A game-theoretic security framework for quantum cryptography: Performance analysis and application


In this paper, we analyze quantum key distribution (QKD) protocols through a game-theoretic framework. In particular, we assume parties and adversaries are “rational.” Unlike other game theoretic models, we show how important key-rate and noise tolerance computations may be performed through our system allowing for interesting comparisons to the “standard adversarial model” of QKD. We show that, depending on the relative cost of operating devices, increased noise tolerance and improved secure communication rates are possible if one assumes adversaries are rational as opposed to being malicious.

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  1. 1.

    Bennett, C.H., Brassard, G.: Quantum cryptography: public key distribution and coin tossing. In: Proceedings of IEEE International Conference on Computers, Systems and Signal Processing, vol. 175. New York (1984)

  2. 2.

    Scarani, V., Bechmann-Pasquinucci, H., Cerf, N.J., Dušek, M., Lütkenhaus, N., Peev, M.: The security of practical quantum key distribution. Rev. Mod. Phys. 81, 1301–1350 (2009)

    ADS  Article  Google Scholar 

  3. 3.

    Pirandola, S., Andersen, U.L., Banchi, L., Berta, M., Bunandar, D., Colbeck, R., Englund, D., Gehring, T., Lupo, C., Ottaviani, C., et al.: Advances in quantum cryptography (2019). arXiv preprint arXiv:1906.01645

  4. 4.

    Katz, J.: Bridging game theory and cryptography: Recent results and future directions. In: Theory of Cryptography Conference, pp. 251–272. Springer, Berlin (2008)

  5. 5.

    Krawec, W.O., Miao, F.: Game theoretic security framework for quantum key distribution. In: International Conference on Decision and Game Theory for Security, pp. 38–58. Springer, Berlin (2018)

  6. 6.

    Damgård, I.B., Fehr, S., Salvail, L., Schaffner, C.: Cryptography in the bounded-quantum-storage model. SIAM J. Comput. 37(6), 1865–1890 (2008)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Wehner, S., Schaffner, C., Terhal, B.M.: Cryptography from noisy storage. Phys. Rev. Lett. 100(22), 220502 (2008)

    ADS  Article  Google Scholar 

  8. 8.

    Damgård, I.B., Fehr, S., Renner, R., Salvail, L., Schaffner, C.: A tight high-order entropic quantum uncertainty relation with applications. In: Annual International Cryptology Conference, pp. 360–378. Springer, Berlin (2007)

  9. 9.

    Manshaei, M., Zhu, Q., Alpcan, T., Basar, T., Hubaux, J.: Game theory meets network security and privacy. ACM Comput. Surv. 45(3), 25:1–25:39 (2013)

    Article  Google Scholar 

  10. 10.

    Halpern, J., Teague, V.: Rational secret sharing and multiparty computation. In: Proceedings of the Thirty-Sixth Annual ACM Symposium on Theory of Computing, pp. 623–632. ACM, New York (2004)

  11. 11.

    Asharov, G., Lindell, Y.: Utility dependence in correct and fair rational secret sharing. In: Advances in Cryptology-CRYPTO 2009, pp. 559–576. Springer, Berlin (2009)

  12. 12.

    Kol, G., Naor, M.: Cryptography and Game Theory: Designing Protocols for Exchanging Information. In: Theory of Cryptography Conference, pp. 320–339. Springer, Berlin (2008)

  13. 13.

    Miao, F., Zhu, Q., Pajic, M., Pappas, G.J.: A hybrid stochastic game for secure control of cyber-physical systems. Automatica 93, 55–63 (2018)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Pajic, M., Tabuada, P., Lee, I., Pappas, G.J.: Attack-resilient state estimation in the presence of noise. In: 2015 54th IEEE Conference on Decision and Control (CDC), pp. 5827–5832 (2015)

  15. 15.

    Miao, F., Zhu, Q., Pajic, M., Pappas, G.J.: Coding schemes for securing cyber-physical systems against stealthy data injection attacks. IEEE Trans. Control Netw. Syst. 4(1), 106–117 (2016)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Zhu, Q., Basar, T.: Game-theoretic methods for robustness, security, and resilience of cyberphysical control systems: games-in-games principle for optimal cross-layer resilient control systems. Control Syst. IEEE 35(1), 46–65 (2015)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Zhu, M., Martinez, S.: Stackelberg-game analysis of correlated attacks in cyber-physical systems. Am. Control Conf. (ACC) 2011, 4063–4068 (2011)

    Google Scholar 

  18. 18.

