Advertisement

Computing on quantum shared secrets for general quantum access structures

  • Roozbeh BassirianEmail author
  • Sadra Boreiri
  • Vahid Karimipour
Article
  • 103 Downloads

Abstract

Quantum secret sharing is a method for sharing a secret quantum state among a number of individuals such that certain authorized subsets of participants can recover the secret shared state by collaboration and other subsets cannot. In this paper, we first propose a method for sharing a quantum secret in a basic (2, 3) threshold scheme, only by using qubits and the 7-qubit CSS code. Based on this (2, 3) scheme, we propose a new (nn) scheme, and we also construct a quantum secret sharing scheme for any quantum access structure by induction. Secondly, based on the techniques of performing quantum computation on 7-qubit CSS codes, we introduce a method that authorized subsets can perform universal quantum computation on this shared state, without the need for recovering it. This generalizes recent attempts for doing quantum computation on (nn) threshold schemes.

Keywords

Quantum cryptography Quantum computation Quantum secret sharing Access structures Error correction Seven qubit code Transversal computation 

Notes

Acknowledgements

We thank the referees, specially one of them, whose very careful reading of the manuscript and very valuable comments led to a much better presentation of this article. This work was done with a support from the research council of the Sharif University of Technology, and with the financial support from Sharif University of Technology under Grant No. G951418, and with partial support from Iran National Science Foundation under the Grant INSF-96011347.

