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Security of practical private randomness generation
Open accessAbstract
Measurements on entangled quantum systems necessarily yield outcomes that are intrinsically unpredictable if they violate a Bell inequality. This property can be used to generate certified randomness in a device-independent way, i.e., without making detailed assumptions about the internal working of the quantum devices used to generate the random numbers. Furthermore these numbers are also private; i.e., they appear random not only to the user but also to any adversary that might possess a perfect description of the devices. Since this process requires a small initial random seed to sample the behavior of the quantum devices and to extract uniform randomness from the raw outputs of the devices, one usually speaks of device-independent randomness expansion. The purpose of this paper is twofold. First, we point out that in most real, practical situations, where the concept of device independence is used as a protection against unintentional flaws or failures of the quantum apparatuses, it is sufficient to show that the generated string is random with respect to an adversary that holds only classical side information; i.e., proving randomness against quantum side information is not necessary. Furthermore, the initial random seed does not need to be private with respect to the adversary, provided that it is generated in a way that is independent from the measured systems. The devices, however, will generate cryptographically secure randomness that cannot be predicted by the adversary, and thus one can, given access to free public randomness, talk about private randomness generation. The theoretical tools to quantify the generated randomness according to these criteria were already introduced in S. Pironio et al. [Nature (London) 464, 1021 (2010)], but the final results were improperly formulated. The second aim of this paper is to correct this inaccurate formulation and therefore lay out a precise theoretical framework for practical device-independent randomness generation.
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On the basis of the CHSH inequality, Fehr et al. and Pironio et al. proposed self-testing randomness expansion protocols against classical adversaries.
“On the basis of the Clauser -Horne -Shimony -Holt (CHSH) inequality, 56 Fehr et al. 57 and Pironio et al. 58 proposed self-testing randomness expansion protocols against classical adversaries.”
From Quantum random number generation (2016) · cites this paper as [58]
Section 3.4 (pp. 12–13) presents a Bell-violation-based randomness expansion protocol and proves its security against classical side information. But the Introduction (p. 3) says this paper’s results apply to arbitrary Bell inequalities; it associates a specifically CHSH-based protocol with another work. The claim’s CHSH-specific characterization of this paper’s protocol is therefore not supported. (LLM, conf=0.85)
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Experimental Certification of Random Numbers via Quantum Contextuality
Um, M., Zhang, X., Zhang, J., Wang, Y., Shen, Y., Deng, D. L., Duan, L.-M., & Kim, K. (2013) · Scientific Reports · doi:10.1038/srep01627 · Cited by 72
Miller, C. A., & Shi, Y. (2014) · ACM Symposium on Theory of Computing (STOC) · doi:10.1145/2591796.2591843 · Cited by 72
“Classical security was proved in [31, 16, 32].”
Realistic noise-tolerant randomness amplification using finite number of devices
Brandão, F. G. S. L., Ramanathan, R., Grudka, A., Horodecki, K., Horodecki, M., Horodecki, P., Szarek, T., & Wojewódka–Ściążko, H. (2016) · Nature Communications · doi:10.1038/ncomms11345 · Cited by 69
“It has a similar flavour to previous results [19–21] obtained in the context of the related problem of randomness expansion.”
InfluentialExperimental Low-Latency Device-Independent Quantum Randomness
Zhang, Y., Shalm, L. K., Bienfang, J. C., Stevens, M. J., Mazurek, M. D., Nam, S. W., Abellán, C., Amaya, W., Mitchell, M. W., Fu, H., et al. (2020) · Physical Review Letters · doi:10.1103/physrevlett.124.010505 · Cited by 69
“[2] S.”
BackgroundA Cryptographic Test of Quantumness and Certifiable Randomness from a Single Quantum Device
Brakerski, Z., Christiano, P. F., Mahadev, U., Vazirani, U. V., & Vidick, T. (2021) · Journal of the ACM · doi:10.1145/3441309 · Cited by 83
“There has been considerable research into certifiable quantum random number generation [Col06], [PAM + 10], [FGS11], [PM11], [VV11], [MS16], [BKG + 18].”
