A largely self-contained and complete security proof for quantum key  distribution

Marco Tomamichel; Anthony Leverrier

doi:10.22331/q-2017-07-14-14

A largely self-contained and complete security proof for quantum key distribution

Marco Tomamichel¹ and Anthony Leverrier²

¹Centre for Quantum Software and Information, University of Technology Sydney, Australia
²Inria Paris, France

Published:	2017-07-14, volume 1, page 14
Eprint:	arXiv:1506.08458v3
Doi:	https://doi.org/10.22331/q-2017-07-14-14
Citation:	Quantum 1, 14 (2017).

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Abstract

In this work we present a security analysis for quantum key distribution, establishing a rigorous tradeoff between various protocol and security parameters for a class of entanglement-based and prepare-and-measure protocols. The goal of this paper is twofold: 1) to review and clarify the stateof-the-art security analysis based on entropic uncertainty relations, and 2) to provide an accessible resource for researchers interested in a security analysis of quantum cryptographic protocols that takes into account finite resource effects. For this purpose we collect and clarify several arguments spread in the literature on the subject with the goal of making this treatment largely self-contained.
More precisely, we focus on a class of prepare-and-measure protocols based on the Bennett-Brassard (BB84) protocol as well as a class of entanglement-based protocols similar to the Bennett-Brassard-Mermin (BBM92) protocol. We carefully formalize the different steps in these protocols, including randomization, measurement, parameter estimation, error correction and privacy amplification, allowing us to be mathematically precise throughout the security analysis. We start from an operational definition of what it means for a quantum key distribution protocol to be secure and derive simple conditions that serve as sufficient condition for secrecy and correctness. We then derive and eventually discuss tradeoff relations between the block length of the classical computation, the noise tolerance, the secret key length and the security parameters for our protocols. Our results significantly improve upon previously reported tradeoffs.

Featured image: Maximal secret key rate $l/m$ as a function of the block length $m$ for different error thresholds $\delta$, optimized over all protocols.

► BibTeX data

@article{Tomamichel2017largelyself,
  doi = {10.22331/q-2017-07-14-14},
  url = {https://doi.org/10.22331/q-2017-07-14-14},
  title = {A largely self-contained and complete security proof for quantum key  distribution},
  author = {Tomamichel, Marco and Leverrier, Anthony},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {1},
  pages = {14},
  month = jul,
  year = {2017}
}

► References

[1] C.H. Bennett and G. Brassard. Quantum Cryptography: Public Key Distribution and Coin Tossing. In Proc. IEEE International Conference on Computers, Systems and Signal Processing 1984, volume 1, pages 175–179, Bangalore, 1984.

[2] A.K. Ekert. Quantum Cryptography Based on Bell's Theorem. Physical Review Letters, 67 (6): 661–663, 1991. 10.1103/PhysRevLett.67.661.
https://doi.org/10.1103/PhysRevLett.67.661

[3] C. Bennett, G. Brassard, and N. Mermin. Quantum Cryptography Without Bell's Theorem. Physical Review Letters, 68 (5): 557–559, 1992. 10.1103/PhysRevLett.68.557.
https://doi.org/10.1103/PhysRevLett.68.557

[4] H.-K. Lo and H.F. Chau. Unconditional Security of Quantum Key Distribution over Arbitrarily Long Distances. Science, 283 (5410): 2050–2056, 1999. 10.1126/science.283.5410.2050.
https://doi.org/10.1126/science.283.5410.2050

[5] P.W. Shor and J. Preskill. Simple Proof of Security of the BB84 Quantum Key Distribution Protocol. Physical Review Letters, 85 (2): 441–444, 2000. 10.1103/PhysRevLett.85.441.
https://doi.org/10.1103/PhysRevLett.85.441

[6] D. Mayers. Unconditional Security in Quantum Cryptography. Journal of the ACM, 48 (3): 351–406, 2001. 10.1145/382780.382781.
https://doi.org/10.1145/382780.382781

[7] M. Koashi. Unconditional Security of Quantum Key Distribution and the Uncertainty Principle. Journal of Physics: Conference Series, 36 (1): 98–102, 2006. 10.1088/1742-6596/36/1/016.
https://doi.org/10.1088/1742-6596/36/1/016

[8] H. Maassen and J. Uffink. Generalized Entropic Uncertainty Relations. Physical Review Letters, 60 (12): 1103–1106, 1988. 10.1103/PhysRevLett.60.1103.
https://doi.org/10.1103/PhysRevLett.60.1103

[9] W. Heisenberg. Über den Anschaulichen Inhalt der Quantentheoretischen Kinematik und Mechanik. Zeitschrift für Physik, 43 (3-4): 172–198, mar 1927.

[10] R. Renner. Security of Quantum Key Distribution. PhD thesis, ETH Zurich, 2005. URL http://arxiv.org/abs/quant-ph/0512258.
arXiv:quant-ph/0512258

[11] L.C. Comandar, M. Lucamarini, B. Fröhlich, J.F. Dynes, A.W. Sharpe, S.W.-B. Tam, Z.L. Yuan, R.V. Penty, and A.J. Shields. Quantum key distribution without detector vulnerabilities using optically seeded lasers. Nature Photonics, 10 (5): 312–315, 2016. 10.1038/nphoton.2016.50.
https://doi.org/10.1038/nphoton.2016.50

[12] P. Jouguet, S. Kunz-Jacques, A. Leverrier, P. Grangier, and E. Diamanti. Experimental demonstration of long-distance continuous-variable quantum key distribution. Nature Photonics, 7 (5): 378–381, 2013. 10.1038/nphoton.2013.63.
https://doi.org/10.1038/nphoton.2013.63

[13] M. Tomamichel, C.C.W. Lim, N. Gisin, and R. Renner. Tight Finite-Key Analysis for Quantum Cryptography. Nature Communications, 3: 634, 2012. 10.1038/ncomms1631.
https://doi.org/10.1038/ncomms1631

[14] M. Hayashi and T. Tsurumaru. Concise and Tight Security Analysis of the Bennett-Brassard 1984 Protocol with Finite Key Lengths. New Journal of Physics, 14 (9): 093014, 2012. 10.1088/1367-2630/14/9/093014.
https://doi.org/10.1088/1367-2630/14/9/093014

