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

This paper introduces a novel joint validation framework that establishes the first mathematical and empirical correlation between the Mauduit-Sarkozy second-order correlation measure ( $C_2$ ) and Maurer's Universal Statistical Test Z-score, demonstrating their combined utility for assessing the cryptographic quality of ring-oscillator TRNGs through Markov chain analysis and OpenTRNG implementations.

Original authors: Miguel Alcocer, Ana Isabel Gómez, Domingo Gomez-Perez

Published 2026-03-17

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Miguel Alcocer, Ana Isabel Gómez, Domingo Gomez-Perez

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: Making "True" Randomness

Imagine you are a casino owner. You need dice that are perfectly fair so no one can cheat. In the digital world, computers need "random numbers" to create secret keys for encryption (like the password that locks your bank account).

Most computers are terrible at making random numbers because they are predictable machines. To fix this, engineers use Ring Oscillator TRNGs (True Random Number Generators). Think of these as digital pendulums or metronomes that swing back and forth. Because of tiny, unpredictable vibrations in the hardware (like thermal noise or "jitter"), these pendulums never swing at exactly the same speed. That tiny, chaotic wobble is the source of true randomness.

The Problem: How Do We Know They Are Random?

Just because a pendulum wobbles doesn't mean the resulting numbers are good enough for cryptography. We need to test them. The paper tackles a specific problem: We have different ways to test randomness, but nobody knew if they were actually measuring the same thing.

The authors looked at two famous "rulers" used to measure randomness:

The Maurer Test: This checks how "compressible" the data is. If a sequence is truly random, it's like a scrambled egg—you can't squeeze it into a smaller box without losing information. If it's predictable, you can compress it easily.
The Correlation Measure ( $C_2$ ): This checks if the numbers are "friends" with each other. If you see a 1, does it make it more likely you'll see another 1 a moment later? In a truly random sequence, the numbers should be strangers; they shouldn't know what the previous number was.

The Gap: Until now, researchers used these rulers separately. They didn't know if a high score on the Maurer Test guaranteed a low score on the Correlation Test.

The Solution: A "Joint Validation Framework"

The authors built a bridge between these two rulers. They created a new framework that links them together, along with a third concept called Markov Chains (which is just a fancy way of saying "predicting the next step based on the current step").

Here is the analogy:

Imagine you are trying to predict the weather.
Markov Chains are like looking at the clouds right now to guess if it will rain tomorrow.
Correlation ( $C_2$ ) measures how much the weather today influences the weather tomorrow.
Maurer's Test measures how chaotic the weather patterns are over a whole month.

The paper proves mathematically that if the weather is truly chaotic (high Maurer score), then today's weather cannot predict tomorrow's weather (low Correlation). They showed that these two tests are actually looking at the same underlying truth from different angles.

The Experiment: The "Counter" vs. The "Sampler"

The team used a tool called OpenTRNG (an open-source simulator) to test different designs. They compared two ways of reading the "wobbly" pendulums:

Direct Sampling (The Old Way): Taking a snapshot of the pendulum at a specific time.
- Analogy: Trying to guess the outcome of a coin flip by looking at it while it's still spinning. It's messy and often biased.
- Result: The numbers were predictable (high correlation). The test failed.
Counting Periods (The New Way): Instead of taking a snapshot, they counted how many times the pendulum swung between two ticks of a master clock.
- Analogy: Instead of looking at the spinning coin, you count how many times it spins before it stops. This smooths out the tiny errors.
- Result: This method was a huge success. The numbers became much more random. The "Correlation" dropped to near zero, and the "Maurer Test" passed with flying colors.

The Key Findings

They are Best Friends: The study found a strong positive correlation between the Maurer Test score and the Correlation score. If one test says the numbers are good, the other almost certainly will too. This means engineers can use these tests together as a "unified metric" to quickly check if a design is safe.
The "Counting" Trick Works: The method of counting oscillator swings (rather than just sampling bits) is superior. It reduced the predictability of the numbers by about 40 times compared to the old method.
Beating the Theory: The best designs they tested were even better than the theoretical math predicted. They achieved a level of randomness that was 2 to 3 times better than the "Schmidt bound" (a famous mathematical limit for how random a sequence can be).

Why Does This Matter?

For the people building secure systems (like banks, governments, and crypto wallets), this paper is a gift.

Simpler Testing: Instead of running dozens of complex, unrelated tests, they now have a unified way to check quality. If the "Correlation" is low and the "Maurer Z-score" is low, the system is likely secure.
Better Designs: It confirms that counting oscillator swings is the "gold standard" for building these random number generators.
Early Warning: If a new design fails one of these tests, the authors know immediately that the other test will likely fail too, saving time and resources.

In short: The authors took two different ways of measuring chaos, proved they are mathematically linked, and showed that a specific "counting" technique creates the most chaotic (and therefore most secure) random numbers possible.

1. Problem Statement

True Random Number Generators (TRNGs) based on Ring Oscillators (ROs) are critical for cryptographic applications, relying on physical entropy sources like thermal jitter. However, ensuring their cryptographic quality requires rigorous statistical validation.