BackgroundLin, P., Rosset, D., Zhang, Y., Bancal, J.-D., & Liang, Y.-C. (2018) · Physical Review A · doi:10.1103/physreva.97.032309 · Cited by 61
Security of Device-Independent Quantum Key Distribution in the Bounded-Quantum-Storage Model
Pironio, S., Masanes, L., Leverrier, A., & Acín, A. (2013) · Physical Review X · doi:10.1103/physrevx.3.031007 · Cited by 55
Self-citationSemi-device-independent framework based on natural physical assumptions
van Himbeeck, T., Woodhead, E., Cerf, N. J., García−Patrón, R., & Pironio, S. (2017) · Dépôt institutionnel de l'Université libre de Bruxelles (Université Libre de Bruxelles) · Cited by 54
Self-citationDevice-Independent Randomness Generation in the Presence of Weak Cross-Talk
Silman, J., Pironio, S., & Massar, S. (2013) · Physical Review Letters · doi:10.1103/physrevlett.110.100504 · Cited by 49
Self-citationA Framework for Quantum-Secure Device-Independent Randomness Expansion
Brown, P. J., Ragy, S., & Colbeck, R. (2019) · IEEE Transactions on Information Theory · doi:10.1109/tit.2019.2960252 · Cited by 53
Efficient randomness certification by quantum probability estimation
Zhang, Y., Fu, H., & Knill, E. H. (2020) · Physical Review Research · doi:10.1103/physrevresearch.2.013016 · Cited by 49
“The value for the error and that the certification did not take into account the success probability or quantum side information were clarified in subsequent papers [6, 7].”
BackgroundCertified Quantum Random Numbers from Untrusted Light
Drahi, D., Walk, N., Hoban, M. J., Fedorov, A. K., Shakhovoy, R., Feimov, A., Kurochkin, Y., Kolthammer, W. S., Nunn, J., & Barrett, J. (2020) · Goldsmiths (University of London) · doi:10.17169/refubium-29280 · Cited by 46
Vazirani, U. V., & Vidick, T. (2012) · Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences · doi:10.1098/rsta.2011.0336 · Cited by 81
Certified Quantum Random Numbers from Untrusted Light
Drahi, D., Walk, N., Hoban, M. J., Fedorov, A. K., Shakhovoy, R., Feimov, A., Kurochkin, Y., Kolthammer, W. S., Nunn, J., Barrett, J., et al. (2020) · Physical Review X · doi:10.1103/physrevx.10.041048 · Cited by 63
“It should be noted that this extractor still requires a perfectly random seed of length d and thus any protocol that makes use of leftover hashing can technically only be regarded as a randomness expansion protocol [52, 53].”
BackgroundQuantum to Classical Randomness Extractors
Berta, M., Fawzi, O., & Wehner, S. (2014) · IEEE Transactions on Information Theory · doi:10.1109/tit.2013.2291780 · Cited by 45
“This is exactly the setting of the recent proofs [36], [68], [87] of [22], [67] where such violations were used to certify the creation of random bits using quantum measurements as a black box.”
BackgroundPhysical security in the post-quantum era
Chowdhury, S., Covic, A., Acharya, R. Y., Dupee, S., Ganji, F., & Forte, D. J. (2021) · Journal of Cryptographic Engineering · doi:10.1007/s13389-021-00255-w · Cited by 60
“expands the randomness. A similar method has been devised in [Vazirani and Vidick 2012], where the length of the seed could be reduced to (log 2 n)3. It has been shown that the results presented in [Pironio and Massar 2013; Vazirani and Vidick 2012] can be extended to a more general 6Shot noise can be produced from qu…”
BackgroundMethod(Nearly) optimal P values for all Bell inequalities
Elkouss, D., & Wehner, S. (2016) · npj Quantum Information · doi:10.1038/npjqi.2016.26 · Cited by 37
Device-independent randomness generation from several Bell estimators
Nieto-Silleras, O., Bamps, C., Silman, J., & Pironio, S. (2018) · New Journal of Physics · doi:10.1088/1367-2630/aaaa06 · Cited by 35
“We follow the approach introduced in [3, 14].”
