Modelling 1/f Noise in TRNGs via Fractional Brownian Motion

TL;DR

This paper introduces fractional Brownian motion as a theoretical framework to model 1/f noise in true random number generators, providing analytical tools for understanding and calibrating oscillator-based TRNGs.

Contribution

It offers a comprehensive model capturing phase noise spectral densities and provides closed-form solutions for variance, entropy, and parameter estimation, bridging physical noise modeling with cryptographic needs.

Findings

Derived closed-form solutions for phase noise variance growth.

Provided explicit formulas for min-entropy under Gaussian assumptions.

Developed asymptotic methods for Allan variance estimation.

Abstract

Security of oscillatory true random number generators remains not fully understood due to insufficient understanding of complex $1/f^\alpha$ phase noise. To bridge this gap, we introduce fractional Brownian motion as a comprehensive theoretical framework, capturing power-law spectral densities from white to flicker frequency noise. Our key contributions provide closed-form tractable solutions: (1) a quasi-renewal property showing conditional variance grows with power-law time dependence, enabling tractable leakage analysis; (2) closed-form min-entropy expressions under Gaussian phase posteriors; and (3) asymptotically unbiased Allan variance parameter estimation. This framework bridges physical modelling with cryptographic requirements, providing both theoretical foundations and practical calibration for oscillator-based TRNGs.