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IEEE International Symposium on Information Theory (ISIT)

Impact of channel sparsity and correlated eavesdropping on secret key generation from multipath channel randomness

Tzu-Han Chou, Stark Christiaan Draper, A.M. Sayeed

Published Jun 1, 2010• 50 citations
Publisher

Abstract

We study secret key generation from reciprocal multipath wireless channels modeled as multiple parallel fading channels. We consider two channel characteristics that heavily impact secret key capacity: channel sparsity and correlation between main and eavesdropper's channels. We propose a joint model to capture the channel sparsity and correlated eavesdropping. For the case without eavesdroppers, we show that at each transmitter SNR γ there is an optimal sparsity 0 ≤ ρopt≤ 1, which specifies the fraction of non-zero channel coefficients, that yields the maximum secret key capacity. The reduction in secret key capacity due to eavesdropping is due to two sources. First is the overlap between the main and eavesdropper channels, i.e., the pattern of non-zero subchannels common to both. The second is the correlation between the channel coefficients of the overlapping channels. We show that when the power of the training signal is uniformly distributed over the non-zero channels, there is a cutoff SNR γcbelow which the secret key rate is zero, but non-zero (and increasing in γ) when γ > γc. We also show that in the low SNR regime, the optimal input signal is peaky (a non-uniform training signal) by which the secret key capacity is non-zero at all γ > 0.

DOI
10.1109/isit.2010.5513556

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We study fundamental limits on the generation of secret keys based on the randomness inherent to reciprocal wireless multipath channels. Estimates of the common channel at the two ends of a link are jointly Gaussian sources from which secret keys can be generated. The key generation problem is cast as an equivalent communication problem to characterize the secret key capacity. We analyze the low-SNR regime to quantify the minimum energy per secret key bit required for reliable key acquisition. Our results show that, in contrast to the low SNR behavior of conventional channel capacity, there is a non-zero SNR ¿* that achieves the minimum energy per key bit. A time-sharing scheme is proposed to achieve the minimum energy per key bit at any SNR below ¿*. We also investigate the reliability of secret key generation via error exponent analysis. In particular, our results yield a tight upper bound on the minimum energy required to generate a finite-length key with a specified probability of error in key acquisition.
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.
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This paper investigates the performance of key generation between two nodes assisted by a relay in the presence of correlated eavesdropping channels. A cooperative jamming scheme is utilized to impose superimposed channel measurements on the relay and eavesdropper. Both lower and upper bounds on key capacities for both secret key (SK) and private key (PK) generation are evaluated, where the lower bounds are derived by using minimum mean square error and zero forcing methods for channel estimation, and the upper bounds are derived by formulating several enhanced discrete memoryless source (DMS) models. The analytical expressions are further simplified in the high signal-to-noise ratio (SNR) regime. We discover that one of the two legitimate channels should specialize in playing a role of jamming the relay or eavesdropper. We also demonstrate that the derived lower and upper bounds are tight when the eavesdropping channels are lowly or highly correlated. When the eavesdropping channels are uncorrelated, the SK and PK capacities can be determined since the corresponding upper and lower bounds are equal. Moreover, at high SNRs, a constant gap exists between the SK/PK upper and lower bounds as the correlation coefficient becomes one.
Secret key generation from reciprocal multi-antenna channels is an interesting alternative to cryptographic key management in wireless systems without infrastructure access. In this work, we study the secret key rate for the basic source model with a MIMO channel. First, we derive an expression for the secret key rate under spatial correlation modelled by the Kronecker model and with spatial precoding at both communication nodes. Next, we analyze the result for uncorrelated antennas to understand the optimal precoding for this special case, which is equal power allocation. Then, the impact of correlation is characterized using Majorization theory. Surprisingly for small SNR, spatial correlation increases the secret key rate. For high SNR, the maximum secret key rate is achieved for uncorrelated antennas. The results indicate that a solid system design for reciprocal MIMO key generation is required to establish the secret key rate gains.