[15] V. Scarani and R. Renner. Quantum Cryptography with Finite Resources: Unconditional Security Bound for Discrete-Variable Protocols with One-Way Postprocessing. Physical Review Letters, 100 (20), 2008. 10.1103/PhysRevLett.100.200501.
https://doi.org/10.1103/PhysRevLett.100.200501

[16] R. Renner. Symmetry of Large Physical Systems Implies Independence of Subsystems. Nature Physics, 3 (9): 645–649, 2007. 10.1038/nphys684.
https://doi.org/10.1038/nphys684

[17] M. Christandl, R. König, and R. Renner. Postselection Technique for Quantum Channels with Applications to Quantum Cryptography. Physical Review Letters, 102 (2), 2009. 10.1103/PhysRevLett.102.020504.
https://doi.org/10.1103/PhysRevLett.102.020504

[18] L. Sheridan, T.P. Le, and V. Scarani. Finite-Key Security Against Coherent Attacks in Quantum Key Distribution. New Journal of Physics, 12: 123019, 2010.

[19] C. Pfister, N. Lütkenhaus, S. Wehner, and P.J. Coles. Sifting Attacks in Finite-Size Quantum Key Distribution. New Journal of Physics, 18 (5): 053001, 2016. 10.1088/1367-2630/18/5/053001.
https://doi.org/10.1088/1367-2630/18/5/053001

[20] M. Tomamichel, S. Fehr, J. Kaniewski, and S. Wehner. A Monogamy-of-Entanglement Game with Applications to Device-Independent Quantum Cryptography. New Journal of Physics, 15 (10): 103002, 2013. 10.1088/1367-2630/15/10/103002.
https://doi.org/10.1088/1367-2630/15/10/103002

[21] M. Tomamichel and R. Renner. Uncertainty Relation for Smooth Entropies. Physical Review Letters, 106 (11): 110506, 2011. 10.1103/PhysRevLett.106.110506.
https://doi.org/10.1103/PhysRevLett.106.110506

[22] M. Tomamichel. Quantum Information Processing with Finite Resources — Mathematical Foundations, volume 5 of SpringerBriefs in Mathematical Physics. Springer International Publishing, 2016. ISBN 978-3-319-21890-8. 10.1007/978-3-319-21891-5.
https://doi.org/10.1007/978-3-319-21891-5

[23] C.W. Helstrom. Quantum Detection and Estimation Theory. Academic Press, New York, NY, 1976.

[24] M. Tomamichel, R. Colbeck, and R. Renner. Duality Between Smooth Min- and Max-Entropies. IEEE Transactions on Information Theory, 56 (9): 4674–4681, 2010. 10.1109/TIT.2010.2054130.
https://doi.org/10.1109/TIT.2010.2054130

[25] J.L. Carter and M.N. Wegman. Universal Classes of Hash Functions. Journal of Computer and System Sciences, 18 (2): 143–154, 1979. 10.1016/0022-0000(79)90044-8.
https://doi.org/10.1016/0022-0000(79)90044-8

[26] M.N. Wegman and J.L. Carter. New Hash Functions and their Use in Authentication and Set Equality. Journal of Computer and System Sciences, 22 (3): 265–279, 1981. 10.1016/0022-0000(81)90033-7.
https://doi.org/10.1016/0022-0000(81)90033-7

[27] A. Rényi. On Measures of Information and Entropy. In Proc. 4th Berkeley Symposium on Mathematical Statistics and Probability, volume 1, pages 547–561, Berkeley, California, USA, 1961. University of California Press.

[28] R. König, R. Renner, and C. Schaffner. The Operational Meaning of Min- and Max-Entropy. IEEE Transactions on Information Theory, 55 (9): 4337–4347, 2009. 10.1109/TIT.2009.2025545.
https://doi.org/10.1109/TIT.2009.2025545

[29] S. Winkler, M. Tomamichel, S. Hengl, and R. Renner. Impossibility of Growing Quantum Bit Commitments. Physical Review Letters, 107 (9): 090502, 2011. ISSN 0031-9007. 10.1103/PhysRevLett.107.090502.
https://doi.org/10.1103/PhysRevLett.107.090502

[30] H.-K. Lo, H.F. Chau, and M. Ardehali. Efficient Quantum Key Distribution Scheme and a Proof of Its Unconditional Security. Journal of Cryptology, 18(2):133–165, 2004 10.1007/s00145-004-0142-y.
https://doi.org/10.1007/s00145-004-0142-y

[31] D. Frauchiger, R. Renner, and M. Troyer. True randomness from realistic quantum devices, 2013. URL http://arxiv.org/abs/1311.4547.
arXiv:1311.4547

[32] C. Portmann and R. Renner. Cryptographic Security of Quantum Key Distribution, 2014. URL http://arxiv.org/abs/1409.3525.
arXiv:1409.3525

[33] L. Lydersen, C. Wiechers, C. Wittmann, D. Elser, J. Skaar, and V. Makarov. Hacking Commercial Quantum Cryptography Systems by Tailored Bright Illumination. Nature Photonics, 4 (10): 686–689, 2010. 10.1038/nphoton.2010.214.
https://doi.org/10.1038/nphoton.2010.214

[34] M. Tomamichel and Esther Hänggi. The Link Between Entropic Uncertainty and Nonlocality. Journal of Physics A: Mathematical and Theoretical, 46 (5): 055301, 2013. 10.1088/1751-8113/46/5/055301.
https://doi.org/10.1088/1751-8113/46/5/055301

[35] C.C.W. Lim, C. Portmann, M. Tomamichel, R. Renner, and Nicolas Gisin. Device-Independent Quantum Key Distribution with Local Bell Test. Physical Review X, 3 (3): 031006, 2013. 10.1103/PhysRevX.3.031006.
https://doi.org/10.1103/PhysRevX.3.031006

[36] I. Devetak and A. Winter. Distillation of Secret Key and Entanglement from Quantum States. Proceedings of the Royal Society A, 461 (2053): 207–235, 2005. 10.1098/rspa.2004.1372.
https://doi.org/10.1098/rspa.2004.1372

