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Correlation Bounds and Markov Analysis for Ring-Oscillator TRNGs: A Joint Validation Framework

Miguel Alcocer, Ana Isabel Gómez, Domingo Gomez-Perez

Abstract

True Random Number Generators (TRNGs) based on ring oscillators require rigorous statistical validation to ensure cryptographic quality. While the Mauduit-Sárközy $k$-th order correlation measure $C_k$ provides theoretical bounds on pseudorandomness, and Maurer's Universal Statistical Test offers empirical entropy assessment, no prior work has correlated these metrics. This paper presents the first joint validation framework linking Maurer's Z-score to off-peak 2nd-order correlation $C_2$. We also derive the mathematical relationship between the previous two measures and high-order Markov chain transition probabilities in counter-based TRNGs over oscillator sampling architectures. Our results are validated computationally using OpenTRNG implementations, and demonstrate that practical implementations achieve Schmidt's improved bound. The initial results suggest a strong positive correlation between Maurer Z-score and $C_2$. Therefore, the results suggest a unified metric for TRNG quality-assessment can be achieve as a combination of these metrics, simplifying the study of new designs.

Correlation Bounds and Markov Analysis for Ring-Oscillator TRNGs: A Joint Validation Framework

Abstract

True Random Number Generators (TRNGs) based on ring oscillators require rigorous statistical validation to ensure cryptographic quality. While the Mauduit-Sárközy -th order correlation measure provides theoretical bounds on pseudorandomness, and Maurer's Universal Statistical Test offers empirical entropy assessment, no prior work has correlated these metrics. This paper presents the first joint validation framework linking Maurer's Z-score to off-peak 2nd-order correlation . We also derive the mathematical relationship between the previous two measures and high-order Markov chain transition probabilities in counter-based TRNGs over oscillator sampling architectures. Our results are validated computationally using OpenTRNG implementations, and demonstrate that practical implementations achieve Schmidt's improved bound. The initial results suggest a strong positive correlation between Maurer Z-score and . Therefore, the results suggest a unified metric for TRNG quality-assessment can be achieve as a combination of these metrics, simplifying the study of new designs.
Paper Structure (6 sections, 2 theorems, 17 equations, 2 figures)

This paper contains 6 sections, 2 theorems, 17 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Theoretical Analysis
  4. Computational Results and Discussion
  5. Results and Discussion
  6. Conclusion and future work

Key Result

Theorem 1

For a binary sequence $S$ of length $N$, suppose that $T(X)$ is the empirical transition matrix of a Markov chain of memory $k$. Under the assumption that $2^{k-2}\cdot C_k(S) < 1$, all transition probabilities satisfy, for any binary vector $V$ of length $k$.

Figures (2)

  • Figure 1: Ring oscillator composed of three inverters $(V_1,V_2,V_3)$ generating an oscillating clock signal $s$. Source almaraz2025current
  • Figure 2: Maurer Z-score vs. off-peak $C_2(S)$ for 58 OpenTRNG configurations. Color indicates XOR accumulation depth ($n=1$ to 16). Dashed line: Schmidt bound in Equation \ref{['eq:C_bound']}.

Theorems & Definitions (4)

  • Theorem 1
  • proof
  • Theorem 2
  • proof