induces zero MASR is presented analytically in the noisy case. Numerical results are provided to illustrate the huge disaster in the respect of secrecy rate.

DOI
10.1109/glocom.2017.8254502

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Globecom•Dec 1, 2010•1 citation
We in this paper introduce an advanced eavesdropper that aims to paralyze the artificial-noise-aided secure communications. We consider the M-1-2 Gaussian MISO wiretap channel, which consists of a M-antenna transmitter, a single-antenna receiver, and a two-antenna eavesdropper. This type of eavesdropper, by adopting an optimal Grassmann manifold (OGM) filtering structure, can reduce the maximum achievable secrecy rate (MASR) to be zero by using only two receive antennas, regardless of the number of antennas at the transmitter. Specifically, the eavesdropper exploits linear filters to serially recover the legitimate information symbols and intends to find the optimal filter that minimizes the mean-square error (MSE) in estimating the symbols. During the process, a convex semidefinite programming (SDP) problem with constraints on the filter matrix can be formulated and solved. Interestingly, the resulted optimal filters constitute a complex Grassmann manifold on the matrix space. Based on the filters, a novel expression of MASR is derived and further verified to be zero under the noiseless environment. Besides this, an achievable variable region (AVR) that induces zero MASR is presented analytically in the noisy case. Numerical results are provided to illustrate the huge disaster in the respect of secrecy rate.
We consider secure communications over MISO wiretap channels, in the presence of a passive eavesdropper with multiple antennas. In this scenario, an artificial-noise-resistant (ANR) eavesdropping behavior is introduced to invalidate the artificial noise (AN) scheme proposed by Goel et al. This novel eavesdropper, by exploiting the statistical filter derived from the collected signals, can completely eliminate the influence of AN by using only two receive antennas. In particular, the received signals are firstly syphered to eliminate the power influence of AN by a linear weight which is generated from the statistical estimation for the signal covariance matrix. In comparison with the original legitimate signals, the weighted signals are however imposed by a phase difference which can be then erased by a weight vector inferred from the available information at the eavesdropper. Based on the two filtering processes, we derive a novel expression of achievable secrecy rate and give an analytical expression for the zero-secrecy-rate distance of eavesdropper to the transmitter. Finally, we characterize the expression of maximum achievable secrecy rate (MASR) and show that the optimal power allocation strategy under an ANR eavesdropping behavior is transformed into no allocation of transmission power to AN. Numerical results are presented to illustrate the damage caused by the investigated eavesdropping behavior. Interestingly, the consequence of AN elimination is not influenced even under large number of transmit antennas.
In this paper, we study the achievable secrecy rate of the multiple input single output (MISO) wiretap channel, which consists of a transmitter, a legitimate receiver and multiple eavesdroppers. The transmitter is equipped with multiple antennas, while the receiver and all the eavesdroppers each have a single antenna. We assume that the transmitter has perfect channel state information (CSI) for their links to legitimate receiver, but it has only imperfect CSI for their channels to eavesdroppers. Optimal transmit covariance matrix is obtained based on worst-case secrecy rate maximization, under power constraints. The maximum secrecy rate computation can be formulated as a non-convex optimal problem, which cannot be solved directly. In this paper, we transform non-convex optimal problem into a quasi-convex optimal problem which can be efficiently solved with the bisection method. The simulation result and analysis demonstrate the effectiveness of our methods.
This paper considers the problem of filter design with secrecy constraints, where two legitimate parties (Alice and Bob) communicate in the presence of an eavesdropper (Eve) over a Gaussian multiple-input-multiple-output (MIMO) wiretap channel. This problem involves designing, subject to a power constraint, the transmit and the receive filters which minimize the mean-squared error (MSE) between the legitimate parties whilst assuring that the eavesdropper MSE remains above a certain threshold. We consider a general MIMO Gaussian wiretap scenario, where the legitimate receiver uses a linear zero-forcing (ZF) filter and the eavesdropper receiver uses either a ZF or an optimal linear Wiener filter. We provide a characterization of the optimal filter designs by demonstrating the convexity of the optimization problems. We also provide generalizations of the filter designs from the scenario where the channel state is known to all the parties to the scenario where there is uncertainty in the channel state. A set of numerical results illustrates the performance of the novel filter designs, including the robustness to channel modeling errors. In particular, we assess the efficacy of the designs in guaranteeing not only a certain MSE level at the eavesdropper, but also in limiting the error probability at the eavesdropper. We also assess the impact of the filter designs on the achievable secrecy rates. The penalty induced by the fact that the eavesdropper may use the optimal nonlinear receive filter rather than the optimal linear one is also explored in the paper.
This paper considers the problem of filter design with secrecy constraints, where two legitimate parties (Alice and Bob) communicate in the presence of an eavesdropper (Eve), over a Gaussian multiple-input multiple-output (MIMO) wiretap channel. This problem involves the design of transmit and receive filters which minimize the mean-square error (MSE) between the legitimate parties, whilst assuring that the eavesdropper MSE remains above a certain level. We characterize the form of the optimal transmit filter when both the legitimate receiver and the eavesdropper employ Zero-Forcing (ZF) filters. By capitalizing on the dual problem, we also show that the original matrix optimization problem can be reduced to a simple scalar optimization problem, whose solution can be readily computed by employing a simple bisection method. Numerical results illustrate the main conclusions.
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 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 achieves ergodic secrecy rate subject to a power constraint where the full information on the legitimate channel is known to the transmitter, but only statistical information on the eavesdropper channel is available at the transmitter. Existing results address the case in which the eavesdropper channel has independent and identically distributed Gaussian entries with zero-mean, i.e., the channel has trivial covariance. This paper addresses the general case where eavesdropper channel has nontrivial covariance. A set of equations describing the optimal input covariance matrix are proposed. Based on this framework, we show that the optimal input covariance has always rank one. Numerical results are presented to illustrate the algorithm.