[37] D. Elkouss, A. Leverrier, R. Alleaume, and J.J. Boutros. Efficient Reconciliation Protocol for Discrete-Variable Quantum Key Distribution. In Proc. IEEE ISIT 2009, pages 1879–1883, 2009. 10.1109/ISIT.2009.5205475.
https://doi.org/10.1109/ISIT.2009.5205475

[38] M. Tomamichel, J. Martinez-Mateo, C. Pacher, and D. Elkouss. Fundamental Finite Key Limits for Information Reconciliation in Quantum Key Distribution, 2014. URL http://arxiv.org/abs/1401.5194.
arXiv:1401.5194

[39] M. Tomamichel. A Framework for Non-Asymptotic Quantum Information Theory. PhD thesis, ETH Zurich, 2012. URL http://arxiv.org/abs/1203.2142.
arXiv:1203.2142

[40] R.J. Serfling. Probability Inequalities for the Sum in Sampling without Replacement. Annals of Statistics, 2 (1): 39–48, 1974.

[41] J.H. van Lint. Introduction to Coding Theory. Graduate Texts in Mathematics. Springer, third edition, 1999.

[42] H.-K. Lo, X. Ma, and K. Chen. Decoy State Quantum Key Distribution. Physical Review Letters, 94 (23), 2005. 10.1103/PhysRevLett.94.230504.
https://doi.org/10.1103/PhysRevLett.94.230504

[43] J. Hasegawa, M. Hayashi, T. Hiroshima, and A. Tomita. Security analysis of decoy state quantum key distribution incorporating finite statistics, 2007. URL http://arxiv.org/abs/0707.3541.
arXiv:0707.3541

[44] C.C.W. Lim, M. Curty, N. Walenta, F. Xu, and H. Zbinden. Concise security bounds for practical decoy-state quantum key distribution. Physical Review A, 89 (2): 022307, 2014. 10.1103/PhysRevA.89.022307.
https://doi.org/10.1103/PhysRevA.89.022307

[45] D. Bruss. Optimal Eavesdropping in Quantum Cryptography with Six States. Physical Review Letters, 81 (14): 3018–3021, 1998. 10.1103/PhysRevLett.81.3018.
https://doi.org/10.1103/PhysRevLett.81.3018

[46] F. Dupuis, O. Fawzi, and R. Renner. Entropy accumulation, 2016. URL http://arxiv.org/abs/1607.01796.
arXiv:1607.01796

[47] R. Arnon-Friedman, R. Renner, and T. Vidick. Simple and tight device-independent security proofs, 2016. URL http://arxiv.org/abs/1607.01797.
arXiv:1607.01797

[48] R. Bhatia. Matrix Analysis. Graduate Texts in Mathematics. Springer, 1997. ISBN 0-387-94846-5.

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[1] Masahito Hayashi, "Quantum-Inspired Secure Wireless Communication Protocol Under Spatial and Local Gaussian Noise Assumptions", IEEE Access 10, 29040 (2022).

[2] Michel Boyer, Gilles Brassard, Nicolas Godbout, Rotem Liss, and Stéphane Virally, "Simple and Rigorous Proof Method for the Security of Practical Quantum Key Distribution in the Single-Qubit Regime Using Mismatched Basis Measurements", Quantum Reports 5 1, 52 (2023).

[3] Marc Fischlin and Evangelos Gkoumas, Lecture Notes in Computer Science 16207, 264 (2026) ISBN:978-3-032-10535-6.

[4] Keegan Yao, Walter O. Krawec, and Jiadong Zhu, "Quantum Sampling for Finite Key Rates in High Dimensional Quantum Cryptography", IEEE Transactions on Information Theory 68 5, 3144 (2022).

[5] Amir Arqand, Thomas A. Hahn, and Ernest Y.-Z. Tan, "Generalized Rényi Entropy Accumulation Theorem and Generalized Quantum Probability Estimation", Physical Review X 15 4, 041013 (2025).

[6] Carles Roch i Carceller, Hanwool Lee, Jonatan Bohr Brask, Kieran Flatt, and Joonwoo Bae, "Sequential semi-device-independent quantum randomness certification", Physical Review Research 8 2, 023186 (2026).

[7] M. K. Bochkov and A. S. Trushechkin, "Security of quantum key distribution with detection-efficiency mismatch in the single-photon case: Tight bounds", Physical Review A 99 3, 032308 (2019).

[8] Mustafa Gündoǧan, Jasminder S. Sidhu, Markus Krutzik, and Daniel K. L. Oi, "Time-delayed single satellite quantum repeater node for global quantum communications", Optica Quantum 2 3, 140 (2024).

[9] Anqi Huang, Akihiro Mizutani, Hoi-Kwong Lo, Vadim Makarov, and Kiyoshi Tamaki, "Characterization of State-Preparation Uncertainty in Quantum Key Distribution", Physical Review Applied 19 1, 014048 (2023).

[10] Dimiter Ostrev, "QKD parameter estimation by two-universal hashing", Quantum 7, 894 (2023).

[11] Víctor Zapatero and Marcos Curty, "Finite-key security of passive quantum key distribution", Physical Review Applied 21 1, 014018 (2024).

[12] Yue Jiang, Rui Zhong, Hu-Lin Zhang, Zheng Wang, Yi-Fei Liu, Ren-Hao Fan, Dong-Xiang Qi, Wen-Jie Tang, Ziyu Wang, Ruwen Peng, and Mu Wang, "A Multichannel Metasurface for Multiprotocol Quantum Key Distributions", Nano Letters 25 18, 7442 (2025).

[13] Anurag Anshu, Mario Berta, Rahul Jain, and Marco Tomamichel, "A minimax approach to one-shot entropy inequalities", Journal of Mathematical Physics 60 12, 122201 (2019).

[14] Yanbao Zhang, Akshay Seshadri, and Emanuel Knill, Optica Quantum 2.0 Conference and Exhibition QTu3A.20 (2023) ISBN:978-1-957171-27-2.

[15] Devashish Tupkary and Norbert Lütkenhaus, "Using Cascade in quantum key distribution", Physical Review Applied 20 6, 064040 (2023).

