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arXiv:1207.1473 · Physical Review A

Postprocessing for quantum random-number generators: Entropy evaluation and randomness extraction

Xiongfeng Ma, Feihu Xu, He Xu, Xiaoqing Tan, Bing Qi, Hoi‐Kwong Lo

Published Jun 21, 2013• 241 citations
arXivHTMLPDF

Abstract

Quantum random-number generators (QRNGs) can offer a means to generate information-theoretically provable random numbers, in principle. In practice, unfortunately, the quantum randomness is inevitably mixed with classical randomness due to classical noises. To distill this quantum randomness, one needs to quantify the randomness of the source and apply a randomness extractor. Here, we propose a generic framework for evaluating quantum randomness of real-life QRNGs by min-entropy, and apply it to two different existing quantum random-number systems in the literature. Moreover, we provide a guideline of QRNG data postprocessing for which we implement two information-theoretically provable randomness extractors: Toeplitz-hashing extractor and Trevisan's extractor.

DOI
10.1103/physreva.87.062327

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Quantum random-number generators (QRNGs) can offer a means to generate information-theoretically provable random numbers, in principle. In practice, unfortunately, the quantum randomness is inevitably mixed with classical randomness due to classical noises. To distill this quantum randomness, one needs to quantify the randomness of the source and apply a randomness extractor. Here, we propose a generic framework for evaluating quantum randomness of real-life QRNGs by min-entropy, and apply it to two different existing quantum random-number systems in the literature. Moreover, we provide a guideline of QRNG data postprocessing for which we implement two information-theoretically provable randomness extractors: Toeplitz-hashing extractor and Trevisan's extractor.
Randomness is widely used in various fields, including encryption, statistical analysis, and numerical simulations. Randomness is also a fundamental resource in science and engineering. For such applications, we usually need to provide unbiased and independent random bits. This raises the issue of where to get these supposed random bits. Quantum random number generators (QRNGs) generate real random numbers based on the inherent randomness of quantum measurements. In practice, unfortunately, quantum randomness is inevitably mixed with classical randomness due to classical noise. Also, randomness is often correlated and biased. It is necessary to process the resulting raw bits sequence and convert them to good-quality output values that are as close to uniform distribution as possible. Random extractors are required for this. We analyze the randomness obtained by quantum random number generators as well as various examples of post-processing. We discuss the types of randomness extractors. Also, we use information-theoretically secure extractors and present improved novel randomness extraction.
Theoretically, Quantum random number generators (QRNGs) can produce truly random numbers that cannot be predicted in advance and have numerous uses. But in practice, it cannot escape the problem of unwanted classical noise, which reduces the quality of the random bits they produce. Therefore, we need to post-process data from a QRNG’s entropy source to improve the statistical quality of the classical output sequence generated from the quantum entropy source. In this work, we implement and compare six extraction primitives: Von Neumann, Multi-level Peres, Toeplitz-hash, Circulant-hash, Trevisan’s construction, and SHA-256 distillation. Using a large classical random bit stream generated by an experimental QRNG, we evaluate these algorithms based on extraction efficiency (η), computational complexity, and seed-entropy requirements. The findings show that although deterministic approaches, such as Multi-level Peres, outperform von Neumann maps, algebraic approaches, especially Circulant extractors, offer higher throughput (η = 0.90) and lower seed consumption by leveraging the cyclicity of the group Zn. The Trevisan extractor provides strong theoretical security with reduced seed growth, but at the cost of higher computational complexity. In contrast, the SHA-256-based method offers fast practical post-processing but provides no information-theoretic entropy guarantees. Statistical validation via autocorrelation and inter-sequence cross-correlation confirms that all evaluated methods successfully decouple hardware-induced dependencies
Randomness extraction algorithms play an essential role in Quantum Random Number Generators (QRNGs), where they are used to suppress unwanted classical noise and distill true randomness from their biased output. By employing the SHA-512 hash function and Toeplitz matrix multiplication, we analyse two suitable constructions based on different principles and reach postprocessing rates of 8.69 Mbps and 3.68 Mbps, respectively. Finally, we develop a length-compatible Toeplitz-hashing algorithm able to achieve rates of 143.29 Mbps in a parallelized GPU implementation.
Abstract Random Number Generators (RNGs) are devices whose utility spans from cryptography to gambling. Depending on the source of the random seed, RNGs are classified as Pseudo-Random Number Generators (PRNGs), which are based on mathematical algorithms, True Random Number Generators (TRNGs) sourced through seemingly random physical processes, and finally, Quantum Random Number Generators (QRNGs) which harness the intrinsic randomness of measured outcomes of quantum states. To ensure that the output of a QRNG is private and nearly uniform, the raw data undergoes a strong extraction process such as Toeplitz or Trevisan Hashing. The extraction process needs to be extremely efficient to achieve high-speed random number generation. The present work demonstrates state-of-the-art methods for performing information-theoretically provable randomness extraction using variations of the Toeplitz Hashing algorithm. These algorithms are implemented on various hardware for comprehensive analysis. Subsequently, these methods are applied to two raw data sets from a PRNG source and a quantum source. A new benchmark of 19.5 Gbps for Randomness Extraction using the Fast Fourier Transform-based Toeplitz Strong Extractor is demonstrated in this work. The implementation of the algorithm on a GPU-based system demonstrates notably enhanced speed, marking a significant leap beyond the existing state-of-the-art implementations.
Random numbers are of vital importance in today’s world and used for example in many cryptographical protocols to secure the communication over the internet. The generators producing these numbers are Pseudo Random Number Generators (PRNGs) or True Random Number Generators (TRNGs). A subclass of TRNGs are the Quantum based Random Number Generators (QRNGs) whose generation processes are based on quantum phenomena. However, the achievable quality of the numbers generated from a practical implementation can differ from the theoretically possible. To ease this negative effect post-processing can be used, which contains the use of extractors. They extract as much entropy as possible from the original source and produce a new output with better properties. The quality and the different properties of a given output can be measured with the help of statistical tests. In our work we examined the effect of different extractors on two QRNG outputs and found that witg the right extractor we can improve their quality.
Quantum Random Number Generators (QRNGs) serve as high entropy sources for Quantum Key Distribution (QKD) systems. The raw data from quantum sources however require post-processing to generate a nearly uniform distribution. This work presents a state-of-the-art implementation of the Toeplitz Strong Extractor on an FPGA, achieving a benchmark extraction speed of 26.57 Gbps. A detailed implementation flow of the post-processing on FPGA is presented along with the various execution speeds obtained based on different randomness extraction ratios. Raw data from an in-house phase noise-based QRNG is processed on the FPGA using the implementation and the output is validated via the NIST STS 2.1.2 statistical randomness tests suite.