BackgroundMethodResultInfluentialSelf-citationProvably-secure quantum randomness expansion with uncharacterised homodyne detection
Wang, C., Primaatmaja, I. W., Ng, H. J., Haw, J. Y., Ho, R., Zhang, J., Zhang, G., & Lim, C. C. W. (2023) · Nature Communications · doi:10.1038/s41467-022-35556-z · Cited by 37
Algorithmic Pseudorandomness in Quantum Setups
Bendersky, A., de la Torre, G., Senno, G., Figueira, S. D., & Acín, A. (2016) · Physical Review Letters · doi:10.1103/physrevlett.116.230402 · Cited by 32
“For the loophole to apply, the boxes should communicate between rounds and adapt accordingly, as, for instance, studied in the context of the memory loophole [15,16].”
BackgroundAgresti, I., Poderini, D., Guerini, L., Mancusi, M., Carvacho, G., Aolita, L., Cavalcanti, D., Chaves, R., & Sciarrino, F. (2020) · Communications Physics · doi:10.1038/s42005-020-0375-6 · Cited by 34
Certifying quantum randomness by probability estimation
Zhang, Y., Knill, E. H., & Bierhorst, P. (2018) · Physical Review A · doi:10.1103/physreva.98.040304 · Cited by 34
Randomness versus nonlocality in the Mermin-Bell experiment with three parties
Woodhead, E., Bourdoncle, B., & Acín, A. (2018) · Quantum · doi:10.22331/q-2018-08-17-82 · Cited by 31
Device-independent bit commitment based on the CHSH inequality
Aharon, N., Massar, S., Pironio, S., & Silman, J. (2016) · New Journal of Physics · doi:10.1088/1367-2630/18/2/025014 · Cited by 31
Self-citationEntropy Bounds for Multiparty Device-Independent Cryptography
Grasselli, F., Murta, G., Kampermann, H., & Bruß, D. (2021) · PRX Quantum · doi:10.1103/prxquantum.2.010308 · Cited by 26
“[48] S.”
MethodPan, A. K. (2021) · Physical Review A · doi:10.1103/physreva.104.022212 · Cited by 28
Semi-definite programming and quantum information
Mironowicz, P. (2024) · Journal of Physics A Mathematical and Theoretical · doi:10.1088/1751-8121/ad2b85 · Cited by 30
Generation of quantum randomness by probability estimation with classical side information
Knill, E., Zhang, Y., & Bierhorst, P. (2020) · Physical Review Research · doi:10.1103/physrevresearch.2.033465 · Cited by 27
“[11, 12]) with pairs of ions located in separate traps.”
