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arXiv:2605.09556

Stream randomness extraction against quantum side information

Chun-Yang Luan, Xiang-Jie Lie, Lin Cheng, Gang-Xi Wang, Cheng-Kang Pan, Xiang Zhang, Xingjian Zhang

quant-phSubmitted May 10, 2026• Announced May 12, 2026• 0 citations
v1 · May 10, 2026v2 · Jun 5, 2026
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Abstract

Randomness extraction is indispensable for quantum random number generators, serving to eliminate bias and potential information leakage from raw measurement data. Conventional extractors operate in a block-wise fashion, requiring the complete accumulation of raw data before processing. To circumvent the latency and buffering overheads that hinder real-time random number generation, recent work introduced a stream-cipher implementation for the randomness extractor based on the Toeplitz matrix hashing. In this work, we generalize this stream-processing paradigm to the broader family of randomness extractors based on (almost dual) universal$_2$ random hashing. Specifically, we shift the computational burden from a time-consuming block-wise post-processing stage into an offline pre-processing stage that generates a pseudo-random mask. This allows the raw data to be processed by the mask on the fly using a simple bitwise exclusive-OR operation. Crucially, we prove that this stream implementation strictly preserves the security guarantees of the original block-wise protocols. We detail the transformation of three typical constructions -- based on standard Toeplitz, circulant, and modified Toeplitz matrices -- from block to stream implementations, and benchmark their practical performance using realistic quantum experimental data. We anticipate our framework will enhance the efficiency of real-time quantum cryptographic systems.

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Randomness extraction is indispensable for quantum random number generators, serving to eliminate bias and potential information leakage from raw measurement data. Conventional extractors operate in a block-wise fashion, requiring the complete accumulation of raw data before processing. To circumvent the latency and buffering overheads that hinder real-time random number generation, recent work introduced a stream-cipher implementation for the randomness extractor based on the Toeplitz matrix hashing. In this work, we generalize this stream-processing paradigm to the broader family of randomness extractors based on (almost dual) universal2_22​ random hashing. Specifically, we shift the computational burden from a time-consuming block-wise post-processing stage into an offline pre-processing stage that generates a pseudo-random mask. This allows the raw data to be processed by the mask on the fly using a simple bitwise exclusive-OR operation. Crucially, we prove that this stream implementation strictly preserves the security guarantees of the original block-wise protocols. We detail the transformation of three typical constructions -- based on standard Toeplitz, circulant, and modified Toeplitz matrices -- from block to stream implementations, and benchmark their practical performance using realistic quantum experimental data. We anticipate our framework will enhance the efficiency of real-time quantum cryptographic systems.
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.
Abstract Randomness extraction is a key problem in cryptography and theoretical computer science. With the recent rapid development of quantum cryptography, quantum‐proof randomness extraction has also been widely studied, addressing the security issues in the presence of a quantum adversary. In contrast with conventional quantum‐proof randomness extractors characterizing the input raw data as min‐entropy sources, it is found that the input raw data generated by a large class of trusted‐device quantum random number generators can be characterized as the so‐called reverse block source. This fact enables us to design improved extractors. Two novel quantum‐proof randomness extractors for reverse block sources that realize real‐time block‐wise extraction are proposed specifically. In comparison with the general min‐entropy randomness extractors, the designs achieve a significantly higher extraction speed and a longer output data length with the same seed length. In addition, they enjoy the property of online algorithms, which process the raw data on the fly without waiting for the entire input raw data to be available. These features make the design an adequate choice for the real‐time post‐processing of practical quantum random number generators. Applying the extractors to the raw data generated by a widely used quantum random number generator, a simulated extraction speed as high as 300 Gbps is achieved.
Quantum random number generators (QRNG) based on continuous variable (CV) quantum fluctuations offer great potential for their advantages in measurement bandwidth, stability and integrability. More importantly, it provides an efficient and extensible path for significant promotion of QRNG generation rate. During this process, real-time randomness extraction using information theoretically secure randomness extractors is vital, because it plays critical role in the limit of throughput rate and implementation cost of QRNGs. In this work, we investigate parallel and real-time realization of several Toeplitz-hashing extractors within one field-programmable gate array (FPGA) for parallel QRNG. Elaborate layout of Toeplitz matrixes and efficient utilization of hardware computing resource in the FPGA are emphatically studied. Logic source occupation for different scale and quantity of Toeplitz matrices is analyzed and two-layer parallel pipeline algorithm is delicately designed to fully exploit the parallel algorithm advantage and hardware source of the FPGA. This work finally achieves a real-time post-processing rate of QRNG above 8 Gbps. Matching up with integrated circuit for parallel extraction of multiple quantum sideband modes of vacuum state, our demonstration shows an important step towards chip-based parallel QRNG, which could effectively improve the practicality of CV QRNGs, including device trusted, device-independent, and semi-device-independent schemes.
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.
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.
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