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IEEE Wireless Communications and Networking Conference

Secret Key Capacity of Key Generation in the Slow-Fading Wireless Channels

Yuang Zhang, Liquan Chen, Yufan Song, Yan Lv

Published Mar 24, 2025• 0 citations
Publisher

Abstract

This paper investigates the secret key capacity of key generation in the slow-fading wireless channel by considering the temporal correlation between channel measurements in the previous and current frame. We derive the analytical expression of secret key capacity in the slow-fading channel, and propose the design guideline for practical key generation schemes in slow fading channel environments. We systematically analyze the impact of all corresponding parameters on secret key capacity, including sampling period, sampling delay, eavesdroppers' location, signal-to-noise ratio, Doppler spread, and pilot length. Theoretical and simulation results indicate that secret key capacity in the slow fading channel is affected by frame-to-frame correlation and increasing transmission power or sampling period can suppress this effect.

DOI
10.1109/wcnc61545.2025.10978619

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This paper investigates the secret key capacity of key generation in the slow-fading wireless channel by considering the temporal correlation between channel measurements in the previous and current frame. We derive the analytical expression of secret key capacity in the slow-fading channel, and propose the design guideline for practical key generation schemes in slow fading channel environments. We systematically analyze the impact of all corresponding parameters on secret key capacity, including sampling period, sampling delay, eavesdroppers' location, signal-to-noise ratio, Doppler spread, and pilot length. Theoretical and simulation results indicate that secret key capacity in the slow fading channel is affected by frame-to-frame correlation and increasing transmission power or sampling period can suppress this effect.
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.
This letter examines the optimal pilot length for key generation in correlated wireless channels. We first deduce the secret key capacity as a function of the pilot length N and other system parameters. Then, derive two fixed-point equations to identify the optimal N with and without an eavesdropper. Our analysis and numerical results confirm the existence of the optimal N that maximizes the secret key capacity, and both results for optimal N are consistent. Furthermore, the optimal N generally decreases with average signal-to-noise ratio and maximum Doppler shift. Our examination can provide useful guidelines in designing practical key generation systems.
Wireless key generation in slow fading channels is challenging because of the limited channel variation and randomness. This paper proposes a novel artificial randomness (AR) assisted approach for fast key generation in slow fading environments. It integrates user-designed randomness into the channel probing to form a fast-changing combined channel to realize information-theory security. The analytical expressions of secret key capacity are derived. We find that it is possible to improve secret key capacity by introducing AR when legitimate users have a better channel condition than that of eavesdropper. We also find that the improved secret key capacity is proportional to the channel probing number and is bounded by the noise variance and channel condition. Simulation and experimental results show that AR approach can generate secret key effectively in slow fading environments by carefully designing probing numbers. Compared to existing work in literature, the proposed approach does not rely on multiple antennas or extra helpers, and it can be applied in both single antenna and multi-antenna systems.
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.
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.
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.