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arXiv:1208.3790 · IEEE Journal on Selected Areas in Communications

Secret Key Generation from Sparse Wireless Channels: Ergodic Capacity and Secrecy Outage

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

Published Aug 23, 2013• 23 citations
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Abstract

This paper investigates generation of a secret key from a reciprocal wireless channel. In particular we consider wireless channels that exhibit sparse structure in the wideband regime and study the impact of sparsity on the secret key capacity. We explore this problem in two steps. First, we study key generation from a state-dependent discrete memoryless multiple source. The state of the source captures the effect of channel sparsity. Secondly, we consider a wireless channel model that captures channel sparsity and correlation between the legitimate users' channel and the eavesdropper's channel. Such dependency can significantly reduce the secret key capacity. According to system delay requirements, two performance measures are considered: (i) ergodic secret key capacity and (ii) outage probability. We show that in the wideband regime when a white sounding sequence is adopted, a sparser channel can achieve a higher ergodic secret key rate than a richer channel can. For outage performance, we show that if the users generate secret keys at a fraction of the ergodic capacity, the outage probability will decay exponentially in signal bandwidth. Moreover, a larger exponent is achieved by a richer channel.

DOI
10.1109/jsac.2013.130909

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This paper investigates generation of a secret key from a reciprocal wireless channel. In particular we consider wireless channels that exhibit sparse structure in the wideband regime and study the impact of sparsity on the secret key capacity. We explore this problem in two steps. First, we study key generation from a state-dependent discrete memoryless multiple source. The state of the source captures the effect of channel sparsity. Secondly, we consider a wireless channel model that captures channel sparsity and correlation between the legitimate users' channel and the eavesdropper's channel. Such dependency can significantly reduce the secret key capacity. According to system delay requirements, two performance measures are considered: (i) ergodic secret key capacity and (ii) outage probability. We show that in the wideband regime when a white sounding sequence is adopted, a sparser channel can achieve a higher ergodic secret key rate than a richer channel can. For outage performance, we show that if the users generate secret keys at a fraction of the ergodic capacity, the outage probability will decay exponentially in signal bandwidth. Moreover, a larger exponent is achieved by a richer channel.
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.
This letter investigates the secret key capacity of key generation from correlated wireless channels in a source model. We systematically study a practical scenario by taking into account all relevant parameters including sampling delay, eavesdroppers' location, qualities of legitimate and eavesdropping channels, Doppler spread, and pilot length. Our findings indicate that secret key capacity is determined by the cross correlation of the channel measurements, and a better legitimate channel is not necessary when the correlation between legitimate channels is higher than correlation between legitimate and eavesdropping channels. We also find that it is possible to tune the secret key capacity by carefully designing the sampling delay, pilot length, and channel qualities. This letter offers practical design guidelines on secure key generation systems.
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.
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.
Standard cryptography is expected to poorly fit IoT applications and services, as IoT devices can hardly cope with the computational complexity often required to run encryption algorithms. In this framework, physical layer security is often claimed as an effective solution to enforce secrecy in IoT systems. It relies on wireless channel characteristics to provide a mechanism for secure communications, with or even without cryptography. Among the different possibilities, an interesting solution aims at exploiting the random-like nature of the wireless channel to let the legitimate users agree on a secret key, simultaneously limiting the eavesdropping threat thanks to the spatial decorrelation properties of the wireless channel. The actual reliability of the channel-based key generation process depends on several parameters, as the actual correlation between the channel samples gathered by the users and the noise always affecting the wireless communications. The sensitivity of the key generation process can be expressed by the secrecy key rate, which represents the maximum number of secret bits that can be achieved from each channel observation. In this work, the secrecy key rate value is computed by means of simulations carried out under different working conditions in order to investigate the impact of major channel parameters on the SKR values. In contrast to previous works, the secrecy key rate is computed under a line-of-sight wireless channel and considering different correlation levels between the legitimate users and the eavesdropper.
Physical layer secret key generation exploits the inherent randomness of wireless channels. However, for wireless channels with long coherence time, key generation rate can be extremely low due to lack of randomness in channels. In this paper, a novel key generation scheme is proposed for a general wireless network where channels are static and each transmitter is equipped with two antennas. It integrates opportunistic beamforming and frequency diversity to achieve fast secret key generation. In this scheme, channel fluctuations are first induced by controlling amplitude and phase of each symbol in the training sequence on each antenna. Thus, key generation rate is significantly increased. However, the induced channel fluctuations lead to correlation between the legitimate channel and the eavesdropping channel, which compromises key secrecy. To this end, frequency diversity is then exploited to ensure that key secrecy grows with the key size. The secret key generation scheme is investigated in both narrowband and wideband systems, and its performance is evaluated through both theoretical analysis and simulations. Performance results have validated randomness and secrecy of secret keys and also illustrate that the proposed scheme can generate secret keys at a rate of 2Kb/s for narrowband systems and 20Kb/s for wideband systems.