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IEEE International Conference on Wireless Information Technology and Systems

Optimizing the common-randomness secret key rate in beamformed communication systems

Britton T. Quist, Michael A. Jensen

Published Nov 1, 2012• 2 citations
Publisher

Abstract

Generating key bits with multiple antenna systems has the potential to dramatically increase the rate at which key bits can be generated. For beamforming systems, this work considers how to probe the channel to maximize the mutual information between channel estimates. For these systems, we provide an optimal probing strategy that uses the eigenvectors of the channel correlation matrix as probing vectors and then optimally allocates the transmission energy to each vector. Since this strategy cannot be implemented in practice, we also propose a practical channel estimation scheme that achieves near-optimal performance. Simulations demonstrate the performance advantage of the proposed strategy relative to that achieved using traditional channel estimation in which the total energy is equally allocated to all probing vectors.

DOI
10.1109/icwits.2012.6417769

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Generating key bits with multiple antenna systems has the potential to dramatically increase the rate at which key bits can be generated. For beamforming systems, this work considers how to probe the channel to maximize the mutual information between channel estimates. For these systems, we provide an optimal probing strategy that uses the eigenvectors of the channel correlation matrix as probing vectors and then optimally allocates the transmission energy to each vector. Since this strategy cannot be implemented in practice, we also propose a practical channel estimation scheme that achieves near-optimal performance. Simulations demonstrate the performance advantage of the proposed strategy relative to that achieved using traditional channel estimation in which the total energy is equally allocated to all probing vectors.
Establishing secret keys from the commonly-observed randomness of reciprocal wireless propagation channels has recently received considerable attention. In this work we propose improved strategies for channel estimation between MIMO or beamforming systems for secret key generation. The amount of mutual information that can be extracted from the channel matrix estimates is determined by the quality of channel matrix estimates. By allocating increased energy to channel estimation for higher gain beamforming combinations at the expense of low-gain combinations, key establishment performance can be increased. Formalizing the notion of preferential energy allocation to the most efficient excitations is the central theme of this dissertation. For probing with beamforming systems, we formulate a theoretically optimal probing strategy that upper bounds the number of key bits that can be generated from reciprocal channel observations. Specifically, we demonstrate that the eigenvectors of the channel spatial covariance matrix should be used as beamformer weights during channel estimation and we optimize the energy allocated to channel estimation for each beamformer weight under a total energy constraint. The optimal probing strategy is not directly implementable in practice, and therefore we propose two different modifications to the optimal algorithm based on a Kronecker approximation to the spatial covariance matrix. Though these approximations are suboptimal, they each perform well relative to the upper bound. To explore how effective an array is at extracting all of the information available in the propagation environment connecting two nodes, we apply the optimal beamformer probing strategy to a vector current basis function expansion on the array volume. We prove that the resulting key rate is a key rate spatial bound that upper bounds the key rate achievable by any set of antenna arrays probing the channel with the same total energy constraint. For MIMO systems we assume the channel is separable with a Kronecker model, and then for that model we propose an improved probing strategy that iteratively optimizes the energy allocation for each node using concave maximization. The performance of this iterative approach is better than that achieved using the traditional probing strategy in many realistic probing scenarios.
Establishing secret keys from the commonly-observed randomness of reciprocal wireless propagation channels has recently received considerable attention. We consider such key establishment between two multiantenna nodes that use beamforming for communication, showing that the upper bound on the number of key bits that can be generated from the channel observations can be maximized by properly probing the channel. Specifically, we demonstrate that the eigenvectors of the channel spatial covariance matrix should be used as beamformer weights during channel estimation and we optimize the energy allocated to channel estimation for each beamformer weight under a total energy constraint. Finally, by assuming that the channel covariance is separable, we illustrate implementation of the technique for practical beamforming systems and more advanced signal models that incorporate antenna mutual coupling.
Keys for symmetric encryption can be established by generating key bits from the mutually-observed randomness present within estimates of a reciprocal wireless channel. We consider this approach for key establishment in multiple-input multiple-output (MIMO) systems and provide a channel estimation strategy that increases the key rate, or the mutual information between the channel estimates at the two nodes. The technique applies MIMO transmit and receive beamformers derived from the eigenvectors of the covariance matrix and then iteratively optimizes the power allocated to each transmit beamformer. Simulation results demonstrate the iterative behavior and the achieved performance of the proposed algorithm.
The performance of techniques for establishing secret encryption keys from the reciprocal electromagnetic propagation between two wireless radios is generally measured by comparing the number of key bits generated to the key rate, or the information theoretic bound on the number of key bits that can be determined for a single independent channel estimate. While typically this bound is computed by assuming that the radios have no information regarding the channel, recent work has demonstrated an approach for computing this bound for nodes using beamformed antenna arrays that know the spatial covariance of a wide-sense stationary channel. Since this bound does not directly indicate how to perform practical channel estimation to obtain estimates that can achieve the bound, the prior work demonstrates an approximation that indicates a practical channel estimation procedure while yielding a mutual information near that of the key rate in rich multipath fading environments. This paper demonstrates appreciable degradation in the mutual information achieved using this approximation when the channel includes a line-of-sight component and proposes a modified approximation technique that performs well over a broad range of propagation conditions.
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.
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.