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Allerton Conference on Communication, Control, and Computing

The compound secrecy capacity of a class of non-degraded MIMO Gaussian channels

Rafael F. Schaefer, Sergey L. Loyka

Published Sep 1, 2014• 6 citations
Publisher

Abstract

Secrecy capacity of a class of non-degraded compound MIMO Gaussian channels is obtained. Earlier results established for isotropic uncertainty sets are extended to broader class of (non-isotropic) sets, which bound not only the gain but also the eigendirections of the eavesdropper channel. When a maximum element exists in the uncertainty set, a saddle-point exists so that the compound and worst-case channel capacities coincide and signaling on the worst-case channel also works for the whole class of channels. The case of additive uncertainty in the legitimate channel, in addition to the unknown eavesdropper channel of a bounded spectral norm, is also studied. Its compound secrecy capacity and the optimal signaling are established in a closed-form, revealing the saddle-point property. The optimal signaling is Gaussian and on the eigenvectors of the legitimate channel and the worst-case eavesdropper is isotropic. The eigenmode power allocation somewhat resembles the standard water-filling but is not identical to it.

DOI
10.1109/allerton.2014.7028564

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EURASIP Journal on Wireless Communications and Networking•Oct 20, 2009•309 citations
Secrecy capacity of a class of non-degraded compound MIMO Gaussian channels is obtained. Earlier results established for isotropic uncertainty sets are extended to broader class of (non-isotropic) sets, which bound not only the gain but also the eigendirections of the eavesdropper channel. When a maximum element exists in the uncertainty set, a saddle-point exists so that the compound and worst-case channel capacities coincide and signaling on the worst-case channel also works for the whole class of channels. The case of additive uncertainty in the legitimate channel, in addition to the unknown eavesdropper channel of a bounded spectral norm, is also studied. Its compound secrecy capacity and the optimal signaling are established in a closed-form, revealing the saddle-point property. The optimal signaling is Gaussian and on the eigenvectors of the legitimate channel and the worst-case eavesdropper is isotropic. The eigenmode power allocation somewhat resembles the standard water-filling but is not identical to it.
Strong secrecy capacity of compound wiretap channels is studied. The known lower bounds for the secrecy capacity of compound finite-state memoryless channels under discrete alphabets are extended to arbitrary uncertainty sets and continuous alphabets under the strong secrecy criterion. The conditions under which these bounds are tight are given. Under the saddle-point condition, the compound secrecy capacity is shown to be equal to that of the worst-case channel. Based on this, the compound Gaussian multiple-input multiple-output wiretap channel is studied under the spectral norm constraint and without the degradedness assumption. First, it is assumed that only the eavesdropper channel is unknown, but is known to have a bounded spectral norm (maximum channel gain). The compound secrecy capacity is established in a closed form and the optimal signaling is identified. The compound capacity equals the worst-case channel capacity and thus establishing the saddlepoint property; the optimal signaling is Gaussian and on the eigenvectors of the legitimate channel and the worst-case eavesdropper is isotropic. The eigenmode power allocation somewhat resembles the standard water-filling but is not identical to it. More general uncertainty sets are considered and the existence of a maximum element is shown to be sufficient for a saddle-point to exist, so that signaling on the worst-case channel achieves the compound capacity of the whole class of channels. The case of rank-constrained eavesdropper is considered and the respective compound secrecy capacity is established. Subsequently, the case of additive uncertainty in the legitimate channel, in addition to the unknown eavesdropper channel, is studied. Its compound secrecy capacity and the optimal signaling are established in a closed form as well, revealing the same saddle-point property. When a saddle-point exists under strong secrecy, strong and weak secrecy compound capacities are equal.
The compound MIMO Gaussian wiretap channel is studied, where the channel to the legitimate receiver is known and the eavesdropper channel is not known to the transmitter but is known to have a bounded spectral norm (channel gain). The compound secrecy capacity is established without the de-gradedness assumption and the optimal signaling is identified: the compound capacity equals the worst-case channel capacity thus establishing the saddle-point property, the optimal signaling is Gaussian and on the eigenvectors of the legitimate channel and the worst-case eavesdropper is isotropic. The eigenmode power allocation somewhat resembles the standard water-filling but is not identical to it.
