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Quantifying Jitter Transfer for Differential Measurement: Enhancing Security of Oscillator-Based TRNGs

David Lubicz, Maciej Skorski

TL;DR

This work tackles accurate entropy-rate estimation for oscillator-based TRNGs by addressing the imbalance between differential jitter measurements (which filter global noise) and the need for per-oscillator jitter parameters to compute entropy rates. It develops a phase-domain jitter-transfer framework, deriving exact distributions for rising edges and phase differences (including inverse-Gaussian and normal-inverse Gaussian forms) and establishing a normal-approximation in the small-jitter regime. The authors present a transferable mapping $\sigma'^2 = (f_1/f_0)^2\sigma_0^2 + \sigma_1^2$ that relates pair jitter to individual jitter, and provide practical methods to recover the per-oscillator jitters from differential measurements, with a conditioning bound $\kappa_\infty \le (1+L^2)(1+\tfrac{3}{2}L^2)$ dependent on frequency ratios. They validate the theory with simulations and FPGA experiments, showing two measurement schemes—one simple but assumption-driven and a robust alternate design with an extra flip-flop—and demonstrate applicability to MO-TRNGs and entropy-rate computation per MR4712007.

Abstract

The aim of this paper is to describe a way to improve the reliability of the measurement of the statistical parameters of the phase noise in a multi-ring oscillator-based TRNG. This is necessary to guarantee that the entropy rate is within the bounds prescribed by standards or security specifications. According to the literature, to filter out global noises which may strongly affect the measurement of the phase noise parameters, it is necessary to perform a differential measure. But a differential measurement only returns the parameters of the phase noise resulting of the composition of the noises of two oscillators whereas jitters parameters of individual oscillators are required to compute the entropy rate of a multi-ring oscillator-based TRNG. In this paper, we revisit the "jitter transfer principle" in conjunction with a tweaked design of an oscillator based TRNG to enjoy the precision of differential measures and, at the same time, obtain jitter parameters of individual oscillators. We show the relevance of our method with simulations and experiments with hardware implementations.

Quantifying Jitter Transfer for Differential Measurement: Enhancing Security of Oscillator-Based TRNGs

TL;DR

This work tackles accurate entropy-rate estimation for oscillator-based TRNGs by addressing the imbalance between differential jitter measurements (which filter global noise) and the need for per-oscillator jitter parameters to compute entropy rates. It develops a phase-domain jitter-transfer framework, deriving exact distributions for rising edges and phase differences (including inverse-Gaussian and normal-inverse Gaussian forms) and establishing a normal-approximation in the small-jitter regime. The authors present a transferable mapping that relates pair jitter to individual jitter, and provide practical methods to recover the per-oscillator jitters from differential measurements, with a conditioning bound dependent on frequency ratios. They validate the theory with simulations and FPGA experiments, showing two measurement schemes—one simple but assumption-driven and a robust alternate design with an extra flip-flop—and demonstrate applicability to MO-TRNGs and entropy-rate computation per MR4712007.

Abstract

The aim of this paper is to describe a way to improve the reliability of the measurement of the statistical parameters of the phase noise in a multi-ring oscillator-based TRNG. This is necessary to guarantee that the entropy rate is within the bounds prescribed by standards or security specifications. According to the literature, to filter out global noises which may strongly affect the measurement of the phase noise parameters, it is necessary to perform a differential measure. But a differential measurement only returns the parameters of the phase noise resulting of the composition of the noises of two oscillators whereas jitters parameters of individual oscillators are required to compute the entropy rate of a multi-ring oscillator-based TRNG. In this paper, we revisit the "jitter transfer principle" in conjunction with a tweaked design of an oscillator based TRNG to enjoy the precision of differential measures and, at the same time, obtain jitter parameters of individual oscillators. We show the relevance of our method with simulations and experiments with hardware implementations.

Paper Structure

This paper contains 20 sections, 7 theorems, 41 equations, 4 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. The jitter's transfer principle
  3. Preliminaries
  4. Analytical Distributions of Rising Edges and Sampled Phases
  5. Normal Approximation
  6. Main results
  7. Applications
  8. Finding Oscillators' Volatilities
  9. Motivation
  10. Principle
  11. Verification for 3-ROs
  12. General Case
  13. Implementations and Discussion
  14. Conclusion
  15. Proofs
  16. ...and 5 more sections

Key Result

theorem 1

The phase of the sampled signal at the $k$-th rising edge of the clock signal is distributed as the normal variance-mean mixture where the mixing distribution $T_k$ is the time it takes to reach the $k$-th rising edge of $s_0$ and is given by the inverse normal distribution and the $k$-th clock period is distributed as The change in the sampled oscillator phase during one clock period is given

Figures (4)

  • Figure 1: Jitter Transfer Principle: both signals jittered (left) versus jitter transferred (right). The oscillators in this toy example have frequencies $f_0 = 5,f_1=7$ and volatilities $\sigma_0=\sigma_1 = 0.02$ (seen in standard error bands around rising edges).
  • Figure 2: Quality of the normal approximation in the Jitter Transfer Principle (global view). This example assumes similar frequencies and volatilities: $f_1\approx f_0$, $\sigma_1\approx \sigma_0$, and studies various jitter levels defined as $\sigma[T_i]/T_i = \sqrt{\sigma_i^2/f_i}$.
  • Figure 3: Quality of the normal approximation in the Jitter Transfer Principle (discrepancy). This example assumes similar frequencies and volatilities: $f_1\approx f_0$, $\sigma_1\approx \sigma_0$, and studies various jitter levels defined as $\sigma[T_i]/T_i = \sqrt{\sigma_i^2/f_i}$.
  • Figure 4: Schema of a MO-TRNG that allows to measure individual jitters

Theorems & Definitions (10)

  • theorem 1
  • corollary 1
  • corollary 2: Relating Phase Noise and Time Jitter
  • theorem 2
  • remark 1
  • corollary 3: Jitter Transfer Principle baudet_security_2011
  • remark 2: Period Jitter Transfer
  • remark 3: Time Jitter Transfer
  • theorem 3
  • corollary 4: Solutions are stable under comparable frequencies