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IEEE International Workshop on Signal Processing Advances in Wireless Communications (SPAWC)

Secure key agreement over reciprocal fading channels in the low SNR regime

Mattias Andersson, Ashish J. Khisti, Mikael Skoglund

Published Jun 1, 2013• 4 citations
Publisher

Abstract

We study the low SNR scaling of the non-coherent secret-key agreement capacity over a reciprocal, block-fading channel. For the restricted class of strategies, where one of the nodes is constrained to transmit pilot-only symbols, we show that the secret-key capacity scales as SNR ·log T if T ≤ 1/SNR, where T denotes the coherence period, and as SNR·log(1/SNR) otherwise. Our upper bound is inspired by the genie-aided argument of Borade and Zheng (IT-Trans 2010). Our lower bound is based on bursty communication, channel training, and secret message transmission.

DOI
10.1109/spawc.2013.6612135

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IEEE Conference on Communications and Network Security•Sep 1, 2015•1 citation
We study the low SNR scaling of the non-coherent secret-key agreement capacity over a reciprocal, block-fading channel. For the restricted class of strategies, where one of the nodes is constrained to transmit pilot-only symbols, we show that the secret-key capacity scales as SNR ·log T if T ≤ 1/SNR, where T denotes the coherence period, and as SNR·log(1/SNR) otherwise. Our upper bound is inspired by the genie-aided argument of Borade and Zheng (IT-Trans 2010). Our lower bound is based on bursty communication, channel training, and secret message transmission.
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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 study a two-terminal secret-key generation problem over a two-way, approximately reciprocal, block-fading channel. The channel gains between the legitimate terminals are not revealed to any terminal, whereas the channel gains of the eavesdropper are revealed perfectly to the eavesdropper. We study a separation based scheme that involves a training phase followed by a communication phase within each block. The training phase generates correlated estimates of the channel state sequence between the two terminals, whereas the communication phase generates correlated source sequences between the two terminals. A portion of the secret-key is generated from the correlated channel state sequences by creating omniscience between the legitimate terminals and the remainder of the secret-key is generated from the correlated source sequences. An upper bound on the secret-key capacity is also established by reducing the setup to a coherent channel by providing genie-aided side information to the receivers. We observe that the difference between the upper and lower bound decreases as 1/T in the high signal-to-noise-ratio (SNR) regime. Numerical results indicate that the proposed scheme achieves significant gains over the commonly used training-only schemes even for moderate SNR.
We study the Rayleigh fading non-coherent capacity of secret-key agreement with public discussion, where neither the sender nor the receivers have access to instantaneous channel state information (CSI) of any channel. We present two results. At high Signal-to-Noise Ratio (SNR), the secret-key capacity is bounded in SNR, regardless of the number of antennas at each terminal. Second, for a system with a single antenna at both the legitimate and the eavesdropper terminals and an arbitrary number of transmit antennas, the secret-key capacity-achieving input distribution is discrete, with a finite number of mass points. Numerically we observe that at low-SNR, the capacity achieving distribution has two mass points with one of them at the origin.
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 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.