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International Conference on Communications

Semi-Blind Key-Agreement over MIMO Fading Channels

Francesco Renna, Matthieu R. Bloch, Nicola Laurenti

Published Jun 1, 2011• 1 citation
Publisher

Abstract

We analyze the fundamental limits of secret-key agreement over MIMO quasi-static fading channels. In the low-power and high-power regimes, we establish closed-form expressions for secret-key capacity. In the low-power regime, we show that the optimal signaling strategy is independent of the eavesdropper's fading realization. The low-power secret-key capacity is achieved by transmitting along the direction corresponding to the maximal eigenvalue of the legitimate channel. By combining this signaling strategy with reconciliation and privacy amplification, one obtains a semi-blind key-distillation strategy in which the knowledge of the eavesdropper's fading is required for privacy amplification alone.

DOI
10.1109/icc.2011.5963438

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We analyze the fundamental limits of secret-key agreement over MIMO quasi-static fading channels. In the low-power and high-power regimes, we establish closed-form expressions for secret-key capacity. In the low-power regime, we show that the optimal signaling strategy is independent of the eavesdropper's fading realization. The low-power secret-key capacity is achieved by transmitting along the direction corresponding to the maximal eigenvalue of the legitimate channel. By combining this signaling strategy with reconciliation and privacy amplification, one obtains a semi-blind key-distillation strategy in which the knowledge of the eavesdropper's fading is required for privacy amplification alone.
In this paper, we study the fundamental limits of secret-key agreement over MIMO quasi-static fading channels. We provide closed-form expressions for the secret-key capacity in both the asymptotic high-power and low-power regimes. The optimal signaling strategy for the low-power regime is shown to be independent of the eavesdropper's channel and secret-key capacity is achieved by transmitting random Gaussian symbols along the direction corresponding to the maximal eigenvalue of the legitimate channel matrix. Hence, by beamforming and waterfilling over the main channel alone, one obtains a semi-blind key-agreement strategy in which the knowledge of the eavesdropper's channel is only required for privacy amplification. We also derive the probability that a target secret-key rate is not achieved by the optimal low-power signaling when assuming only statistical CSI about the eavesdropper's channel.
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.
We consider secret-key agreement with public discussion over Rayleigh fast-fading channels with transmit, receive and eavesdropper correlation. The legitimate receiver along with the eavesdropper are assumed to have perfect channel knowledge while the transmitter has only knowledge of the correlation matrices. We analyze the secret-key capacity in the low signal-to-noise ratio (SNR) regime. We derive closed-form expressions for the first and the second derivatives of the secret-key capacity with respect to SNR at SNR= 0, for arbitrary correlation matrices and number of transmit, receive and eavesdropper antennas. Moreover, we identify optimal transmission strategies achieving these derivatives. For instance, we prove that achieving the first and the second derivatives requires a uniform power distribution between the eigenvectors spanning the maximal-eigenvalue eigenspace of the transmit correlation matrix. We also compare the optimal transmission scheme to a simple uniform power allocation. Finally, we express the minimum energy required for sharing a secret-key bit as well as the wideband slope in terms of the system parameters.
In Part I of this two-part paper on confidential communication over wireless channels, we studied the fundamental security limits of quasi-static fading channels from the point of view of outage secrecy capacity with perfect and imperfect channel state information. In Part II, we develop a practical secret key agreement protocol for Gaussian and quasi-static fading wiretap channels. The protocol uses a four-step procedure to secure communications: establish common randomness via an opportunistic transmission, perform message reconciliation, establish a common key via privacy amplification, and use of the key. We introduce a new reconciliation procedure that uses multilevel coding and optimized low density parity check codes which in some cases comes close to achieving the secrecy capacity limits established in Part I. Finally, we develop new metrics for assessing average secure key generation rates and show that our protocol is effective in secure key renewal.
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.
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.