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Applied Sciences

Statistical Validation of a Physical Prime Random Number Generator Based on Quantum Noise

Maurício Jonas Ferreira, Nuno A. Silva, Armando N. Pinto, Nelson Jesus Muga

Published Nov 23, 2023• 4 citations
Publisher

Abstract

Random prime numbers are an essential resource for many asymmetric cryptographic protocols. However, despite the emerging popularity of quantum random number generators (QRNGs) as sources of secure randomness, physical prime number generators have not yet been explored. In this work, we experimentally implement and characterize a vacuum-based probabilistic prime number generation scheme with an error probability of 3.5×10−15. By removing the quantum source (QS), an additional scheme based on electronic noise is derived, and a comparative analysis for increasing prime lengths is made. We observed that the QS significantly outperforms the classical scheme for small prime generation, where increases up to 585.0% in the diversity of unique primes obtained are seen. Moreover, we propose a length-agnostic statistical test for prime number sequences and apply it to the output of the uniformized randomness source, which was successful in revealing underlying biases in the output prime distributions. The resultant sequences were subsequently submitted to the NIST statistical test suite, where the quantum and classical sources passed, respectively, 86.96% and 45.34% of the total test set applied.

DOI
10.3390/app132312619

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Random prime numbers are an essential resource for many asymmetric cryptographic protocols. However, despite the emerging popularity of quantum random number generators (QRNGs) as sources of secure randomness, physical prime number generators have not yet been explored. In this work, we experimentally implement and characterize a vacuum-based probabilistic prime number generation scheme with an error probability of 3.5×10−15. By removing the quantum source (QS), an additional scheme based on electronic noise is derived, and a comparative analysis for increasing prime lengths is made. We observed that the QS significantly outperforms the classical scheme for small prime generation, where increases up to 585.0% in the diversity of unique primes obtained are seen. Moreover, we propose a length-agnostic statistical test for prime number sequences and apply it to the output of the uniformized randomness source, which was successful in revealing underlying biases in the output prime distributions. The resultant sequences were subsequently submitted to the NIST statistical test suite, where the quantum and classical sources passed, respectively, 86.96% and 45.34% of the total test set applied.
Quantum random number generators (QRNGs) are an increasingly popular method to obtain information-theoretically provable random keys for cryptographic applications. Here, we experimentally demonstrate a probable prime generation protocol using a vacuum-based QRNG and compare its output with the one yielded by an electronic noise source. The quantum source significantly outperformed the classical scheme for lengths smaller than 128 bits, with increases of up to 837.1% in the number of unique primes being reported. By mapping the input noise to a uniform distribution, this figure was additionally increased by up to 844.3% and the output statistical distribution for small primes significantly improved.
Random prime number generation is crucial for the implementation of several encryption and signature protocols in cryptographic applications. Recently, Quantum Random Number Generators (QRNGs) have emerged as a solution to obtain information-theoretically secure randomness, setting them apart from algorithmic generators. Nonetheless, their output is still prone to deterioration under a flawed implementation and thus requires continuous statistical validation. This process is uniquely challenging for prime number generators since they cannot be submitted to the traditional testing suites, which are primarily designed to test uniform pseudorandom sources. Here, we enable direct statistical testing of prime Random Number Generators (RNGs) by extending a validation framework based on an equiprobable binning of the prime distribution with a predictive machine learning model, which can independently learn correlations in the prime output. The validity of this framework is then verified through extensive cryptanalysis of a prime QRNG based on quadrature measurements of the vacuum state. The developed model successfully identifies flawed configurations of the prime QRNG for output lengths up to 128 bits. This assessment was additionally extended to a purely classical scheme derived from electronic noise measurements. Although no increase in the model’s predictive capacity is seen, our testing framework can nonetheless demonstrate that this classical source produces a conclusively biased prime distribution. In turn, the implemented QRNG remains resilient against this type of cryptanalysis.
Quantum random number generators represent one of the most promising techniques for producing truly and unique random numbers, essential in many applications such as quantum communication and cryptography protocols. In this work, we report an experimental implementation of vacuum-based quantum random number generator. By measuring the quantum fluctuations of the radiation ground state, a mixture of a perfectly Gaussian distributed random signal—originating from the intrinsic randomness in the quantum measurement— and an undesired classical electronic noise is obtained. Using an adequate post-processing, the undesired classical contribution of the electronic noise is removed to produce final, quasi-uniformly distributed random bit sequences. The high-quality character of the generated random bits is confirmed by applying the statistical tests of the NIST SP 800-22 test battery.