BackgroundRandomness in post-selected events
Le, T. P., de la Torre, G., Bancal, J.-D., Pironio, S., & Scarani, V. (2016) · New Journal of Physics · doi:10.1088/1367-2630/18/3/035007 · Cited by 26
Self-citationAn optical chip for self-testing quantum random number generation
Leone, N., Rusca, D., Azzini, S., Fontana, G., Acerbi, F., Gola, A. G., Tontini, A., Massari, N., Zbinden, H., & Pavesi, L. (2020) · APL Photonics · doi:10.1063/5.0022526 · Cited by 21
Randomness Expansion Secured by Quantum Contextuality
Um, M., Zhao, Q., Zhang, J., Wang, P., Wang, Y., Qiao, M., Zhou, H., Ma, X., & Kim, K. (2020) · Physical Review Applied · doi:10.1103/physrevapplied.13.034077 · Cited by 22
Pironio, S., Masanes, L., Leverrier, A., & Acín, A. (2012) · arXiv (Cornell University) · doi:10.48550/arxiv.1211.1402 · Cited by 27
Self-citationSemi-device-independent random number generation with flexible assumptions
Pivoluska, M., Plesch, M., Farkas, M., Ružičková, N., Flegel, C., Valencia, N. H., McCutcheon, W., Malik, M., & Aguilar, E. A. (2021) · npj Quantum Information · doi:10.1038/s41534-021-00387-1 · Cited by 24
Quantum Sampling for Finite Key Rates in High Dimensional Quantum Cryptography
Yao, K., Krawec, W. O., & Zhu, J. (2022) · IEEE Transactions on Information Theory · doi:10.1109/tit.2022.3141874 · Cited by 19
Robust amplification of Santha-Vazirani sources with three devices
Mironowicz, P., Gallego, R., & Pawłowski, M. (2015) · Physical Review A · doi:10.1103/physreva.91.032317 · Cited by 26
Quantum Random Number Generation
Kollmitzer, C., Schauer, S., Raß, S., & Rainer, B. (2020) · Quantum science and technology · doi:10.1007/978-3-319-72596-3 · Cited by 22
Fault-tolerant quantum random-number generator certified by Majorana fermions
Deng, D.-L., & Duan, L.-M. (2013) · Physical Review A · doi:10.1103/physreva.88.012323 · Cited by 17
Certified Quantum Random-Number Generator Based on Single-Photon Entanglement
Leone, N., Azzini, S., Mazzucchi, S., Moretti, V., & Pavesi, L. (2022) · Physical Review Applied · doi:10.1103/physrevapplied.17.034011 · Cited by 21
“[12] S.”
BackgroundBoosting device-independent cryptography with tripartite nonlocality
Grasselli, F., Murta, G., Kampermann, H., & Bruß, D. (2023) · Quantum · doi:10.22331/q-2023-04-13-980 · Cited by 17
Quantum randomness protected against detection loophole attacks
Mironowicz, P., Cañas, G., Cariñe, J., Gómez, E. S., Barra, J. F., Cabello, A., XAVIER, G. B., Lima, G., & Pawłowski, M. (2021) · Quantum Information Processing · doi:10.1007/s11128-020-02948-3 · Cited by 21
Chowdhury, S., Covic, A., Acharya, R. Y., Dupee, S., Ganji, F., & Forte, D. J. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2005.04344 · Cited by 15
“expands the randomness. A similar method has been devised in [Vazirani and Vidick 2012], where the length of the seed could be reduced to (log 2 n)3. It has been shown that the results presented in [Pironio and Massar 2013; Vazirani and Vidick 2012] can be extended to a more general 6Shot noise can be produced from qu…”
BackgroundMethodRobust Randomness Amplifiers: Upper and Lower Bounds
Coudron, M., Vidick, T., & Yuen, H. (2013) · Lecture notes in computer science · doi:10.1007/978-3-642-40328-6_33 · Cited by 14
Certified Randomness in Tight Space
Fyrillas, A., Bourdoncle, B., Maïnos, A., Emeriau, P.-E., Start, K., Margaria, N., Morassi, M., Lemaître, A., Sagnes, I., Stepanov, P., et al. (2024) · PRX Quantum · doi:10.1103/prxquantum.5.020348 · Cited by 18
“In that Appendix, we also describe how our idea can be combined with a protocol for randomness certification against classical side information such as the ones introduced in [31–33], because in the trusted provider scenario this restriction on the side information is reasonable.”
MethodQuantum Probability Estimation for Randomness with Quantum Side InformationPreprint
Knill, E. H., Zhang, Y., & Fu, H. (2018) · arXiv (Cornell University) · doi:10.48550/arxiv.1806.04553 · Cited by 26
“That the claim did not take into account probability of success or quantum side information was clarified in subsequent papers [41, 42].”