[16] A. S. Trushechkin, E. O. Kiktenko, and A. K. Fedorov, "Practical issues in decoy-state quantum key distribution based on the central limit theorem", Physical Review A 96 2, 022316 (2017).

[17] Guillermo Currás-Lorenzo, Margarida Pereira, Shlok Nahar, and Devashish Tupkary, "Security of quantum key distribution with source and detector imperfections through phase-error estimation", npj Quantum Information 12 1, 129 (2026).

[18] Mugelan RK and Swetha NG, "Entropy-Driven Privacy Amplification with Adaptive Chaotic Statistical Conditioning for Quantum Key Distribution", (2026).

[19] Aman Gupta, Ravi Singh Adhikari, Anju Rani, Xiaoyu Ai, and Robert Malaney, 2026 International Conference on Quantum Communications, Networking, and Computing (QCNC) 107 (2026) ISBN:979-8-3315-6110-9.

[20] Anton Trushechkin, "Security of quantum key distribution with detection-efficiency mismatch in the multiphoton case", Quantum 6, 771 (2022).

[21] Andrej Kržič, Sakshi Sharma, Christopher Spiess, Uday Chandrashekara, Sebastian Töpfer, Gregor Sauer, Luis Javier González-Martín del Campo, Teresa Kopf, Stefan Petscharnig, Thomas Grafenauer, Roland Lieger, Bernhard Ömer, Christoph Pacher, René Berlich, Thomas Peschel, Christoph Damm, Stefan Risse, Matthias Goy, Daniel Rieländer, Andreas Tünnermann, and Fabian Steinlechner, "Towards metropolitan free-space quantum networks", npj Quantum Information 9 1, 95 (2023).

[22] Xia Zhang, Xue Wang, Haitao Ma, Ming Liu, and Junjing Xing, "Tighter sum unitary uncertainty relation", Laser Physics 34 12, 125203 (2024).

[23] Anne Broadbent and Rabib Islam, Lecture Notes in Computer Science 12552, 92 (2020) ISBN:978-3-030-64380-5.

[24] Yuki Takeuchi, Yuichiro Matsuzaki, Koichiro Miyanishi, Takanori Sugiyama, and William J. Munro, "Quantum remote sensing with asymmetric information gain", Physical Review A 99 2, 022325 (2019).

[25] Michael Epping, Hermann Kampermann, Chiara macchiavello, and Dagmar Bruß, "Multi-partite entanglement can speed up quantum key distribution in networks", New Journal of Physics 19 9, 093012 (2017).

[26] Prabhanjan Ananth, Alexander Poremba, and Vinod Vaikuntanathan, Lecture Notes in Computer Science 14372, 93 (2023) ISBN:978-3-031-48623-4.

[27] Shlok Nahar, Devashish Tupkary, and Norbert Lütkenhaus, "Imperfect detectors for adversarial tasks with applications to quantum key distribution", Quantum 10, 2044 (2026).

[28] H. F. Chau, "Security of finite-key-length measurement-device-independent quantum key distribution using an arbitrary number of decoys", Physical Review A 102 1, 012611 (2020).

[29] Anton S. Trushechkin, Evgeniy O. Kiktenko, Dmitry A. Kronberg, and Aleksey K. Fedorov, "Security of the decoy state method for quantum key distribution", Uspekhi Fizicheskih Nauk 191 01, 93 (2021).

[30] Christopher Portmann and Renato Renner, "Security in quantum cryptography", Reviews of Modern Physics 94 2, 025008 (2022).

[31] Zhongrui Huang, Hong Lai, and Linchun Wan, "An Advanced Collaborative Routing Algorithm for Optimizing Entanglement and Resource Efficiency in Quantum Networks", International Journal of Theoretical Physics 64 1, 18 (2025).

[32] Khodakhast Bibak, "Quantum key distribution using universal hash functions over finite fields", Quantum Information Processing 21 4, 121 (2022).

[33] Éloi Descamps and Damian Markham, "Noisy certification of continuous-variable graph states", Physical Review A 111 2, 022413 (2025).

[34] Rahul Jain and Srijita Kundu, "A Direct Product Theorem for Quantum Communication Complexity with Applications to Device-Independent Cryptography", SIAM Journal on Computing 54 4, 964 (2025).

[35] S. Pirandola, U. L. Andersen, L. Banchi, M. Berta, D. Bunandar, R. Colbeck, D. Englund, T. Gehring, C. Lupo, C. Ottaviani, J. L. Pereira, M. Razavi, J. Shamsul Shaari, M. Tomamichel, V. C. Usenko, G. Vallone, P. Villoresi, and P. Wallden, "Advances in quantum cryptography", Advances in Optics and Photonics 12 4, 1012 (2020).

[36] Neha Sharma and Vikas Saxena, 2022 8th International Conference on Signal Processing and Communication (ICSC) 36 (2022) ISBN:978-1-6654-5430-8.

[37] Snigdha Kashyap, Bharat Bhushan, Avinash Kumar, and Parma Nand, Studies in Big Data 108, 1 (2022) ISBN:978-981-19-0923-8.

[38] Renato Renner and Ramona Wolf, "Quantum Advantage in Cryptography", AIAA Journal 61 5, 1895 (2023).

[39] Hamza Sohail, Burhan Ul Islam Khan, Nur Fatin Liyana Mohd Rosely, Khang Wen Goh, Dwi Sudarno Putra, Abdul Raouf Khan, and Mesith Chaimanee, "High-dimensional GHZ-based multi-party quantum key agreement: rigorous security, fairness, and loss tolerance", EPJ Quantum Technology 13 1, 48 (2026).

[40] Ernest Y.-Z. Tan, Pavel Sekatski, Jean-Daniel Bancal, René Schwonnek, Renato Renner, Nicolas Sangouard, and Charles C.-W. Lim, "Improved DIQKD protocols with finite-size analysis", Quantum 6, 880 (2022).

[41] Fabio Banfi, Ueli Maurer, Christopher Portmann, and Jiamin Zhu, Lecture Notes in Computer Science 11891, 282 (2019) ISBN:978-3-030-36029-0.

[42] Khodakhast Bibak, Bruce M. Kapron, and Venkatesh Srinivasan, "Authentication of variable length messages in quantum key distribution", EPJ Quantum Technology 9 1, 8 (2022).

