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IEEE International Symposium on Information Theory (ISIT)

Minimum energy per bit for secret key acquisition over multipath wireless channels

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

Published Jun 1, 2009• 29 citations
Publisher

Abstract

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.

DOI
10.1109/isit.2009.5205918

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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.
Secure wireless communications is a challenging problem due to the shared nature of the wireless medium. Most existing security protocols apply cryptographic techniques for bit scrambling at the application layer by exploiting a shared secret key between pairs of communicating nodes. However, more recent research argues that multipath propagation - a salient feature of wireless channels - provides a physical resource for secure communications. In this context, we propose a protocol that exploits the inherent randomness in multipath wireless channels for generating secret keys through channel estimation and quantization. Our approach is particularly attractive in wideband channels which exhibit a large number of statistically independent degrees of freedom (DoF), thereby enabling the generation of large, more-secure, keys. We show that the resulting keys are distinct for distinct pairwise links with a probability that increases exponentially with the key-size/channel DoF. We also characterize the probability that the two users sharing a common link generate the same key. This characterization is used to analyze the energy consumption in successful acquisition of a secret key by the two users. For a given key size, our results show that there is an optimum transmit power, and an optimum quantization strategy, that minimizes the energy consumption. The proposed approach to secret key generation through channel quantization also obviates the problem of key pre-distribution inherent to many existing cryptographic approaches.
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.
Recently, multipath in wireless channels has shown an interesting aspect. It has been found that wireless channels may provide a common reciprocal source of randomness for any two communicating nodes. This seems particularly interesting for secret key generation in wireless networks. In this paper, we consider the randomness property of wireless channels and present an intelligent algorithm for key generation. We analyze the multipath channel and derive optimal parameters enabling efficient key extraction leveraging the richness of a wireless environment. Finally, we test our algorithm in a typical outdoor wireless channel. Our simulation results show that more than 70 secret bits can be extracted from a single channel observation.
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.
Providing extra security in wireless communication networks using channel impulse response as a source of common randomness is an appealing direction in prospective systems. While some theoretical and experimental work has demonstrated its potential, a number of issues remain unexplored. This paper is concerned with the reliability of secret key generation under practical estimation requirements. We provide an analytical expression which relates the channel SNRs and degree of reciprocity to probability of key mismatch. In the next step we show how this affects key reconciliation and secrecy leaking. Finally, we consider how quantization with a protective band can help to trade the secret key rate for increased reliability and security of the scheme.
Wireless channel reciprocity can be successfully exploited as a common source of randomness for the generation of a secret key by two legitimate users willing to achieve confidential communications over a public channel. This paper presents an analytical framework to investigate the theoretical limits of secret-key generation when wireless multi-dimensional Gaussian channels are used as source of randomness. The intrinsic secrecy content of wide-sense stationary wireless channels in frequency, time and spatial domains is derived through asymptotic analysis as the number of observations in a given domain tends to infinity. Some significant case studies are presented where single and multiple antenna eavesdroppers are considered. In the numerical results, the role of signal-to-noise ratio, spatial correlation, frequency and time selectivity is investigated.