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Modelling 1/f Noise in TRNGs via Fractional Brownian Motion

Maciej Skorski

TL;DR

Oscillator-based TRNGs exhibit complex 1/f^α phase noise that challenges security guarantees. The paper adopts fractional Brownian motion as a unified phase-noise model, linking time-domain leakage, spectral behavior, and entropy through analytically tractable expressions. Key contributions include a quasi-renewal variance formula, exact min-entropy under Gaussian posteriors, and asymptotically unbiased Allan-variance parameter estimation, plus a parameter-estimation framework and open-source repository. This framework enables rigorous, leakage-aware security analysis and practical calibration for oscillator-based TRNGs without heavy Monte Carlo simulations.

Abstract

Security of oscillatory true random number generators remains not fully understood due to insufficient understanding of complex $1/f^α$ 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.

Modelling 1/f Noise in TRNGs via Fractional Brownian Motion

TL;DR

Oscillator-based TRNGs exhibit complex 1/f^α phase noise that challenges security guarantees. The paper adopts fractional Brownian motion as a unified phase-noise model, linking time-domain leakage, spectral behavior, and entropy through analytically tractable expressions. Key contributions include a quasi-renewal variance formula, exact min-entropy under Gaussian posteriors, and asymptotically unbiased Allan-variance parameter estimation, plus a parameter-estimation framework and open-source repository. This framework enables rigorous, leakage-aware security analysis and practical calibration for oscillator-based TRNGs without heavy Monte Carlo simulations.

Abstract

Security of oscillatory true random number generators remains not fully understood due to insufficient understanding of complex 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.

Paper Structure

This paper contains 22 sections, 13 theorems, 67 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Contribution
  4. Related Work
  5. Prelimaries
  6. Results
  7. Time Domain Properties
  8. Spectral Properties
  9. Leakage Resilience
  10. Bit Bias and Entropy Rate
  11. Quasi-Stationarity and Allan Variance
  12. Conclusion
  13. Proofs
  14. Proof of \ref{['thm:covariance']}
  15. Proof of \ref{['thm:diff_spectrum']}
  16. ...and 7 more sections

Key Result

lemma 1

For any $0\leqslant s,t$ and $H>0$ we have In particular, the variance is given by:

Figures (8)

  • Figure 1: From Gaussian Processes to Security Guarantees
  • Figure 2: Correlation function for various Hurst parameters
  • Figure 3: Power spectral density for various Hurst exponents.
  • Figure 4: Periodic (wrapped) normal distribution $Y=\mathsf{Norm}(\mu,\sigma^2) \bmod 2\pi$.
  • Figure 5: Bit bias vs. phase noise variance
  • ...and 3 more figures

Theorems & Definitions (15)

  • lemma 1: Covariance of Fractional Brownian Motion
  • corollary 1: Scale-Ratio Coordinates Covariance Representation
  • corollary 2: Correlation
  • theorem 1: Spectrum of FBM
  • theorem 2: Spectrum of Differenced Process
  • corollary 3: Fractional Frequency Spectrum
  • theorem 3: Renewal and Predictive Posterior
  • corollary 4: Drift Independence
  • remark 1: Conditional Renewal
  • theorem 4: Distribution of Periodic Gaussian
  • ...and 5 more