References

  1. 1.
    Shamir, A.: How to share a secret. Commun. ACM 22(11), 612–613 (1979)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Blakley, G.R., et al.: Safeguarding cryptographic keys. Proc. Natl. Comput. Conf. 48, 313–317 (1979)Google Scholar
  3. 3.
    Hillery, M., Bužek, V., Berthiaume, A.: Quantum secret sharing. Phys. Rev. A 59(3), 1829 (1999)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Karlsson, A., Koashi, M., Imoto, N.: Quantum entanglement for secret sharing and secret splitting. Phys. Rev. A 59(1), 162 (1999)ADSCrossRefGoogle Scholar
  5. 5.
    Smith, A.D.: Quantum secret sharing for general access structures. arXiv preprint arXiv:quant-ph/0001087 (2000)
  6. 6.
    Cleve, R., Gottesman, D., Lo, H.-K.: How to share a quantum secret. Phys. Rev. Lett. 83, 648–651 (1999)ADSCrossRefGoogle Scholar
  7. 7.
    Gottesman, D.: Theory of quantum secret sharing. Phys. Rev. A 61, 042311 (2000)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    Imai, H., Müller-Quade, J., Nascimento, A.C., Tuyls, P., Winter, A.: A quantum information theoretical model for quantum secret sharing schemes. arXiv preprint arXiv:quant-ph/0311136 (2003)
  9. 9.
    Bai, C.-M., Li, Z.-H., Xu, T.-T., Li, Y.-M.: A generalized information theoretical model for quantum secret sharing. Int. J. Theor. Phys. 55(11), 4972–4986 (2016)CrossRefGoogle Scholar
  10. 10.
    Bennett, C.H., Bessette, F., Brassard, G., Salvail, L., Smolin, J.: Experimental quantum cryptography. J. Cryptol. 5(1), 3–28 (1992)CrossRefGoogle Scholar
  11. 11.
    Gröblacher, S., Jennewein, T., Vaziri, A., Weihs, G., Zeilinger, A.: Experimental quantum cryptography with qutrits. New J. Phys. 8(5), 75 (2006)ADSCrossRefGoogle Scholar
  12. 12.
    Liang, W.-Y., Li, M., Yin, Z.-Q., Chen, W., Wang, S., An, X.-B., Guo, G.-C., Han, Z.-F.: Simple implementation of quantum key distribution based on single-photon bell-state measurement. Phys. Rev. A 92, 012319 (2015)ADSCrossRefGoogle Scholar
  13. 13.
    Broadbent, A., Fitzsimons, J., Kashefi, E.: Proceedings of the 50th Annual IEEE Symposium on Foundations of Computer Science (2009)Google Scholar
  14. 14.
    Barz, S., Kashefi, E., Broadbent, A., Fitzsimons, J.F., Zeilinger, A., Walther, P.: Demonstration of blind quantum computing. Science 335(6066), 303–308 (2012)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    Ouyang, Y., Tan, S.-H., Fitzsimons, J.F.: Quantum homomorphic encryption from quantum codes. Phys. Rev. A 98, 042334 (2018)ADSCrossRefGoogle Scholar
  16. 16.
    Karimipour, V., Asoudeh, M.: Quantum secret sharing and random hopping: using single states instead of entanglement. Phys. Rev. A 92, 030301 (2015)ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    Bagherinezhad, S., Karimipour, V.: Quantum secret sharing based on reusable Greenberger–Horne–Zeilinger states as secure carriers. Phys. Rev. A 67, 044302 (2003)ADSCrossRefGoogle Scholar
  18. 18.
    Song, X.-L., Liu, Y.-B., Deng, H.-Y., Xiao, Y.-G.: (t, n) threshold d-level quantum secret sharing. Sci. Rep. 7(1), 6366 (2017)ADSCrossRefGoogle Scholar
  19. 19.
    Gordon, G., Rigolin, G.: Generalized quantum-state sharing. Phys. Rev. A 73(6), 062316 (2006)ADSCrossRefGoogle Scholar
  20. 20.
    Qin, H., Dai, Y.: Proactive quantum secret sharing. Quantum Inf. Process. 14(11), 4237–4244 (2015)ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    Ouyang, Y., Tan, S.-H., Zhao, L., Fitzsimons, J.F.: Computing on quantum shared secrets. Phys. Rev. A 96(5), 052333 (2017)ADSCrossRefGoogle Scholar
  22. 22.
    Fortescue, B., Gour, G.: Reducing the quantum communication cost of quantum secret sharing. IEEE Trans. Inf. Theory 58(10), 6659–6666 (2012)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Bai, C.-M., Li, Z.-H., Si, M.-M., Li, Y.-M.: Quantum secret sharing for a general quantum access structure. Eur. Phys. J. D 71(10), 255 (2017)ADSCrossRefGoogle Scholar
  24. 24.
    Knill, E., Laflamme, R., Viola, L.: Theory of quantum error correction for general noise. Phys. Rev. Lett. 84, 2525–2528 (2000)ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    Gottesman, D.: Stabilizer codes and quantum error correction. arXiv preprint arXiv:quant-ph/9705052 (1997)
  26. 26.
    Wootters, W.K., Zurek, W.H.: A single quantum cannot be cloned. Nature 299(5886), 802–803 (1982)ADSCrossRefGoogle Scholar
  27. 27.
    Calderbank, A.R., Shor, P.W.: Good quantum error-correcting codes exist. Phys. Rev. A 54, 1098–1105 (1996)ADSCrossRefGoogle Scholar
  28. 28.
    Gottesman, D., Chuang, I.L.: Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations. Nature 402(6760), 390 (1999)ADSCrossRefGoogle Scholar
  29. 29.
    Zhou, X., Leung, D.W., Chuang, I.L.: Methodology for quantum logic gate construction. Phys. Rev. A 62, 052316 (2000)ADSCrossRefGoogle Scholar
  30. 30.
    Bennett, C.H., Brassard, G., Mermin, N.D.: Quantum cryptography without Bell’s theorem. Phys. Rev. Lett. 68, 557–559 (1992)ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    Dunjko, V., Fitzsimons, J.F., Portmann, C., Renner, R.: Composable security of delegated quantum computation. In: International Conference on the Theory and Application of Cryptology and Information Security, pp. 406–425. Springer, Berlin (2014)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Computer EngineeringSharif University of TechnologyTehranIran
  2. 2.Department of PhysicsSharif University of TechnologyTehranIran

Personalised recommendations