The secrecy capacity of compound wiretap channels is studied. Using the known lower bounds for non-degraded discrete memoryless channels and establishing the converse, the secrecy capacity is established under the less capable and noisier conditions. The compound capacity is shown to be equal to the worst-case channel capacity when there exists a saddle- point in the mutual information difference. Earlier results on the secrecy capacity of compound Gaussian MIMO wiretap channels under the spectral norm constraint are extended to the case of rank-deficient eavesdroppers (which may model a massive MIMO base station) without the degradedness assumption. Its compound secrecy capacity is established in a closed form and the optimal signaling is identified: the compound capacity equals the worst-case channel capacity thus establishing the saddle-point property; the optimal signaling is Gaussian and on the eigenvectors of the legitimate channel and the worst-case eavesdropper is omnidirectional (i.e. isotropic over the sub-space spanned by the active eigenmodes of the legitimate channel). The case of additive uncertainty in the legitimate channel, in addition to the unknown eavesdropper channel, is also studied and its compound secrecy capacity is established.
Secrecy rate maximization in Gaussian MIMO wiretap channels is considered. While the optimality of Gaussian signaling and a general expression for the secrecy capacity have been well established, closed-form solutions for the optimal transmit covariance matrix are known for some special cases only, while the general case remains an open problem. This chapter reviews known closed-form solutions and presents a numerical algorithm for the general case with guaranteed convergence to the global optimum. The known solutions include full-rank and rank-1 cases (which, when combined, provide a complete solution for the case of two transmit antennas), the case of identical right singular vectors for the eavesdropper and legitimate channels, and the cases of weak, isotropic, and omnidirectional eavesdroppers, which also provide lower and upper bounds to the general case. Necessary optimality conditions and a tight upper bound for the rank of the optimal covariance matrix in the general case are discussed. Sufficient and necessary conditions for the optimality of three popular signaling strategies over MIMO channels, namely, isotropic and zero-forcing signaling as well as water-filling over the legitimate channel eigenmodes, are presented. The chapter closes with a detailed description of a numerical globally convergent algorithm to solve the general case, and gives some illustrative examples. Introduction Due to their high spectral efficiency, wireless MIMO (multiple input, multiple output) systems are widely adopted by academia and industry. The broadcast nature of wireless channels stimulated significant interest in their security aspects and the Gaussian MIMO wiretap channel (WTC) has emerged as a popular model to study information theoretic secrecy aspects of wireless systems [1]. A number of results have been obtained for this model, including the proof of optimality of Gaussian signaling [1–4], which is far from trivial and significantly more involved than that of the regular (no wiretap) MIMO channel. Once the functional form of the optimal input is established, the only unknown is its covariance matrix since the mean is always zero. This latter part has not been solved yet in the general case; only a number of special cases have been settled.
Optimal signalling over the wire-tap MIMO Gaussian channel is studied under the total transmit power constraint. The recent results are extended in several directions, including a rank-deficient solution for the optimal covariance, lower and upper capacity bounds for the general case, and characterization of optimality of the isotropic signaling. An isotropic eavesdropper model is studied, which provides (tight) upper and lower capacity bounds for the non-isotropic case and also serves as the worst-case scenario. The optimal signaling for this model is obtained in an explicit form and its properties are studied, including the high and low-SNR behavior, the conditions for the eavesdropper to be negligible and the capacity saturation effect.
This paper considers the compound wiretap channel, which generalizes Wyner's wiretap model to allow the channels to the (legitimate) receiver and to the eavesdropper to take a number of possible states. No matter which states occur, the transmitter guarantees that the receiver decodes its message and that the eavesdropper is kept in full ignorance about the message. The compound wiretap channel can also be viewed as a multicast channel with multiple eavesdroppers, in which the transmitter sends information to all receivers and keeps the information secret from all eavesdroppers. For the discrete memoryless channel, lower and upper bounds on the secrecy capacity are derived. The secrecy capacity is established for the degraded channel and the semideterministic channel with one receiver. The parallel Gaussian channel is further studied. The secrecy capacity and the secrecy degree of freedom ( ) are derived for the degraded case with one receiver. Schemes to achieve the for the case with two receivers and two eavesdroppers are constructed to demonstrate the necessity of a prefix channel in encoder design. Finally, the multi-antenna (i.e., MIMO) compound wiretap channel is studied. The secrecy capacity is established for the degraded case and an achievable is given for the general case.