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Filter Design with Secrecy Constraints: The Degraded Parallel Gaussian Wiretap Channel

Miguel R. D. Rodrigues, Pedro D. M. Almeida

Published Jan 1, 2008• 18 citations
Publisher

Abstract

We introduce the problem of filter design with secrecy constraints in the classical wiretap scenario, where two legitimate parties, Alice and Bob, communicate in the presence of an eavesdropper, Eve. In particular, we consider the design of transmit and receive filters that minimize the mean-squared error (MSE) between the legitimate parties whilst guaranteeing a certain eavesdropper MSE level subject to a total average power constraint, in the situation where the main channel and the wiretap channel consist of a bank of parallel independent degraded Gaussian channels (a scenario representative of orthogonal frequency division multiplexing (OFDM) communications systems). We derive the form of the optimal receive filter and the optimal transmit filter, which are diagonal. In the regime of low available power, we demonstrate that the diagonal elements of the optimal transmit filter satisfy a waterfilling type of expression. In contrast, in the regime of high available power the diagonal elements of the optimal transmit filter satisfy a simple expression involving the linear minimum mean-squared error (LMMSE) that obeys a simple operational interpretation. A range of numerical results corroborate the conclusions.

DOI
10.1109/glocom.2008.ecp.666

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We introduce the problem of filter design with secrecy constraints in the classical wiretap scenario, where two legitimate parties, Alice and Bob, communicate in the presence of an eavesdropper, Eve. In particular, we consider the design of transmit and receive filters that minimize the mean-squared error (MSE) between the legitimate parties whilst guaranteeing a certain eavesdropper MSE level subject to a total average power constraint, in the situation where the main channel and the wiretap channel consist of a bank of parallel independent degraded Gaussian channels (a scenario representative of orthogonal frequency division multiplexing (OFDM) communications systems). We derive the form of the optimal receive filter and the optimal transmit filter, which are diagonal. In the regime of low available power, we demonstrate that the diagonal elements of the optimal transmit filter satisfy a waterfilling type of expression. In contrast, in the regime of high available power the diagonal elements of the optimal transmit filter satisfy a simple expression involving the linear minimum mean-squared error (LMMSE) that obeys a simple operational interpretation. A range of numerical results corroborate the conclusions.
This paper considers the problem of Alter design with secrecy constraints, where two legitimate parties, Alice and Bob, communicate in the presence of an eavesdropper, Eve, over multiple-input multiple-output (MIMO) Gaussian channels. In particular, we consider the design of transmit and receive filters that minimize the mean-squared error (MSE) between the legitimate parties subject to a certain eavesdropper MSE level, in the situation where the eavesdropper MIMO channel is a degraded version of the main MIMO channel. We characterize the form of the optimal receive filters as well as the form of optimal transmit filter in different scenarios. We also put forth an iterative algorithm to obtain the optimal values of the transmit and receive filters. Finally, we present a set of numerical results that illustrate the conclusions.
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. In particular, this problem involves the design of the transmit and the receive filters, subject to a power constraint, which minimize the mean-squared error (MSE) between the legitimate parties whilst assuring that the eavesdropper MSE remains above a certain level. We consider a general Multiple-Input Multiple-Output (MIMO) Gaussian wiretap scenario, where a Zero-Forcing (ZF) filter is used at the legitimate receiver, whilst the eavesdropper uses an optimal linear receive filter, and characterize the receive and transmit filters in various power regimes. Finally, we present a set of numerical results that support some of the main conclusions.
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.
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 multiple-input multiple-output (MIMO) Gaussian channels. In particular, we consider the design of transmit and receive filters that minimize the mean-squared error (MSE) between the legitimate parties subject to a certain eavesdropper MSE level, in the situation where the eavesdropper MIMO channel is a degraded version of the main MIMO channel. We analyze the penalty in terms of MSE in the eavesdropper channel due to the assumption that optimal linear receive filters are used, while the eavesdropper employs nonlinear conditional mean estimation instead. This penalty is also shown to be negligible in regions of operational interest. We present a set of numerical results to illustrate the main conclusions.
In this paper, we investigate the problem of jointly designing optimum linear precoder and decoder with secrecy constraints, where two legitimate parties communicate in the presence of an eavesdropper over multiple-input multiple-output (MIMO) Gaussian channels. In particular, we focus on optimizing the linear transceiver to guarantee a certain quality of service (QoS) level for the legitimate receiver in terms of mean square error (MSE), whilst employing the artificial noise to maximally degrade the MSE for the eavesdropper. We assume that the transmitter may not or may have the channel state information (CSI) for the eavesdropper, and then propose two design formulations. With the aid of majorization theory, we characterize the forms of optimal transmit precoder when both the legitimate receiver and the eavesdropper employ the optimal linear MMSE receiver. Numerical results illustrate main conclusions.