Quantum random number generators (QRNGs) are currently in high demand across a large number of cryptographic applications as secure sources of true randomness. In this work, we characterize the conditions from which randomness can be extracted in a QRNG based on homodyne measurements of vacuum fluctuations by assessing the impact of experimental limitations, such as the digitizer resolution or the presence of excess local oscillator (LO) noise due to an unbalanced detection. Moreover, we propose an estimation method to quantify the excess entropy contribution introduced by an unbalanced detection and analyze the implementation of the post-processing algorithm. Finally, we submitted the generated numbers to a set of statistical tests to assess the quality of its output randomness and verified that it passes the standard libraries.
Random numbers are an essential resource to many applications, including cryptography and Monte Carlo simulations. Quantum random number generators (QRNGs) represent the ultimate source of randomness, as the numbers are obtained by sampling a physical quantum process that is intrinsically probabilistic. However, they are yet to be widely employed to replace deterministic pseudo random number generators (PRNG) for practical applications. QRNGs are regarded as interesting devices. However, they are slower than PRNGs for simulations and are typically seen as clumsy laboratory prototypes, prone to failures, and unreliable for cryptographic applications. Here, we overcome these limitations and demonstrate a compact and self-contained QRNG capable of generating random numbers at a pace of 8 Gbit/s uninterruptedly for 71 days. During this period, the physical parameters of the quantum process were monitored in real time by self-checking functions implemented in the generator itself. At the same time, the output random numbers were analyzed with the most stringent suites of statistical tests. The analysis shows that the QRNG under test sustained the continuous operation without physical instabilities or hardware failures. At the same time, the output random numbers were analyzed with the most stringent suites of statistical tests, which were passed during the whole operation time. This extensive trial demonstrates the reliability of a robustly designed QRNG and paves the way to its use in practical applications based on randomness.
This study introduces a quantum-enhanced approach to password generation using two distinct quantum randomness sources: classical measurement of superposition states and depolarizing noise. These sources overcome the shortcomings of classical pseudo-random number generators. Notably, this work pioneers integrating depolarizing noise as a controlled entropy source alongside superposition, offering a novel alternative to conventional quantum randomness methods. Quantum circuits, implemented via Qiskit, were executed on AerSimulator (idealized simulation) and FakeProvider (hardware-inspired noise model) backends, producing binary sequences through XOR and Concatenation modes. These sequences were converted into passwords of lengths 16 and 25 across datasets of 10K, 100K, and 500K, using Binary Character Mapping and Randomized reseeding methods. Binary entropy’s randomness was verified by the NIST Statistical Test Suite, with all p-values above 0.01. Generated password strength was evaluated through Clustering (Levenshtein distance), Entropy (3.84 to 4.40 bits per character), Hamming Distance, Uniqueness (100 percent), and ZXCVBN scoring (Score 4). Results from both backend-based password datasets were nearly identical, confirming simulated and hardware-based quantum randomness as a reliable and scalable alternative to systems. This work positions quantum noise-driven password generation as a robust cryptographic tool with the potential for real-world security applications.This research investigates the effectiveness of quantum noise, derived from superposition states and depolarizing channels, as a high-entropy source for generating cryptographically secure passwords. Quantum circuits implemented on the AerSimulator and FakeProvider backends produce random binary sequences, which are processed through chaotic mixing and rejection sampling to ensure uniform randomness. These sequences are rigorously evaluated using Shannon entropy, min-entropy and the NIST statistical test suite, alongside password strength assessments via tools like ZXCVBN. Results indicate exceptional randomness, with an average Shannon entropy of 3.84 ± 0.4 bits per character and 100% uniqueness across datasets of 20,000 passwords. The FakeProvider backend slightly outperforms the AerSimulator, achieving full compliance with NIST tests, while both produce passwords with top-tier strength scores (ZXCVBN Score 4). Comparative analyses demonstrate that quantum noise surpasses classical pseudo-random number generators in entropy density and resistance to brute-force and dictionary-based attacks. This study validates quantum noise as a superior entropy source, offering significant potential for enhancing password security in cryptographic applications and paving the way for quantum-enhanced cybersecurity solutions.