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IEEE Military Communications Conference (MILCOM)

An Empirical Upper Bound on Secrecy Capacity of Multipath Fading Channels

Sima Azizi, Arash Komaee

Published Oct 1, 2018• 2 citations
Publisher

Abstract

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.

DOI
10.1109/milcom.2018.8599701

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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.
Symmetric cryptography relies on pairs of identical secret keys shared by the legitimate communicating parties. To implement a symmetric-key algorithm for cryptography, a major concern is to develop secure methods for distribution of the secret key. In securing the wireless fading channels by symmetric-key algorithms, the physical layer properties of the channel can be exploited for distribution of the secret keys. In this approach, the channel state provides a common randomness which is shared by the legitimate users but is mostly unknown to an eavesdropper. by means of signal processing techniques, this common randomness is extracted into random secret keys. This paper establishes an information theoretic upper bound on the rate at which the secret keys can be extracted. Instead of the conventional approach that relies on mathematical models for the wireless channel, this paper adopts an experimental approach to estimate this bound from empirical data. A set of signal processing techniques is developed here to numerically estimate this bound for a pair of received signal strength (RSS) recorded by indoor commercial radios.
Secret-key generation (SKG) at physical layer is considered as a promising solution for lightweight key distribution. On the analysis of the secret-key capacity over multipath fading channels, existing studies generally neglect the channel observation method, which has a direct impact on the secret-key capacity. Moreover, there is a lack of concise and interpretable expressions for the secret-key capacity based on the channel state information. Motivated by the above, we analyze the performance of SKG based on MMSE estimation of channel impulse response over multipath fading channels. The general expression of secret-key capacity is derived. We find that the secret-key capacity depends on the estimation signal-to-noise ratios (SNRs) of channel-tap gains. Then, the condensed-parameter expressions of exact secret-key capacity and the expressions of asymptotic secret-key capacity at high SNR are derived in two specific PDP cases. We show that the bandwidth and the transmission SNR determine the upper bound of secret-key capacity over channels with different degrees of freedom for the flat PDP case. In addition, we find that the asymptotic secret-key capacities under these two PDPs follow a uniform form, which is a linear function of the channel estimation SNR (in dB). Finally, we simulate the channel probing process under different transmission and channel parameters and exploit the simulated channel estimates to estimate the secret-key capacity through numerical methods. The simulation results demonstrate the theoretical analysis and conclusions.
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.
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.
By exploiting multipath fading channels as a source of common randomness, physical layer (PHY) based key generation protocols allow two terminals with correlated observations to generate secret keys with information-theoretical security. The state of the art, however, still suffers from major limitations,e.g., low key generation rate, lower entropy of key bits and a high reliance on node mobility. In this paper, a novel cooperative key generation protocol is developed to facilitate high-rate key generation in narrowband fading channels, where two keying nodes extract the phase randomness of the fading channel with the aid of relay node(s). For the first time, we explicitly consider the effect of estimation methods on the extraction of secret key bits from the underlying fading channels and focus on a popular statistical method - maximum likelihood estimation (MLE). The performance of the cooperative key generation scheme is extensively evaluated theoretically. We successfully establish both a theoretical upper bound on the maximum secret key rate from mutual information of correlated random sources and a more practical upper bound from Cramer-Rao bound (CRB) in estimation theory. Numerical examples and simulation studies are also presented to demonstrate the performance of the cooperative key generation system. The results show that the key rate can be improved by a couple of orders of magnitude compared to the existing approaches.