Towards the Secrecy Capacity of the Gaussian MIMO Wire-Tap Channel: The 2-2-1 Channel (original) (raw)

Secrecy capacity of the 2-2-1 Gaussian MIMO wire-tap channel

2008

We find the secrecy capacity of the 2-2-1 Gaussian MIMO wire-tap channel, which consists of a transmitter and a receiver with two antennas each, and an eavesdropper with a single antenna. We determine the secrecy capacity of this channel by proposing an achievable scheme and then developing a tight upper bound that meets the proposed achievable secrecy rate. We show that, for this channel, Gaussian signalling in the form of beam-forming is optimal, and no pre-processing of information is necessary.

Optimal Signaling for Secure Communications Over Gaussian MIMO Wiretap Channels

IEEE Transactions on Information Theory, 2016

Optimal signalling over the Gaussian MIMO wire-tap channel is studied under the total transmit power constraint. A closed-form solution for an optimal transmit covariance matrix is obtained when the channel is strictly degraded. In combination with the rank-1 solution, this provides the complete characterization of the optimal covariance for the case of two transmit antennas. The cases of weak eavesdropper and high SNR are considered. It is shown that the optimal covariance does not converge to a scaled identity in the high-SNR regime. Necessary optimality conditions and a tight upper bound on the rank of an optimal covariance matrix are established for the general case, along with a lower bound to the secrecy capacity, which is tight in a number of scenarios. I. INTRODUCTION Multiple-input multiple-output (MIMO) architecture has gained prominence in both academia and industry as a spectrally-efficient approach to wireless communications [1]. With wide deployment of wireless networks, security issues have recently gained additional importance, including information-theoretic approach at the physical layer [2]. The physical-layer security in MIMO systems has been recently under active investigation [3]-[10]. It was demonstrated that Gaussian signaling is optimal over the Gaussian MIMO wire-tap channels (MIMO-WTC) [6]-[10] and the optimal transmit covariance has been found for MISO systems [3], the 2-2-1 system [7],

An MMSE Approach to the Secrecy Capacity of the MIMO Gaussian Wiretap Channel

EURASIP Journal on Wireless Communications and Networking, 2009

This paper provides a closed-form expression for the secrecy capacity of the multiple-input multiple output (MIMO) Gaussian wiretap channel, under a power-covariance constraint. Furthermore, the paper specifies the input covariance matrix required in order to attain the capacity. The proof uses the fundamental relationship between information theory and estimation theory in the Gaussian channel, relating the derivative of the mutual information to the minimum mean-square error (MMSE). The proof provides the missing intuition regarding the existence and construction of an enhanced degraded channel that does not increase the secrecy capacity. The concept of enhancement has been used in a previous proof of the problem. Furthermore, the proof presents methods that can be used in proving other MIMO problems, using this fundamental relationship.

Secrecy Rate Maximization in Gaussian MIMO Wiretap Channels

Information Theoretic Security and Privacy of Information Systems

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.

The Gaussian wiretap channel with noisy public feedback: Breaking the high-SNR ceiling

2009 Conference Record of the Forty-Third Asilomar Conference on Signals, Systems and Computers, 2009

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.

The Gaussian Multiple Access Wire-Tap Channel with Collective Secrecy Constraints

2006 IEEE International Symposium on Information Theory, 2006

We consider the Gaussian Multiple Access Wire-Tap Channel (GMAC-WT). In this scenario, multiple users communicate with an intended receiver in the presence of an intelligent and informed wire-tapper who receives a degraded version of the signal at the receiver. We define a suitable security measure for this multi-access environment. We derive an outer bound for the rate region such that secrecy to some pre-determined degree can be maintained. We also find, using Gaussian codebooks, an achievable such secrecy region. Gaussian codewords are shown to achieve the sum capacity outer bound, and the achievable region concides with the outer bound for Gaussian codewords, giving the capacity region when inputs are constrained to be Gaussian. We present numerical results showing the new rate region and compare it with that of the Gaussian Multiple-Access Channel (GMAC) with no secrecy constraints.

The capacity-equivocation region of the MIMO Gaussian wiretap channel

2010 IEEE International Symposium on Information Theory, 2010

We study the Gaussian multiple-input multiple-output (MIMO) wiretap channel, which consists of a transmitter, a legitimate user, and an eavesdropper. In this channel, the transmitter sends a common message to both the legitimate user and the eavesdropper. In addition to this common message, the legitimate user receives a private message, which is desired to be kept hidden as much as possible from the eavesdropper. We obtain the entire capacity-equivocation region of the Gaussian MIMO wiretap channel. This region contains all achievable common message, private message, and private message's equivocation (secrecy) rates. In particular, we show the sufficiency of jointly Gaussian auxiliary random variables and channel input to evaluate the existing single-letter description of the capacity-equivocation region due to Csiszar-Korner.

Achievable rates for the general Gaussian multiple access wire-tap channel with collective secrecy

2006

We consider the General Gaussian Multiple Access Wire-Tap Channel (GGMAC-WT). In this scenario, multiple users communicate with an intended receiver in the presence of an intelligent and informed eavesdropper who is as capable as the intended receiver, but has different channel parameters. We aim to provide perfect secrecy for the transmitters in this multi-access environment. Using Gaussian codebooks, an achievable secrecy region is determined and the power allocation that maximizes the achievable sum-rate is found. Numerical results showing the new rate region are presented. It is shown that the multiple-access nature of the channel may be utilized to allow users with zero single-user secrecy capacity to be able to transmit in perfect secrecy. In addition, a new collaborative scheme is shown that may increase the achievable sum-rate. In this scheme, a user who would not transmit to maximize the sum rate can help another user who (i) has positive secrecy capacity to increase its rate, or (ii) has zero secrecy capacity to achieve a positive secrecy capacity.

Resource Allocation for Parallel Gaussian MIMO Wire-tap Channels

IEEE Communications Letters, 2000

Wire-tap channels are able to provide perfect secrecy when a receiver exhibits a better channel than its wiretapping opponent. In this letter, we extend the perfect secrecy principle to parallel Gaussian multiple input multiple output (MIMO) wire-tap channels with independent sub-channels. Assuming Nr ≥ Nt and Ne ≥ Nt, mathematical derivations show that a sub-optimal solution can be given in closed-form considering either secrecy rate maximization subject to a total power constraint or power minimization subject to a secrecy rate constraint. Although sub-optimal for MIMO systems, simulation results show that the proposed algorithm allows to reach higher secrecy rates with substantial gains in power consumption compared to the single input single output (SISO) system.