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IEEE Transactions on Information Theory

Secure Communications With Insecure Feedback: Breaking the High-SNR Ceiling

Tung T. Kim, H. Vincent Poor

Published Jul 21, 2010• 6 citations
Publisher

Abstract

A multiple-antenna Gaussian wiretap channel in which the number of antennas at the source is not greater than that at the eavesdropper is considered. Without feedback, the secrecy capacity over such a channel generally converges to a constant at high signal-to-noise ratio (SNR). A half-duplex secure protocol allowing the destination to actively transfer random keys in the form of known interference is proposed. It is shown that using multiple antennas at the destination is instrumental in achieving a secrecy rate that grows linearly with logSNR. The pre-log factor of the secrecy rate, i.e., the number of secure degrees of freedom, is characterized, revealing an interesting interplay between the numbers of antennas at the three communication nodes. The relationship of the achievable secure degrees of freedom to those obtained in the case without feedback is finally discussed.

DOI
10.1109/tit.2010.2050798

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arXiv:0708.4219•Aug 30, 2007•33 citations
A multiple-antenna Gaussian wiretap channel in which the number of antennas at the source is not greater than that at the eavesdropper is considered. Without feedback, the secrecy capacity over such a channel generally converges to a constant at high signal-to-noise ratio (SNR). A half-duplex secure protocol allowing the destination to actively transfer random keys in the form of known interference is proposed. It is shown that using multiple antennas at the destination is instrumental in achieving a secrecy rate that grows linearly with logSNR. The pre-log factor of the secrecy rate, i.e., the number of secure degrees of freedom, is characterized, revealing an interesting interplay between the numbers of antennas at the three communication nodes. The relationship of the achievable secure degrees of freedom to those obtained in the case without feedback is finally discussed.
A multiple-antenna Gaussian wiretap channel in which the number of antennas at the source is not larger than that at the eavesdropper is considered. Without feedback, the secrecy capacity over such a channel generally converges to a constant at high signal-to-noise ratio (SNR). A half-duplex secure protocol allowing the destination to actively broadcast random keys over insecure channels is proposed. It is shown that using multiple antennas at the destination is instrumental in achieving a secrecy rate that grows linearly with log SNR. The pre-log factor of the secrecy rate, i.e. the secure degree of freedom, is characterized, revealing an interesting interplay between the numbers of antennas at the three communication nodes.
Secure communication protocols over a single-relay system in the presence of a multiple-antenna eavesdropper are investigated. When there is no direct link between the source and the destination, an amplify-and-forward (AF) protocol that allows both the destination and the source to jam the eavesdropper is shown to achieve secrecy rates that grow linearly with the transmit power in dB, even if all the legitimate nodes have strictly fewer antennas than the eavesdropper. The number of achievable secure degrees of freedom is quantified and shown to match an upper bound when the number of antennas at the destination is sufficiently large. However, in the presence of a direct link between source and destination, this scaling rule no longer applies. The bottleneck of the AF scheme in the presence of a direct source-destination link is identified, motivating the use of another AF scheme and a novel compress-and-forward protocol that can achieve a positive number of secure degrees of freedom using an insecure feedback link.
Secure communication over a half-duplex relay system in the presence of an external wiretapper is investigated. In such a scenario, conventional relaying protocols generally provide secrecy rates that converge to a finite constant in the high signal-to-noise ratio (SNR) regime. It is shown in this work that strictly superior performance can be achieved by allowing the destination to transfer random keys in the form of known interference to the relay insecurely over the wireless medium. When there is no direct link between the source and the destination, an amplify-and-forward (AF) protocol is shown to achieve a secrecy rate that scales as 1/2 log SNR, if each of the legitimate nodes has a single antenna only. However, in the presence of a direct link between source and destination, this scaling rule no longer applies. The bottleneck of the AF scheme in the presence of a direct source-destination link is identified, motivating the use of multiple antennas at the destination to almost surely guarantee a secure degree of freedom of one half over the channel. The results hold even if the eavesdropper has more antennas than the source and the relay combined.
This letter considers secrecy communication from an information-theoretic perspective, and studies the secrecy capacity of a multi-input multi-output (MIMO) Gaussian wiretap channel with a source (Alice), an eavesdropper (Eve) and a Full-Duplex (FD) legitimate receiver (Bob). Bob can allocate part of his antennas to transmit jamming signals to impair Eve’s channel. Our goal is to identify the secrecy capacity behavior in the high signal-to-noise ratio (SNR) regimes, i.e., the maximal achievable secure degrees of freedom (S.D.o.F). Such S.D.o.F maximization is generally difficult to solve since it requires to face a nonlinear fractional problem. To deal with this issue, we first propose a cooperative secrecy transmission scheme, and prove its optimality in the sense of achieving the maximal S.D.o.F.. By studying this proposed transmission scheme, we obtain the maximal achievable S.D.o.F. in closed form for any given antenna allocation at Bob. Based on this closed-form result, we further analytically derive the optimal antenna allocation at Bob. To the best of our knowledge, this is the first time that the benefit brought by using the FD jamming Bob has been quantified.
We consider the problem of secure communication in wireless fading channels in the presence of quantized version of the receivers channel sate information at transmit side. The transmitter has multiple antennas and is able to simultaneously transmits an information bearing signal and artificial noise. For general values of signal to noise ratio (SNR), we obtain the exact closed-form expression for the distribution function of output signal to interference plus noise ratio at the legitimate receiver and a lower bound of the ergodic secrecy capacity can be obtain by using simple numerical integration. For the interference-limited case, we obtained a closed-form expression for a lower bound of secrecy capacity. Our results apply to arbitrary number of antennas and number of feedback bits. The analytical and numerical results show that the ergodic secrecy capacity lower bound approaches to a constant as the SNR of system grows large. In addition, we find that, for system with reasonable number of feedback bits, the optimal power allocation strategy should allocate most of the available power to the information signal.
The role of multiple antennas for secure communication is investigated within the framework of Wyner's wiretap channel. We characterize the secrecy capacity in terms of generalized eigenvalues when the sender and eavesdropper have multiple antennas, the intended receiver has a single antenna, and the channel matrices are fixed and known to all the terminals, and show that a beamforming strategy is capacity-achieving. In addition, we show that in the high signal-to-noise (SNR) ratio regime the penalty for not knowing eavesdropper's channel is small--a simple ``secure space-time code'' that can be thought of as masked beamforming and radiates power isotropically attains near-optimal performance. In the limit of large number of antennas, we obtain a realization-independent characterization of the secrecy capacity as a function of the number βββ: the number of eavesdropper antennas per sender antenna. We show that the eavesdropper is comparatively ineffective when β<1β<1β<1, but that for β≥2β\ge2β≥2 the eavesdropper can drive the secrecy capacity to zero, thereby blocking secure communication to the intended receiver. Extensions to ergodic fading channels are also provided.