Online Adaptive Randomness Extraction for Physical and Quantum Entropy Sources: Composing Health-Test Soundness with the Leftover Hash Lemma

Physical and quantum random number generators (QRNGs) do not emit uniform bits: detector bias, dead time, after-pulsing, and environmental drift leave their raw output biased, correlated, and partially predictable to an adversary who models the device. Deployable generators therefore include a conditioning (randomness-extraction) stage that distils near-uniform keys from a source of certified min-entropy, and the cryptographic workhorse of that stage is the Leftover Hash Lemma (LHL). This paper analyses two ways in which real deployments depart from the textbook LHL. First, a physical source has no fixed min-entropy: continuous health tests, as standardised in NIST SP 800-90B, produce a runtime lower-bound estimate that varies from block to block and can occasionally be optimistic. Driving the extractor output length from this estimate, we prove an online adaptive extraction theorem whose security bound cleanly composes the estimator soundness error with the extractor error, yielding a (δ+ε)-secure key and separating how often the entropy estimate is optimistic from how uniform the hash output is, as two independently tunable budgets. Second, for settings that publish or reuse the extractor seed — public randomness beacons, reproducible pipelines, and seed-constrained devices — we prove that publishing the seed costs nothing, since this is the defining property of a strong extractor, and that reusing a single seed across m independent source blocks costs at most mε in statistical distance. A corollary bounds the security loss when the extracted key is consumed by a downstream symmetric primitive, and a worked instantiation with realistic QRNG parameters reports the achievable secret-key rate. A reproducible numerical simulation on a synthetic biased source corroborates all four claims. Throughout, we are explicit about the single load-bearing unproven premise — empirical min-entropy certification of the physical device — which belongs to entropy-source validation and lies outside the scope of the analysis.

quantum random number generator, randomness extraction, min-entropy, Leftover Hash Lemma, entropy-source validation

1. Barker, E., Kelsey, J., McKay, K., Roginsky, A., & Turan, M. S. (2022). Recommendation for Random Bit Generator (RBG) Constructions (NIST Special Publication 800-90C, 3rd public draft). National Institute of Standards and Technology. https://csrc.nist.gov/pubs/sp/800/90/c/3pd [Google Scholar] [Crossref]

2. Bennett, C. H., Brassard, G., Crépeau, C., & Maurer, U. M. (1995). Generalized privacy amplification. IEEE Transactions on Information Theory, 41(6), 1915–1923. https://doi.org/10.1109/18.476316 [Google Scholar] [Crossref]

3. Dodis, Y., Ostrovsky, R., Reyzin, L., & Smith, A. (2008). Fuzzy extractors: How to generate strong keys from biometrics and other noisy data. SIAM Journal on Computing, 38(1), 97–139. https://doi.org/10.1137/060651380 [Google Scholar] [Crossref]

4. Hayashi, M., & Tsurumaru, T. (2016). More efficient privacy amplification with less random seeds via dual universal hash function. IEEE Transactions on Information Theory, 62(4), 2213–2232. https://doi.org/10.1109/TIT.2016.2526018 [Google Scholar] [Crossref]

5. Herrero-Collantes, M., & Garcia-Escartin, J. C. (2017). Quantum random number generators. Reviews of Modern Physics, 89(1), 015004. https://doi.org/10.1103/RevModPhys.89.015004 [Google Scholar] [Crossref]

6. Impagliazzo, R., Levin, L. A., & Luby, M. (1989). Pseudo-random generation from one-way functions. In Proceedings of the 21st Annual ACM Symposium on Theory of Computing (pp. 12–24). Association for Computing Machinery. https://doi.org/10.1145/73007.73009 [Google Scholar] [Crossref]

7. Kelsey, J., McKay, K. A., & Turan, M. S. (2015). Predictive models for min-entropy estimation. In Cryptographic Hardware and Embedded Systems — CHES 2015 (Lecture Notes in Computer Science, Vol. 9293, pp. 373–392). Springer. https://doi.org/10.1007/978-3-662-48324-4_19 [Google Scholar] [Crossref]

8. Liu, Y., Zhao, Q., Li, M.-H., Guan, J.-Y., Zhang, Y., Bai, B., Zhang, W., Liu, W.-Z., Wu, C., Yuan, X., Li, H., Munro, W. J., Wang, Z., You, L., Zhang, J., Ma, X., Fan, J., Zhang, Q., & Pan, J.-W. (2018). Device-independent quantum random-number generation. Nature, 562, 548–551. https://doi.org/10.1038/s41586-018-0559-3 [Google Scholar] [Crossref]

9. Ma, X., Yuan, X., Cao, Z., Qi, B., & Zhang, Z. (2016). Quantum random number generation. npj Quantum Information, 2, 16021. https://doi.org/10.1038/npjqi.2016.21 [Google Scholar] [Crossref]

10. Mannalath, V., Mishra, S., & Pathak, A. (2023). A comprehensive review of quantum random number generators: Concepts, classification and the origin of randomness. Quantum Information Processing, 22(12), 439. https://doi.org/10.1007/s11128-023-04175-y [Google Scholar] [Crossref]

11. Radhakrishnan, J., & Ta-Shma, A. (2000). Bounds for dispersers, extractors, and depth-two superconcentrators. SIAM Journal on Discrete Mathematics, 13(1), 2–24. https://doi.org/10.1137/S0895480197329508 [Google Scholar] [Crossref]

12. Renner, R. (2005). Security of quantum key distribution [Doctoral dissertation, ETH Zürich]. https://doi.org/10.3929/ethz-a-005115027 [Google Scholar] [Crossref]

13. Turan, M. S., Barker, E., Kelsey, J., McKay, K., Baish, M., & Boyle, M. (2018). Recommendation for the entropy sources used for random bit generation (NIST Special Publication 800-90B). National Institute of Standards and Technology. https://doi.org/10.6028/NIST.SP.800-90B [Google Scholar] [Crossref]

14. Vadhan, S. P. (2012). Pseudorandomness. Foundations and Trends in Theoretical Computer Science, 7(1–3), 1–336. https://doi.org/10.1561/0400000010 [Google Scholar] [Crossref]

• The Impact of Ownership Structure on Dividend Payout Policy of Listed Plantation Companies in Sri Lanka
• Urban Sustainability in North-East India: A Study through the lens of NER-SDG index
• Performance Assessment of Predictive Forecasting Techniques for Enhancing Hospital Supply Chain Efficiency in Healthcare Logistics
• The Fractured Self in Julian Barnes' Postmodern Fiction: Identity Crisis and Deflation in Metroland and the Sense of an Ending
• Impact of Flood on the Employment, Labour Productivity and Migration of Agricultural Labour in North Bihar