Modelling 1/f Noise in TRNGs via Fractional Brownian Motion

Security of oscillatory true random number generators remains not fully understood due to insufficient understanding of complex $1\mathrm{/}{f}^{\alpha }$ 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.