The Gap: While the Mauduit-Sárközy correlation measure ( $C_k$ ) provides theoretical bounds on pseudorandomness and Maurer's Universal Statistical Test offers empirical entropy assessment, prior research has not established a direct link between these metrics.
The Challenge: There is no unified framework to correlate the theoretical correlation bounds ( $C_k$ ) with empirical entropy tests (Maurer's Z-score) or high-order Markov chain transition probabilities. This lack of correlation makes it difficult to predict the performance of new TRNG designs without running exhaustive, time-consuming test suites (e.g., NIST, AIS31).

2. Methodology

The authors propose a joint validation framework that theoretically and empirically links three key metrics:

Mauduit-Sárközy Correlation ( $C_k$ ): Specifically focusing on the 2nd-order correlation ( $C_2$ ), which measures the maximum Pearson correlation over non-zero lags.
Maurer's Universal Test: An entropy-based test that measures the compressibility of a sequence by analyzing the spacing between repeated $k$ -bit blocks, resulting in a Z-score.
High-Order Markov Chains: Modeling the TRNG output as a Markov process with memory $k$ to analyze transition probabilities.

Experimental Setup:

Framework: The authors utilized OpenTRNG, an open-source framework for reproducible RO-TRNG design.
Architectures: They tested Elementary Ring Oscillators (ERO) and Coherent Sampling Ring Oscillators (COSO).
Extraction Method: They employed a differential counter-based approach (counting oscillator periods rather than direct bit sampling) to isolate independent thermal noise and cancel common-mode jitter.
Variables: The study varied oscillator frequencies ( $f_1, f_2$ ), reference frequencies ( $f_{ref}$ ), and XOR accumulation depths ( $n = 1, 2, 4, 8, 16$ ).
Data: For each of 58 parameter configurations, $N = 10^5$ bits were generated to compute $C_2$ and the Maurer Z-score.

3. Key Contributions

A. Theoretical Derivations

The paper establishes rigorous mathematical relationships between the metrics:

Theorem 1 (Correlation vs. Markov): The authors prove that low $C_k(S)$ guarantees near-uniform $k$ -bit pattern frequencies. They derive an explicit bound showing that the deviation of Markov transition probabilities from the ideal $1/2$ is bounded by a function of $C_k(S)$ . Specifically, for an 8-bit memory chain, the transition probability deviation is proportional to $2^6 \cdot C_8(S)$ .
Theorem 2 (Correlation vs. Maurer): The paper derives theoretical bounds linking the Maurer test function ( $f_{TU}$ ) directly to the correlation measure $C_k(S)$ . This proves that the compressibility measured by Maurer's test is mathematically constrained by the sequence's correlation properties.
Schmidt's Bound Validation: The work validates Schmidt's improved bound for correlation ( $C_k(S) \le \sqrt{2k \ln(N)/N}$ ), showing it is significantly tighter (approx. 3.5x improvement) than the previous Alon constant bound.

B. Empirical Validation

Joint Distribution Analysis: The study computationally validates the theoretical links by plotting the Maurer Z-score against off-peak $C_2$ for various RO configurations.
Counter vs. Sampling: The authors compared counter-based extraction against direct bit sampling. They demonstrated that counting periods yields significantly better statistical properties (lower correlation) than direct sampling.

4. Key Results

Strong Positive Correlation: A Pearson correlation coefficient of $r = 0.95$ was found between the Maurer Z-score and the off-peak $C_2$ measure. This indicates that as the correlation ( $C_2$ ) increases, the Z-score (deviation from randomness) also increases.
Performance Thresholds:
- Configurations with $|Z| < 0.1$ consistently achieved $C_2 < 0.01$ .
- High correlation values ( $C_2 > 0.8$ ) correlated with $Z > 1$ , indicating a failure to meet random expectations.
Impact of XOR Accumulation: Increasing the XOR accumulation depth ( $n$ ) from 1 to 16 moved the system performance closer to the ideal region ( $|Z| \approx 0, C_2 < 0.01$ ).
Counter-Based Superiority:
- Direct Sampling: $C_2 = 0.34$ , $Z = -2.1$ (Failed).
- Counter-Based: $C_2 = 0.009$ , $|Z| < 1$ (Passed).
- This represents a ~40x improvement in the correlation metric, confirming the efficacy of the differential counting method.
Schmidt Bound Achievement: Practical implementations achieved $C_2$ values 2–3 times lower than Schmidt's improved theoretical bound, suggesting that practical designs can outperform generic theoretical limits when optimized.

5. Significance and Conclusion

Unified Validation Framework: The paper proposes a simplified validation protocol for TRNG designers. Instead of running full test suites immediately, designers can compute $C_2$ (via FFT autocorrelation) and Maurer's Z-score.
- Criteria for Quality: Configurations satisfying $C_2 < 0.01$ and $|Z| < 2$ are highly likely to pass comprehensive suites like NIST or AIS31.
Design Guidance: The strong correlation allows for early detection of design flaws. If a design fails one metric, it is statistically probable to fail the other, saving computational resources.
Theoretical Bridge: By connecting combinatorial measures ( $C_k$ ) with probabilistic models (Markov chains) and entropy tests (Maurer), the paper provides a deeper theoretical understanding of how physical jitter translates into cryptographic randomness.
Future Work: The authors plan to explore architecture-specific correlation bounds and extend the framework to multi-source XOR accumulation and higher-order correlations.

In summary, this work bridges the gap between theoretical correlation bounds and empirical entropy testing, providing a robust, mathematically grounded, and computationally efficient method for validating Ring Oscillator TRNGs.