A Formal Approach to Unconditional Security Proofs for Quantum Key Distribution

  • Takahiro Kubota
  • Yoshihiko Kakutani
  • Go Kato
  • Yasuhito Kawano
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6714)


We present an approach to automate Shor-Preskill style unconditional security proof of QKDs. In Shor-Preskill’s proof, the target QKD, BB84, is transformed into another QKD based on an entanglement distillation protocol (EDP), which is more feasible for direct analysis. We formalized heir method as program transformation in a quantum programming language, QPL. The transform is defined as rewriting rules which are sound with respect to the security in the semantics of QPL. We proved that rewriting always terminates for any program and that the normal form is unique under appropriate conditions. By applying the rewriting rules to the program representing BB84, we can obtain the corresponding EDP-based protocol automatically. We finally proved the security of the obtained EDP-based protocol formally in the quantum Hoare logic, which is a system for formal verification of quantum programs. We show also that this method can be applied to B92 by a simple modification.


QKD BB84 B92 unconditional security automatic verification formal methods 


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  1. 1.
    Bennett, C.H., Brassard, G.: Quantum cryptography: Public-key distribution and coin tossing. In: IEEE International Conference on Computers, Systems and Signal Processing, pp. 175–179 (1984)Google Scholar
  2. 2.
    Ekert, A.K.: Quantum cryptography based on bell’s theorem. Phys. Rev. Lett. 67(6), 661–663 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bennet, C.H.: Quantum cryptography using any two nonorthogonal states. Phys. Rev. Lett. 68(21), 3121–3124 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Inoue, K., Waks, E., Yamamoto, Y.: Differential phase shift quantum key distribution. Phys. Rev. Lett. 89(3), 037902 (2002)CrossRefGoogle Scholar
  5. 5.
    Stucki, D., Brunner, N., Gisin, N., Scarani, V., Zbinden, H.: Fast and simple one-way quantum key distribution. Appl. Phys. Lett. 87(19), 194108–194108–3 (2005)CrossRefGoogle Scholar
  6. 6.
    Nagy, M., Akl, S.G.: Entanglement verification with an application to quantum key distribution protocols. Parallel Processing Letters, 227–237 (2010)Google Scholar
  7. 7.
    Mayers, D.: Unconditional security in quantum cryptography. J. ACM 48, 351–406 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Lo, H.-K., Chau, H.F.: Unconditional security of quantum key distribution over arbitrarily long distances. Phys. Rev. Lett. 283(5410), 2050–2056 (1999)Google Scholar
  9. 9.
    Shor, P.W., Preskill, J.: Simple proof of security of the bb84 quantum key distribution protocol. Phys. Rev. Lett. 85(2), 441–444 (2000)CrossRefGoogle Scholar
  10. 10.
    Calderbank, A.R., Shor, P.W.: Good quantum error-correcting codes exist. Phys. Rev. A 54(2), 1098–1105 (1996)CrossRefGoogle Scholar
  11. 11.
    Tamaki, K., Koashi, M., Imoto, N.: Unconditionally secure key distribution based on two nonorthogonal states. Phys. Rev. Lett. 90(16), 167904 (2003)CrossRefGoogle Scholar
  12. 12.
    Hwang, W.-Y., Wang, X.-B., Matsumoto, K., Kim, J., Lee, H.-W.: Shor-preskill-type security proof for quantum key distribution without public announcement of bases. Phys. Rev. A 67(1), 012302 (2003)CrossRefGoogle Scholar
  13. 13.
    Blanchet, B., Jaggard, A.D., Scedrov, A., Tsay, J.-K.: Computationally sound mechanized proofs for basic and public-key kerberos. In: Proceedings of the 2008 ACM Symposium on Information, Computer and Communications Security, ASIACCS 2008, pp. 87–99. ACM, New York (2008)CrossRefGoogle Scholar
  14. 14.
    Neuman, C., Yu, T., Hatman, S., Raeburn, K.: The kerberos network authentication service (v5) (July 2005),
  15. 15.
    Selinger, P.: Towards a quantum programming language. Mathematical Structures in Computer Science 14, 527–586 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kakutani, Y.: A logic for formal verification of quantum programs. In: Datta, A. (ed.) ASIAN 2009. LNCS, vol. 5913, pp. 79–93. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  17. 17.
    Kubota, T.: Formalization and Automation of Unconditional Security Proof of QKD. Master thesis, the University of Tokyo (February 2011)Google Scholar
  18. 18.
    Nagarajan, R., Papanikolaou, N., Bowen, G., Gay, S.: An Automated Analysis of the Security of Quantum Key Distribution. ArXiv Computer Science e-prints (February 2005)Google Scholar
  19. 19.
  20. 20.
    Lo, H.-K.: Proof of unconditional security of six-state quantum key distribution scheme,

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Takahiro Kubota
    • 1
  • Yoshihiko Kakutani
    • 1
  • Go Kato
    • 2
  • Yasuhito Kawano
    • 2
  1. 1.Department of Computer Science, Graduate School of Information Science and TechnologyThe University of TokyoJapan
  2. 2.NTT Communication Science LaboratoriesNTT CorporationJapan

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