BackgroundQuantum sampling and entropic uncertainty
Krawec, W. O. (2019) · Quantum Information Processing · doi:10.1007/s11128-019-2481-5 · Cited by 13
A framework for quantum-secure device-independent randomness expansion
Brown, P. J., Ragy, S., & Colbeck, R. (2020) · White Rose Research Online (University of Leeds, The University of Sheffield, University of York) · Cited by 13
Kavulich, J. T., Deren, B. P. V., & Schlosshauer, M. (2020) · Physics Letters A · doi:10.1016/j.physleta.2020.127032 · Cited by 12
Single-System-Based Generation of Certified Randomness Using Leggett-Garg Inequality
Nath, P. P., Saha, D., Home, D., & Sinha, U. (2024) · Physical Review Letters · doi:10.1103/physrevlett.133.020802 · Cited by 12
Mazzucchi, S., Leone, N., Azzini, S., Pavesi, L., & Moretti, V. (2021) · Physical Review A · doi:10.1103/physreva.104.022416 · Cited by 12
Security of Semi-Device-Independent Random Number Expansion Protocols
Li, D., Wen, Q., Wang, Y., Zhou, Y., & Gao, F. (2015) · Scientific Reports · doi:10.1038/srep15543 · Cited by 9
“Note that we assume that the state preparation system are not entangled with the measurement system or any other party in the following calculation of the amount of generated randomness, which is similar to that in previous work 7,8 .”
BackgroundMethodResultInfluentialRobust Device Independent Randomness AmplificationPreprint
Ramanathan, R., Brandão, F. G. S. L., Grudka, A., Horodecki, K., Horodecki, M., & Horodecki, P. (2013) · arXiv (Cornell University) · doi:10.48550/arxiv.1308.4635 · Cited by 14
“It has a similar flavour to previous results [19–21] obtained in the context of the related problem of randomness expansion.”
InfluentialHu, D., Li, M., Guo, F.-Z., Wang, Y., Dong, H., & Gao, F. (2025) · EPJ Quantum Technology · doi:10.1140/epjqt/s40507-025-00325-6 · Cited by 8
“A violation of Bell inequality suggests the existence of quantum nonlocality, a fundamental resource for device-independent information processing tasks such as quantum key distribution [7–10] and quantunm random number generation [11–17].”
Bounds on semi-device-independent quantum random-number expansion capabilities
Mannalath, V., & Pathak, A. (2022) · Physical Review A · doi:10.1103/physreva.105.022435 · Cited by 8
Tilted Hardy paradoxes for device-independent randomness extraction
Zhao, S., Ramanathan, R., Liu, Y., & Horodecki, P. (2023) · Quantum · doi:10.22331/q-2023-09-15-1114 · Cited by 10
“Apart from being of fundamental interest, this phenomenon of Bell non-locality has led to the powerful idea of device-independent (DI) quantum key distribution [3, 4], randomness generation [5, 6] and certification of quantum systems [7, 8].”
BackgroundLi, D., Zhou, Y., Gao, F., Li, X., & Wen, Q. (2016) · Physical Review A · doi:10.1103/physreva.94.012104 · Cited by 8
Correlations and randomness generation based on energy constraintsPreprint
van Himbeeck, T., & Pironio, S. (2019) · arXiv (Cornell University) · doi:10.48550/arxiv.1905.09117 · Cited by 33
Self-citationSeguinard, A., Piveteau, A., Mironowicz, P., & Bourennane, M. (2023) · New Journal of Physics · doi:10.1088/1367-2630/ad05a6 · Cited by 9
“The first implementations of the concept [5, 10, 11] were able to provide a guarantee that the numbers obtained from a quantum device are not possible to be pre-determined or predicted with certainty.”
