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IEEE Globecom Workshops (GC Wkshps)

Pilot-based secret key agreement for reciprocal correlated MIMÓME block fading channels

Stefano Tomasin, Eduard Axel Jorswieck

Published Dec 1, 2014• 3 citations
Publisher

Abstract

We consider a wireless communication scenario where two agents aim at agreeing on a secret key, while leaving a passive eavesdropper in the darkness. All terminals are equipped with multiple antennas resulting in a multiple input multiple output (MIMO) scenario with reciprocal correlated flat Rayleigh fading channels among antenna couples. Correlation properties are known to all agents. In order to limit complexity we focus only on pilot-based secret key agreement (SKA), resulting in a source model with wiretapper, where pilot sequences are known to all agents. The contribution of the paper is twofold: a) we derive closed form expressions of lower and upper bounds of the secret key achievable rates, and b) we address the training sequence optimization taking into account the spatial channel correlation. A reduced dimensionality problem is obtained when spatial correlation follows a Kronecker model and looser bounds are discussed. Numerical results in a cellular scenario, with reference to the cellular 3GPP MIMO channel model, show significant gains by the training sequence optimization.

DOI
10.1109/glocomw.2014.7063620

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IEEE Transactions on Information Theory•Oct 19, 2016•24 citations
We consider a wireless communication scenario where two agents aim at agreeing on a secret key, while leaving a passive eavesdropper in the darkness. All terminals are equipped with multiple antennas resulting in a multiple input multiple output (MIMO) scenario with reciprocal correlated flat Rayleigh fading channels among antenna couples. Correlation properties are known to all agents. In order to limit complexity we focus only on pilot-based secret key agreement (SKA), resulting in a source model with wiretapper, where pilot sequences are known to all agents. The contribution of the paper is twofold: a) we derive closed form expressions of lower and upper bounds of the secret key achievable rates, and b) we address the training sequence optimization taking into account the spatial channel correlation. A reduced dimensionality problem is obtained when spatial correlation follows a Kronecker model and looser bounds are discussed. Numerical results in a cellular scenario, with reference to the cellular 3GPP MIMO channel model, show significant gains by the training sequence optimization.
We consider secret-key agreement with public discussion over multiple-input multiple-output (MIMO) Rayleigh fast-fading channels under correlated environment. We assume that transmit, legitimate receiver and eavesdropper antennas are correlated. The legitimate receiver and the eavesdropper are assumed to have perfect channel knowledge while the transmitter has only knowledge of the correlation matrices. First, we derive the expression of the secret-key capacity under the considered setup. Then, we prove that the optimal transmit strategy achieving the secret-key capacity consists in transmitting independent Gaussian signals along the eingenvectors of the transmit correlation matrix. The powers allocated to each channel mode are determined as the solution to a numerical optimization problem that we derive. A necessary and sufficient condition for beamforming (i.e., transmitting along the strongest channel mode) to be capacity-achieving is derived. Finally, we analyze the impact of correlation matrices on the system performance and provide closed-form expressions of the gain/loss due to correlation in the high power regime.
Fundamental limits of secret-key agreement over reciprocal wireless channels are investigated. We consider a two-way block-fading channel where the channel gains in the forward and reverse links between the legitimate terminals are correlated. The channel gains between the legitimate terminals are not revealed to any terminal, whereas the channel gains of the eavesdropper are revealed to the eavesdropper. We propose a two-phase transmission scheme, that reserves a certain portion of each coherence block for channel estimation, and the remainder of the coherence block for correlated source generation. The resulting secret-key involves contributions of both channel sequences and source sequences, with the contribution of the latter becoming dominant as the coherence period increases. We also establish an upper bound on the secret-key capacity, which has a form structurally similar to the lower bound. Our upper and lower bounds coincide in the limit of high signal-to-noise-ratio (SNR) and large coherence period, thus establishing the secret-key agreement capacity in this asymptotic regime. Numerical results indicate that the proposed scheme achieves significant gains over training-only schemes, even for moderate SNR and small coherence periods, thus implying the necessity of randomness-sharing in practical secret-key generation systems.
