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IEEE Transactions on Information Theory

Secret-Key Agreement Over Non-Coherent Block-Fading Channels With Public Discussion

Ashish J. Khisti

Published Oct 19, 2016• 24 citations
Publisher

Abstract

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.

DOI
10.1109/tit.2016.2618861

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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.
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.
We study secret-key agreement with public discussion over a flat-fading wiretap channel model. The fading gains are correlated across the receivers and sampled independently at each time. Perfect receiver channel state information (CSI) is assumed, whereas a noisy CSI of the main channel is also available to the transmitter. We propose lower and upper bounds on the capacity. Our lower bound is achieved by a coding scheme that involves a separate binning of the receiver CSI sequence and its channel output sequence. In general it improves upon the joint-binning schemes considered in earlier works. Our upper and lower bounds coincide, establishing the capacity, when either the transmitter has no CSI or when the channel gains of the legitimate receiver and the eavesdropper are statistically independent.
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.