BackgroundClauser-Horne Bell test with imperfect random inputs
Yuan, X., Zhao, Q., & Ma, X. (2015) · Physical Review A · doi:10.1103/physreva.92.022107 · Cited by 8
Higher amounts of loophole-free Bell violation using a heralded entangled source
Zhao, S., Cao, W.-F., Zhen, Y.-Z., Chen, C., Li, L., Liu, N.-L., Xu, F., & Chen, K. (2019) · New Journal of Physics · doi:10.1088/1367-2630/ab4538 · Cited by 6
Generalized Rényi Entropy Accumulation Theorem and Generalized Quantum Probability Estimation
Arqand, A., Hahn, T. A., & Tan, E. Y.-Z. (2025) · Physical Review X · doi:10.1103/pgrn-mz9j · Cited by 25
Entropic Uncertainty for Biased Measurements
Krawec, W. O. (2023) · IEEE International Conference on Quantum Computing and Engineering (QCE) · doi:10.1109/qce57702.2023.00138 · Cited by 5
“SI security models offer a nice “middle ground” between fully trusted devices (which have weak security guarantees) and fully device independent models, which offer strong security guarantees [22], [23] but have low bit generation rates with today’s technology [24], [25].”
BackgroundNecessary and Sufficient Condition for Randomness Certification from Incompatibility
Li, Y., Xiang, Y., Tura, J., & He, Q. (2025) · Physical Review Letters · doi:10.1103/zp36-2xg4 · Cited by 7
Reduced gap between observed and certified randomness for semi-device-independent protocols
Wang, Y., Qin, S., Wu, X., Gao, F., & Wen, Q. (2015) · Physical Review A · doi:10.1103/physreva.92.052321 · Cited by 4
Semi-Source Independent Quantum Walk Random Number Generation
Bae, M., & Krawec, W. O. (2021) · IEEE Information Theory Workshop (ITW) · doi:10.1109/itw48936.2021.9611496 · Cited by 4
Effects of measurement dependence on tilted CHSH Bell tests
Huang, X., Li, D., & Zhang, P. (2018) · Quantum Information Processing · doi:10.1007/s11128-018-2060-1 · Cited by 4
Universal security for randomness expansionPreprint
Miller, C. A., & Shi, Y. (2014) · arXiv (Cornell University) · Cited by 6
Efficiently Verifiable Quantum Advantage on Near-Term Analog Quantum Simulators
Liu, Z., Devulapalli, D., Hangleiter, D., Liu, Y.-K., Kollár, A. J., Gorshkov, A. V., & Childs, A. M. (2025) · PRX Quantum · doi:10.1103/prxquantum.6.010341 · Cited by 4
Secure Random Number Generation in Continuous Variable Systems
Haw, J. Y., Assad, S. M., & Lam, P. K. (2020) · Quantum science and technology · doi:10.1007/978-3-319-72596-3_6 · Cited by 4
Device-Independent Quantum Cryptography
Grasselli, F. (2021) · Quantum science and technology · doi:10.1007/978-3-030-64360-7_7 · Cited by 3
Metadata
- Authors
- Pironio, S., & Massar, S.
- Affiliations
- Université Libre de Bruxelles (BE)
- Venue
- Physical Review A
- Publisher
- American Physical Society (APS)
- Published
- 2013-01-30
- Volume
- 87
- Issue
- 1
- ISSN
- 0556-2791 (ISSN-L)
- 1050-2947
- 1094-1622
- OpenAlex
- W1988732892
- Cited by
- 169
- References
- 48
- Citations by year
- 2026: 6 · 2025: 9 · 2024: 6 · 2023: 10 · 2022: 4 · 2021: 12 · 2020: 18 · 2019: 38 · 2018: 12 · 2017: 7 · 2016: 9 · 2015: 10 · 2014: 12 · 2013: 13 · 2012: 3
- FWCI
- 16.19
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Physical Review A (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- Dépôt institutionnel de l'Université libre de Bruxelles (Université Libre de Bruxelles) (submittedVersion, other-oa)
- arXiv (Cornell University)
- Funders
- Agence Nationale de la Recherche (ANR-11-CHRI-0007)
- Topics
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Information and Cryptography
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
- Physical Sciences › Computer Science › Computer Networks and Communications › Cryptography and Data Security
- Keywords
- device-independent randomness generation, private randomness, Bell inequality violation, randomness extraction
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