[43] Gong Zhang, Ignatius William Primaatmaja, Yue Chen, Si Qi Ng, Hong Jie Ng, Marco Pistoia, Xiao Gong, Koon Tong Goh, Chao Wang, and Charles Lim, "Self-Testing Quantum Randomness Expansion on an Integrated Photonic Chip", PRX Quantum 7 2, 020353 (2026).

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[45] Kiyoshi Tamaki, Hoi-Kwong Lo, Akihiro Mizutani, Go Kato, Charles Ci Wen Lim, Koji Azuma, and Marcos Curty, "Security of quantum key distribution with iterative sifting", Quantum Science and Technology 3 1, 014002 (2018).

[46] Akihiro Mizutani, Yuki Takeuchi, and Kiyoshi Tamaki, "Finite-key security analysis of differential-phase-shift quantum key distribution", Physical Review Research 5 2, 023132 (2023).

[47] James Bartusek, Dakshita Khurana, and Alexander Poremba, Lecture Notes in Computer Science 14085, 99 (2023) ISBN:978-3-031-38553-7.

[48] Evgeniy O. Kiktenko, Aleksei O. Malyshev, Maxim A. Gavreev, Anton A. Bozhedarov, Nikolay O. Pozhar, Maxim N. Anufriev, and Aleksey K. Fedorov, "Lightweight Authentication for Quantum Key Distribution", IEEE Transactions on Information Theory 66 10, 6354 (2020).

[49] Alireza Poostindouz and Reihaneh Safavi-Naini, 2021 IEEE International Symposium on Information Theory (ISIT) 1254 (2021) ISBN:978-1-5386-8209-8.

[50] Jarn de Jong, Santiago Scheiner, Naomi R. Solomons, Ziad Chaoui, Damian Markham, and Anna Pappa, "Anonymous and private parameter estimation in networks of quantum sensors", Physical Review Applied 24 5, 054053 (2025).

[51] Florian Kanitschar and Marcus Huber, "Composable finite-size security of high-dimensional quantum-key-distribution protocols", Physical Review Applied 24 5, 054028 (2025).

[52] Sumeet Khatri, "On the design and analysis of near-term quantum network protocols using Markov decision processes", AVS Quantum Science 4 3, 030501 (2022).

[53] Giovanni Chesi and Lorenzo Maccone, Reference Module in Materials Science and Materials Engineering (2026) ISBN:9780128035818.

[54] Federico Grasselli, Gláucia Murta, Jarn de Jong, Frederik Hahn, Dagmar Bruß, Hermann Kampermann, and Anna Pappa, "Secure Anonymous Conferencing in Quantum Networks", PRX Quantum 3 4, 040306 (2022).

[55] Pawel Sniatala, S.S. Iyengar, and Sanjeev Kaushik Ramani, Evolution of Smart Sensing Ecosystems with Tamper Evident Security 107 (2021) ISBN:978-3-030-77763-0.

[56] F Basso Basset, M Valeri, J Neuwirth, E Polino, M B Rota, D Poderini, C Pardo, G Rodari, E Roccia, S F Covre da Silva, G Ronco, N Spagnolo, A Rastelli, G Carvacho, F Sciarrino, and R Trotta, "Daylight entanglement-based quantum key distribution with a quantum dot source", Quantum Science and Technology 8 2, 025002 (2023).

[57] Marie Ioannou, Pavel Sekatski, Alastair A Abbott, Denis Rosset, Jean-Daniel Bancal, and Nicolas Brunner, "Receiver-device-independent quantum key distribution protocols", New Journal of Physics 24 6, 063006 (2022).

[58] A.S. Trushechkin, "On the operational meaning and practical aspects of using the security parameter in quantum key distribution", Quantum Electronics 50 5, 426 (2020).

[59] Devashish Tupkary, Shlok Nahar, Pulkit Sinha, and Norbert Lütkenhaus, "Phase error rate estimation in QKD with imperfect detectors", Quantum 9, 1937 (2025).

[60] V. E. Rodimin, E. O. Kiktenko, V. V. Usova, M. Y. Ponomarev, T. V. Kazieva, A. V. Miller, A. S. Sokolov, A. A. Kanapin, A. V. Losev, A. S. Trushechkin, M. N. Anufriev, N. O. Pozhar, V. L. Kurochkin, Y. V. Kurochkin, and A. K. Fedorov, "Modular Quantum Key Distribution Setup for Research and Development Applications", Journal of Russian Laser Research 40 3, 221 (2019).

[61] Martin Sandfuchs, Marcus Haberland, V. Vilasini, and Ramona Wolf, "Security of differential phase shift QKD from relativistic principles", Quantum 9, 1611 (2025).

[62] Meret Kristen and Jürgen Mottok, Lecture Notes in Computer Science 15655, 218 (2026) ISBN:978-3-032-01822-9.

[63] Yuki Takeuchi, Atul Mantri, Tomoyuki Morimae, Akihiro Mizutani, and Joseph F. Fitzsimons, "Resource-efficient verification of quantum computing using Serfling’s bound", npj Quantum Information 5 1, 27 (2019).

[64] Charles Ci-Wen Lim, Feihu Xu, Jian-Wei Pan, and Artur Ekert, "Security Analysis of Quantum Key Distribution with Small Block Length and Its Application to Quantum Space Communications", Physical Review Letters 126 10, 100501 (2021).

[65] Yu-Ting Lei, Xiang-Jie Li, Xing-Bo Pan, Yun-Rong Zhang, and Gui-Lu Long, "High-Dimensional and Multi-Intensity One-Photon-Interference Quantum Secure Direct Communication", Entropy 27 4, 332 (2025).

[66] G Murta, S B van Dam, J Ribeiro, R Hanson, and S Wehner, "Towards a realization of device-independent quantum key distribution", Quantum Science and Technology 4 3, 035011 (2019).

[67] Hong Jie Ng, Wen Yu Kon, Ignatius William Primaatmaja, Chao Wang, and Charles Lim, "Sampled subblock hashing for large-input randomness extraction", Physical Review Applied 21 3, 034005 (2024).

