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Globecom

Optimal Pilots for Maximal Capacity of Secret Key Generation

Qiping Zhu, Yingbo Hua

Published Dec 1, 2019• 5 citations
Publisher

Abstract

Through a wireless channel, two users can transmit pilot signals to each other, and then utilize the received signals to agree upon a secret key through communications in a public channel without leaking any secret about this key to anyone else. This paper addresses the optimization of the two pilots to maximize the capacity of secret key generation for a MIMO channel with any given receive and transmit correlation matrices. This study shows how the globally optimal pilots can be obtained if the transmission power is either low or high. For an arbitrary transmission power, the algorithm developed is at least locally optimal. Comparison to pilots based on minimum channel estimation errors and uniform power distribution is also presented. It is also shown that the designed pilots meet the requirement for anti- eavesdropping channel estimation.

DOI
10.1109/globecom38437.2019.9013261

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Through a wireless channel, two users can transmit pilot signals to each other, and then utilize the received signals to agree upon a secret key through communications in a public channel without leaking any secret about this key to anyone else. This paper addresses the optimization of the two pilots to maximize the capacity of secret key generation for a MIMO channel with any given receive and transmit correlation matrices. This study shows how the globally optimal pilots can be obtained if the transmission power is either low or high. For an arbitrary transmission power, the algorithm developed is at least locally optimal. Comparison to pilots based on minimum channel estimation errors and uniform power distribution is also presented. It is also shown that the designed pilots meet the requirement for anti- eavesdropping channel estimation.
Physical-layer key generation (PKG) based on channel reciprocity has recently emerged as a new technique to establish secret keys between devices. Most works focus on pairwise communication scenarios with single or small-scale antennas. However, the fifth generation (5G) wireless communications employ massive multiple-input multiple-output (MIMO) to support multiple users simultaneously, bringing serious overhead of reciprocal channel acquisition. This paper presents a multi-user secret key generation in massive MIMO wireless networks. We provide a beam domain channel model, in which different elements represent the channel gains from different transmit directions to different receive directions. Based on this channel model, we analyze the secret key rate and derive a closed-form expression under independent channel conditions. To maximize the sum secret key rate, we provide the optimal conditions for the Kronecker product of the precoding and receiving matrices and propose an algorithm to generate these matrices with pilot reuse. The proposed optimization design can significantly reduce the pilot overhead of the reciprocal channel state information acquisition. Furthermore, we analyze the security under the channel correlation between user terminals (UTs), and propose a low overhead multi-user secret key generation with non-overlapping beams between UTs. Simulation results demonstrate the near-optimal performance of the proposed precoding and receiving matrices design and the advantages of the non-overlapping beam allocation.
This letter examines the optimal pilot length for key generation in correlated wireless channels. We first deduce the secret key capacity as a function of the pilot length N and other system parameters. Then, derive two fixed-point equations to identify the optimal N with and without an eavesdropper. Our analysis and numerical results confirm the existence of the optimal N that maximizes the secret key capacity, and both results for optimal N are consistent. Furthermore, the optimal N generally decreases with average signal-to-noise ratio and maximum Doppler shift. Our examination can provide useful guidelines in designing practical key generation systems.
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.
MIMO and MIMO-OFDM systems are the enabling technologies for the next generation wireless systems. However, the radio signals transmitted are available for reception by a illegitimate user leading to eavesdropping. This raises severe concerns on the security of the transmission. Physical Layer security aims to exploit the wireless channel characteristics and in turn provide secure transmission robust against eavesdropping. In this paper, we investigate the Secret Key Capacity (SKC) in MIMO-OFDM systems. We derive expressions for SKC in MIMO systems and extend further to OFDM waveforms. Numerical simulations are performed for various transmit and receive antenna configurations without any estimation errors.
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 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.