The Error Analysis of the Secret Key Generation Algorithm Using Analog Function Computation

TL;DR

The paper addresses secure key establishment in a decentralized wireless network by leveraging Analog Function Computation (AFC) and Gaussian primes. Each node selects a Gaussian prime and applies pre-/post-processing, yielding a network-wide secret value $scalablekey$ without a central authority, with the secret key expressed as $K=\prod_i p_i$. It develops an error model incorporating thermal noise, channel estimation error, and the Rice $K$-factor, and uses Monte Carlo simulations to quantify the probability of correctly factorizing the norm of the secret key within a tolerance $gamma_t$. Channel modeling includes Rice fading with $K=\nu^2/(2\sigma^2)$, path loss via Friis, and channel reciprocity $h_{ij}=h_{ji}$, analyzing performance under Rayleigh and high-LoS conditions. The results indicate path loss and large-scale fading chiefly limit performance, with 2-node configurations outperforming 3-node ones; increasing the tolerance improves the observed success rate but raises security considerations, underscoring the need for precise channel estimation and practical validation via SDR experiments.

Abstract

This study introduces a decentralized approach to secure wireless communication using a cryptographic secret key generation algorithm among distributed nodes. The system model employs Gaussian prime numbers, ensuring the collaborative generation of a secret key. Pre-processing and post-processing functions enable to generate a secret key across the network. An error model evaluates aspects like thermal noise power and channel estimation errors, while simulations assess the success rate to factorize the norm of the secret key. It is observed that path loss-induced large scale fading emerges as a critical component impacting information and power loss. The robustness of the proposed model under fading channel conditions is evaluated with a success rate. Additionally, it is also observed that the tolerance value set in the factorization algorithms has a significant impact on the success rate. Furthermore, the success rate is compared in two scenarios, one with 2 users and another with 3 users, to provide a comprehensive evaluation of the system performance.

Figures (8)

• Figure 1: System Model
• Figure 2: System Functions
• Figure 3: Success Rate versus Channel Estimation Error Deviation Analysis for a Two-Node.
• Figure 4: Success Rate versus Distance Between Nodes Analysis for a Two-Node.
• Figure 5: Success Rate versus Channel Estimation Error Deviation Analysis for Three-Node.