    Maitra, A., De Joyee, S., Paul, G., Pal, A.K.: Proposal for quantum rational secret sharing. Phys. Rev. A 92(2), 022305 (2015)

    ADS  Article  Google Scholar 

  19. 19.

    Maitra, A., Paul, G., Pal, A.K.: Millionaires problem with rational players: a unified approach in classical and quantum paradigms (2015). arXiv preprint

  20. 20.

    Zhou, L., Sun, X., Su, C., Liu, Z., Choo, K.-K.R.: Game theoretic security of quantum bit commitment. Inf. Sci. 479, 503–514 (2018)

    Article  Google Scholar 

  21. 21.

    Mayers, D.: Unconditionally secure quantum bit commitment is impossible. Phys. Rev. Lett. 78, 3414–3417 (1997)

    ADS  Article  Google Scholar 

  22. 22.

    Dou, Z., Xu, G., Chen, X.-B., Liu, X., Yang, Y.-X.: A secure rational quantum state sharing protocol. Sci. China Inf. Sci. 61(2), 022501 (2018)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Qin, H., Tang, W.K.S., Tso, R.: Establishing rational networking using the DL04 quantum secure direct communication protocol. Quantum Inf. Process. 17(6), 152 (2018)

    ADS  MathSciNet  Article  Google Scholar 

  24. 24.

    Das, B., Roy, U., et al.: Cooperative quantum key distribution for cooperative service-message passing in vehicular ad hoc networks. Int. J. Comput. Appl. 975(8887), 37–42 (2014)

    Google Scholar 

  25. 25.

    Houshmand, M., Houshmand, M., Mashhadi, H.R.: Game theory based view to the quantum key distribution bb84 protocol. In: Intelligent Information Technology and Security Informatics (IITSI), 2010 Third International Symposium on, pp. 332–336. IEEE (2010)

  26. 26.

    Kaur, H., Kumar, A.: Game-theoretic perspective of ping-pong protocol. Phys. A Stat. Mech. Appl. 490, 1415–1422 (2018)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Lucamarini, M., Mancini, S.: Secure deterministic communication without entanglement. Phys. Rev. Lett. 94(14), 140501 (2005)

    ADS  Article  Google Scholar 

  28. 28.

    Boström, K., Felbinger, T.: Deterministic secure direct communication using entanglement. Phys. Rev. Lett. 89(18), 187902 (2002)

    ADS  Article  Google Scholar 

  29. 29.

    Lo, H.-K., Ma, X., Chen, K.: Decoy state quantum key distribution. Phys. Rev. Lett. 94(23), 230504 (2005)

    ADS  Article  Google Scholar 

  30. 30.

    Bennett, C.H.: Quantum cryptography using any two nonorthogonal states. Phys. Rev. Lett. 68, 3121–3124 (1992)

    ADS  MathSciNet  Article  Google Scholar 

  31. 31.

    Lo, H.-K., Chau, H.-F., Ardehali, M.: Efficient quantum key distribution scheme and a proof of its unconditional security. J. Cryptol. 18(2), 133–165 (2005)

    MathSciNet  Article  Google Scholar 

  32. 32.

    Devetak, I., Winter, A.: Distillation of secret key and entanglement from quantum states. Proc. R. Soc. A Math. Phys. Eng. Sci. 461(2053), 207–235 (2005)

    ADS  MathSciNet  MATH  Google Scholar 

  33. 33.

    Shor, P.W., Preskill, J.: Simple proof of security of the BB84 quantum key distribution protocol. Phys. Rev. Lett. 85, 441–444 (2000)

    ADS  Article  Google Scholar 

  34. 34.

    Kraus, B., Branciard, C., Renner, R.: Security of quantum-key-distribution protocols using two-way classical communication or weak coherent pulses. Phys. Rev. A 75(1), 012316 (2007)

    ADS  Article  Google Scholar 

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Correspondence to Walter O. Krawec.

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Han, S., Krawec, W.O. & Miao, F. A game-theoretic security framework for quantum cryptography: Performance analysis and application. Quantum Inf Process 19, 349 (2020). https://doi.org/10.1007/s11128-020-02861-9

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  • Quantum Cryptography
  • Game Theory
  • Quantum Key Distribution
  • Security