[68] Loïc Ferreira, "AKE Protocol Combining PQC and QKD", Abstract_only IACR Communications in Cryptology 2 4, cc2-3-57 (2026).

[69] Jérémy Ribeiro, Gláucia Murta, and Stephanie Wehner, "Fully device-independent conference key agreement", Physical Review A 97 2, 022307 (2018).

[70] Haoran Zhang, Wei Li, Ruihua He, Yan Zhang, Feihu Xu, and Weibo Gao, "Noise-reducing quantum key distribution", Reports on Progress in Physics 88 1, 016001 (2025).

[71] Alex B. Grilo, Lucas Hanouz, and Anne Marin, 2026 International Conference on Quantum Communications, Networking, and Computing (QCNC) 333 (2026) ISBN:979-8-3315-6110-9.

[72] Marc Fischlin, Evangelos Gkoumas, and Gonne Kretschmer, Lecture Notes in Computer Science 16491, 71 (2026) ISBN:978-3-032-22694-5.

[73] Federico Grasselli, Giovanni Chesi, Nathan Walk, Hermann Kampermann, Adam Widomski, Maciej Ogrodnik, Michał Karpiński, Chiara Macchiavello, Dagmar Bruß, and Nikolai Wyderka, "Quantum key distribution with basis-dependent detection probability", Physical Review Applied 23 4, 044011 (2025).

[74] Michal Krelina, "The Prospect of Quantum Technologies in Space for Defence and Security", Space Policy 65, 101563 (2023).

[75] Ernest Y.-Z. Tan, Charles C.-W. Lim, and Renato Renner, "Advantage Distillation for Device-Independent Quantum Key Distribution", Physical Review Letters 124 2, 020502 (2020).

[76] A Pirker, M Zwerger, V Dunjko, H J Briegel, and W Dür, "Simple proof of confidentiality for private quantum channels in noisy environments", Quantum Science and Technology 4 2, 025009 (2019).

[77] Rong Wang and H. F. Chau, "One-Shot Min-Entropy Calculation of Classical-Quantum States and Its Application to Quantum Cryptography", Physical Review Letters 135 2, 020801 (2025).

[78] Akihiro Mizutani, "Quantum key distribution with any two independent and identically distributed states", Physical Review A 102 2, 022613 (2020).

[79] Yang-Guang Shan, Zhen-Qiang Yin, Shuang Wang, Wei Chen, De-Yong He, Guang-Can Guo, and Zheng-Fu Han, "Improved finite-key analysis for side-channel-secure quantum key distribution with de Finetti reduction", Physica Scripta 100 11, 115117 (2025).

[80] Sebastian Philipp Neumann, Alexander Buchner, Lukas Bulla, Martin Bohmann, and Rupert Ursin, "Continuous entanglement distribution over a transnational 248 km fiber link", Nature Communications 13 1, 6134 (2022).

[81] Srijita Kundu and Ernest Y.-Z. Tan, "Composably secure device-independent encryption with certified deletion", Quantum 7, 1047 (2023).

[82] Ronny Mueller, Mujtaba Zahidy, Leif Katsuo Oxenløwe, Søren Forchhammer, and Davide Bacco, "On High-Dimensional Twin-Field Quantum Key Distribution", Quantum 9, 1869 (2025).

[83] Hong-Wei Li, Chen-Peng Hao, Zhi-Jiang Chen, Li Gong, Yi-Fei Lu, Yang Wang, Jia-Ji Li, Chun-Mei Zhang, Rong Wang, Zhen-Qiang Yin, and Qing-Yu Cai, "Security of quantum key distribution with virtual mutually unbiased bases", Science China Physics, Mechanics & Astronomy 67 7, 270313 (2024).

[84] Devashish Tupkary, Ernest Y.-Z. Tan, and Norbert Lütkenhaus, "Security proof for variable-length quantum key distribution", Physical Review Research 6 2, 023002 (2024).

[85] Toyohiro Tsurumaru, Akihiro Mizutani, and Toshihiko Sasaki, "Security loophole in error verification in quantum key distribution", Physical Review Research 7 4, 043245 (2025).

[86] Anne Broadbent and Peter Yuen, "Device-independent oblivious transfer from the bounded-quantum-storage-model and computational assumptions", New Journal of Physics 25 5, 053019 (2023).

[87] Arman Sykot, Mohammad Hasibur Rahman, Rifat Tasnim Anannya, M. R. C. Mahdy, and Liuguo Yin, "Combining Entanglement‐ and Non‐entanglement‐Based Quantum Key Distribution Protocol With GHZ State", Quantum Engineering 2025 1, 6079632 (2025).

[88] Eric Culf and Thomas Vidick, "A monogamy-of-entanglement game for subspace coset states", Quantum 6, 791 (2022).

[89] Ian George, Jie Lin, Thomas van Himbeeck, Kun Fang, and Norbert Lütkenhaus, "Finite-Key Analysis of Quantum Key Distribution with Characterized Devices Using Entropy Accumulation", Quantum 9, 1941 (2025).

[90] Shashank Gupta, Rajesh K Krishnan, Vijayalaxmi Mogiligidda, T Roopak, Manasa Hegde, Sairam Rajamani, and Dilip Singh, 2023 8th International Conference on Frontiers of Signal Processing (ICFSP) 63 (2023) ISBN:979-8-3503-0879-2.

[91] Vaisakh Mannalath, Víctor Zapatero, and Marcos Curty, "Sharp Finite Statistics for Quantum Key Distribution", Physical Review Letters 135 2, 020803 (2025).

[92] Sarnava Datta, Hermann Kampermann, and Dagmar Bruß, "Device-independent secret key rates via a postselected Bell inequality", Physical Review A 105 3, 032451 (2022).

[93] Giacomo Carrara, Gláucia Murta, and Federico Grasselli, "Overcoming Fundamental Bounds on Quantum Conference Key Agreement", Physical Review Applied 19 6, 064017 (2023).

[94] Jerome Wiesemann, Jan Krause, Devashish Tupkary, Norbert Lütkenhaus, Davide Rusca, and Nino Walenta, "A consolidated and accessible security proof for finite-size decoy-state quantum key distribution", Quantum 10, 2037 (2026).

