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International Conference on Wireless Communications and Signal Processing (WCSP)

Secrecy extraction from correlated fading channels: An upper bound

Xiaofu Wu, Yue Song, Chunming Zhao, Xiaohu You

Published Nov 1, 2009• 11 citations
Publisher

Abstract

The reciprocal nature of wireless radio channels can be well employed to extract a secret key between two terminals. The maximum size of a key that can be shared is characterized by the mutual information between the observations of two radios. We consider a correlated fading channel and a closed form for the mutual information is derived. It is found that the secrecy capacity is largely limited by the maximum Doppler frequency.

DOI
10.1109/wcsp.2009.5371757

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IEEE Transactions on Wireless Communications•Sep 9, 2013•12 citations
The reciprocal nature of wireless radio channels can be well employed to extract a secret key between two terminals. The maximum size of a key that can be shared is characterized by the mutual information between the observations of two radios. We consider a correlated fading channel and a closed form for the mutual information is derived. It is found that the secrecy capacity is largely limited by the maximum Doppler frequency.
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.
Extraction of shared secret keys from random state of multipath fading channels is a growing trend in symmetric cryptography. In this context, an essential question is the rate at which the secret keys can be extracted (number of bits per unit of time). This question is addressed in this paper by establishing an empirical upper bound on the extraction rate of the secret bits. A coherent set of signal processing techniques is proposed to numerically estimate this bound from empirical data generated by commercial radios. It is emphasized to solely rely on empirical data rather than mathematical models of fading channels.
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.
We investigate the secrecy capacity of an ergodic fading wiretap channel in which the main channel is correlated with the eavesdropper channel. In this study, the full Channel State Information (CSI) is assumed, and thus the transmitter knows the channel gains of the legitimate receiver and the eavesdropper. By analyzing the resulting secrecy capacity we quantify the loss of the secrecy capacity due to the correlation. In addition, we study the asymptotic behavior of the secrecy capacity as Signal-to-Noise Ratio (SNR) tends to infinity. The capacity of an ordinary fading channel logarithmically increases with SNR. On the contrary, the secrecy capacity converges to a limit which can be an upper bound on the secrecy capacity over the fading wiretap channel. We find a closed form of the upper bound for the correlated Rayleigh wiretap channel which also includes the independent case as a special one. Our work shows that the upper bound is determined by only two channel parameters; the correlation coefficient and the ratio of the main to the eavesdropper channel gains that will be called the average Channel Gain Ratio (CGR). The analysis of the upper bound tells how the two channel parameters affect the secrecy capacity and leads to the conclusion that the excessively large signal power does not provide any advantage in the secrecy capacity and the loss due to the correlation is especially serious in low CGR regime.
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.