We consider secret-key agreement with public discussion over multiple-input multiple-output (MIMO) Rayleigh fast-fading channels under correlated environment. We assume that transmit, legitimate receiver, and eavesdropper antennas are correlated. The legitimate receiver and the eavesdropper are assumed to have perfect channel knowledge while the transmitter has only knowledge of the correlation matrices. First, we derive the expression of the secret-key capacity under the considered setup. We prove that the optimal transmit strategy achieving the secret-key capacity consists in transmitting independent Gaussian signals along the eingenvectors of the transmit correlation matrix. The powers allocated to each channel mode are determined as the solution to a numerical optimization problem. A necessary and sufficient condition for beamforming (i.e., transmitting along the strongest channel mode) to be capacity-achieving is derived. Moreover, we analyze the impact of correlation matrices on the system performance. Finally, we study the system's performance in the two extreme power regimes. In the high-power regime, we provide closed-form expressions of the gain/loss due to correlation. In the low signal-to-noise ratio (SNR) regime, we investigate the energy efficiency of the system by determining the minimum energy required for sharing a secret-key bit and the wideband slope while highlighting the impact of correlation matrices.
Secret key generation from reciprocal multi-antenna channels is an interesting alternative to cryptographic key management in wireless systems without infrastructure access. In this work, we study the secret key rate for the basic source model with a MIMO channel. First, we derive an expression for the secret key rate under spatial correlation modelled by the Kronecker model and with spatial precoding at both communication nodes. Next, we analyze the result for uncorrelated antennas to understand the optimal precoding for this special case, which is equal power allocation. Then, the impact of correlation is characterized using Majorization theory. Surprisingly for small SNR, spatial correlation increases the secret key rate. For high SNR, the maximum secret key rate is achieved for uncorrelated antennas. The results indicate that a solid system design for reciprocal MIMO key generation is required to establish the secret key rate gains.
We study secret-key agreement over a non-coherent block-fading multiple input multiple output (MIMO) wiretap channel. We give an achievable scheme based on training and source emulation and analyze the rate in the high SNR regime. Based on this analysis we find the optimal number of antennas to use for training. Our main result is that if the sum of the number of antennas at Alice and Bob is larger than the coherence time of the channel, the achievable rate does not depend on the number of antennas at Eve. In this case source emulation is not needed, and using only training is optimal. We also consider the case when there is no public channel available. In this case we show that secret-key agreement is still possible by using the wireless channel for discussion, giving the same number of secure degrees of freedom as in the case with a public channel.
Motivated by recent interest in physical-layer secret-key generation over wireless fading channels, we study the non-coherent secret-key generation capacity of a block-fading wireless channel with channel reciprocity and bi-directional (two-way) communication. We assume a non-coherent main channel, i.e., the realization of channel gains on the main channel is not known to any terminal. The eavesdropper is assumed to have both perfect Channel State Information of its own channel and orthogonal observations from the forward and backward channels. As our main result, we establish new upper and lower bounds on the secret-key generation capacity with public discussion, which are structurally similar. The upper bound can be expressed as a sum of three terms-one of the terms arises due to channel reciprocity, while the other two terms correspond to communication on the forward and backward channels, respectively. In the limit of long coherence period, the contribution from channel reciprocity vanishes to zero, whereas the other terms prevail. The lower bound is based on a separation based scheme. In each coherence block, we use the first symbol for training while the remainder of the coherence block is devoted to source emulation, i.e., to generate correlated sources between the terminals. The lower bound expression also consists of three terms and admits an interpretation similar to the upper bound expression. For Rayleigh fading channels, in the high signal-to-noise ratio (SNR) regime, the gap between the upper and lower bounds is shown to vanish inversely with the coherence period. Numerical results indicate significant performance gains over training-only schemes even for moderate values of SNR and small coherence periods.