[95] Marie Ioannou, Maria Ana Pereira, Davide Rusca, Fadri Grünenfelder, Alberto Boaron, Matthieu Perrenoud, Alastair A. Abbott, Pavel Sekatski, Jean-Daniel Bancal, Nicolas Maring, Hugo Zbinden, and Nicolas Brunner, "Receiver-Device-Independent Quantum Key Distribution", Quantum 6, 718 (2022).

[96] Ashutosh Marwah and Frédéric Dupuis, "Smooth Min-entropy Lower Bounds for Approximation Chains", Communications in Mathematical Physics 405 9, 211 (2024).

[97] S. Kirubakaran, R. Priyadharshini, Aswani Ashok, Deepika Rajan, and Paventhan Arumugam, Communications in Computer and Information Science 2837, 33 (2026) ISBN:978-3-032-18318-7.

[98] Mario Mastriani and Sundaraja Sitharama Iyengar, "Satellite quantum repeaters for a quantum Internet", Quantum Engineering 2 4(2020).

[99] Ignatius W. Primaatmaja, Koon Tong Goh, Ernest Y.-Z. Tan, John T.-F. Khoo, Shouvik Ghorai, and Charles C.-W. Lim, "Security of device-independent quantum key distribution protocols: a review", Quantum 7, 932 (2023).

[100] Manuel B. Santos, Ana C. Gomes, Armando N. Pinto, and Paulo Mateus, "Private Computation of Phylogenetic Trees Based on Quantum Technologies", IEEE Access 10, 38065 (2022).

[101] Patrick Andriolo, Esteban Vasquez, Elizabeth Agudelo, Max Riegler, Matej Pivoluska, and Gláucia Murta, "Quantum Key Distribution with Imperfections: Recent Advances in Security Proofs", Brazilian Journal of Physics 56 4, 157 (2026).

[102] Akihiro Mizutani, Go Kato, Koji Azuma, Marcos Curty, Rikizo Ikuta, Takashi Yamamoto, Nobuyuki Imoto, Hoi-Kwong Lo, and Kiyoshi Tamaki, "Quantum key distribution with setting-choice-independently correlated light sources", npj Quantum Information 5 1, 8 (2019).

[103] A S Trushechkin, E O Kiktenko, D A Kronberg, and A K Fedorov, "Security of the decoy state method for quantum key distribution", Physics-Uspekhi 64 1, 88 (2021).

[104] Carlos Pascual-García, Stefan Bäuml, Mateus Araújo, Rotem Liss, and Antonio Acín, "Improved finite-size key rates for discrete-modulated continuous-variable quantum key distribution under coherent attacks", Physical Review A 111 2, 022610 (2025).

[105] O. S. Belova, D. V. Bolotov, S. Yu. Kazantsev, A. V. Klubkov, O. V. Kolesnikov, G. P. Paimtsev, and M. A. Yarygin, "The Influence of Pulsed Electromagnetic Action on a Quantum Key Distribution", Russian Electrical Engineering 96 4, 289 (2025).

[106] Gláucia Murta, Federico Grasselli, Hermann Kampermann, and Dagmar Bruß, "Quantum Conference Key Agreement: A Review", Advanced Quantum Technologies 3 11, 2000025 (2020).

[107] Roman Novak, "Security of Quantum Key Distribution with One-Time-Pad-Protected Error Correction and Its Performance Benefits", Entropy 27 10, 1032 (2025).

[108] Walter O. Krawec, "Security of a High Dimensional Two‐Way Quantum Key Distribution Protocol", Advanced Quantum Technologies 5 10, 2200024 (2022).

[109] Shouvik Ghorai, Philippe Grangier, Eleni Diamanti, and Anthony Leverrier, "Asymptotic Security of Continuous-Variable Quantum Key Distribution with a Discrete Modulation", Physical Review X 9 2, 021059 (2019).

[110] Jiawei Wu, Gui-Lu Long, and Masahito Hayashi, "Quantum Secure Direct Communication with Private Dense Coding Using a General Preshared Quantum State", Physical Review Applied 17 6, 064011 (2022).

[111] Stefan Bäuml, Carlos Pascual-García, Victoria Wright, Omar Fawzi, and Antonio Acín, "Security of discrete-modulated continuous-variable quantum key distribution", Quantum 8, 1418 (2024).

[112] Duarte Mateus and Ricardo Chaves, 2025 IEEE Information Theory Workshop (ITW) 1 (2025) ISBN:979-8-3315-3142-3.

[113] Giulio Malavolta and Michael Walter, Lecture Notes in Computer Science 14926, 126 (2024) ISBN:978-3-031-68393-0.

[114] Daan Leermakers and Boris Škorić, "Two-way unclonable encryption with a vulnerable sender", International Journal of Quantum Information 20 02, 2150037 (2022).

[115] Akihiro Mizutani, Toshihiko Sasaki, Yuki Takeuchi, Kiyoshi Tamaki, and Masato Koashi, "Quantum key distribution with simply characterized light sources", npj Quantum Information 5 1, 87 (2019).

[116] Matheus Sena, Georg Harder, Ronny Döring, Ralf-Peter Braun, Michaela Ritter, Oliver Holschke, Marc Kaplan, and Marc Geitz, "Deploying the Qline System for a QKD Metropolitan Network on the Berlin OpenQKD Testbed", IEEE Photonics Journal 17 1, 1 (2025).

[117] Jerome Wiesemann, Fadri Grünenfelder, Ana Blázquez Coído, Nino Walenta, and Davide Rusca, "Evaluation of quantum key distribution systems against injection-locking attacks", APL Photonics 10 6, 066112 (2025).

[118] Shlok Nahar, Devashish Tupkary, Yuming Zhao, Norbert Lütkenhaus, and Ernest Y.-Z. Tan, "Postselection Technique for Optical Quantum Key Distribution with Improved de Finetti Reductions", PRX Quantum 5 4, 040315 (2024).

[119] Enrique Cervero-Martín and Marco Tomamichel, "Device independent security of quantum key distribution from monogamy-of-entanglement games", Quantum 9, 1652 (2025).

[120] Mina Doosti, Lucas Hanouz, Anne Marin, Elham Kashefi, and Marc Kaplan, 2024 International Conference on Quantum Communications, Networking, and Computing (QCNC) 176 (2024) ISBN:979-8-3503-6677-8.

[121] Lars Kamin, Devashish Tupkary, and Norbert Lütkenhaus, "Improved finite-size effects in quantum key distribution with applications to decoy-state protocols", Physical Review Research 8 1, 013332 (2026).

[122] Ryan Amiri, Petros Wallden, Adrian Kent, and Erika Andersson, "Secure quantum signatures using insecure quantum channels", Physical Review A 93 3, 032325 (2016).

[123] Ittoop Vergheese Puthoor, Ryan Amiri, Petros Wallden, Marcos Curty, and Erika Andersson, "Measurement-device-independent quantum digital signatures", Physical Review A 94 2, 022328 (2016).

[124] Mario Berta, Stephanie Wehner, and Mark M. Wilde, "Entropic uncertainty and measurement reversibility", New Journal of Physics 18 7, 073004 (2016).

[125] Carl A. Miller and Yaoyun Shi, "Robust protocols for securely expanding randomness and distributing keys using untrusted quantum devices", arXiv:1402.0489, (2014).

[126] Kamil Brádler, Mohammad Mirhosseini, Robert Fickler, Anne Broadbent, and Robert Boyd, "Finite-key security analysis for multilevel quantum key distribution", New Journal of Physics 18 7, 073030 (2016).

[127] Akihiro Mizutani, Nobuyuki Imoto, and Kiyoshi Tamaki, "Robustness of the round-robin differential-phase-shift quantum-key-distribution protocol against source flaws", Physical Review A 92 6, 060303 (2015).

[128] Marco Tomamichel, Jesus Martinez-Mateo, Christoph Pacher, and David Elkouss, "Fundamental finite key limits for one-way information reconciliation in quantum key distribution", Quantum Information Processing 16 11, 280 (2017).

[129] Evgeny Kiktenko, Anton Trushechkin, Yury Kurochkin, and Aleksey Fedorov, "Post-processing procedure for industrial quantum key distribution systems", Journal of Physics Conference Series 741 1, 012081 (2016).

[130] Sumeet Khatri, Eneet Kaur, Saikat Guha, and Mark M. Wilde, "Second-order coding rates for key distillation in quantum key distribution", arXiv:1910.03883, (2019).

[131] Thomas Vidick, "Parallel DIQKD from parallel repetition", arXiv:1703.08508, (2017).

[132] James Bartusek, Dakshita Khurana, and Alexander Poremba, "Publicly-Verifiable Deletion via Target-Collapsing Functions", arXiv:2303.08676, (2023).

[133] Ashutosh Marwah and Frédéric Dupuis, "Proving security of BB84 under source correlations", arXiv:2402.12346, (2024).

[134] Prabhanjan Ananth, Alexander Poremba, and Vinod Vaikuntanathan, "Revocable Cryptography from Learning with Errors", arXiv:2302.14860, (2023).

[135] Vladimir L. Kurochkin, Yuriy V. Kurochkin, Alexander V. Miller, Alexander S. Sokolov, and Alan A. Kanapin, "Effect of crosstalk on QBER in QKD in urban telecommunication fiber lines", Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series 10224, 102242U (2016).

[136] Anne Broadbent and Christian Schaffner, "Quantum Cryptography Beyond Quantum Key Distribution", arXiv:1510.06120, (2015).

[137] Corsin Pfister, Norbert Lütkenhaus, Stephanie Wehner, and Patrick J. Coles, "Sifting attacks in finite-size quantum key distribution", New Journal of Physics 18 5, 053001 (2016).

[138] Vishal Singh, Karol Horodecki, Aby Philip, and Mark M. Wilde, "Limiting one-way distillable secret key via privacy testing of extendible states", arXiv:2511.04438, (2025).

[139] Yixin Zhang, "Blockchain of Signature Material Combining Cryptographic Hash Function and DNA Steganography", arXiv:1909.07914, (2019).

[140] Sumeet Khatri, "Towards a General Framework for Practical Quantum Network Protocols", arXiv:2412.20472, (2024).

[141] Marco Tomamichel, Jesus Martinez-Mateo, Christoph Pacher, and David Elkouss, "Fundamental Finite Key Limits for One-Way Information Reconciliation in Quantum Key Distribution", arXiv:1401.5194, (2014).

[142] Arash Atashpendar, "From Information Theory Puzzles in Deletion Channels to Deniability in Quantum Cryptography", arXiv:2003.11663, (2020).

[143] Corsin Pfister, Norbert Lütkenhaus, Stephanie Wehner, and Patrick J. Coles, "Sifting attacks in finite-size quantum key distribution", arXiv:1506.07502, (2015).

[144] Fabio Banfi, Ueli Maurer, Christopher Portmann, and Jiamin Zhu, "Composable and Finite Computational Security of Quantum Message Transmission", arXiv:1908.03436, (2019).

[145] Christopher Portmann, "(Quantum) Min-Entropy Resources", arXiv:1705.10595, (2017).

[146] Jonas Treplin, Philipp Kleinpaß, and Davide Orsucci, "Finite Size Analysis of Decoy-State BB84 with Advantage Distillation", arXiv:2511.21665, (2025).

[147] Ramona Wolf, "Quantum Key Distribution in the Non-Asymptotic Regime", arXiv:1511.06519, (2015).

[148] S. N. Molotkov, "Concatenation of Keys in Quantum Cryptography: How Quantum Entanglement "Penetrates to" Classical Devices", Soviet Journal of Experimental and Theoretical Physics 127 4, 627 (2018).

[149] Janusz Czub, Ryszard Veynar, Wiesław Laskowski, and Marcin Pawłowski, "Optimal pumping strength for BBM92 key distribution protocol", International Journal of Quantum Information 14 8, 1650049-179 (2016).

[150] Yupan Liu, "Computational hardness of estimating quantum entropies via binary entropy bounds", arXiv:2601.03734, (2026).

[151] S. N. Molotkov, "Quantum Key Distribution As a Scheme with Bernoulli Tests", Soviet Journal of Experimental and Theoretical Physics 126 6, 741 (2018).

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