Advertisement

Quantum Information Processing

, Volume 14, Issue 2, pp 715–729 | Cite as

Quantum strongly secure ramp secret sharing

  • Paul Zhang
  • Ryutaroh MatsumotoEmail author
Article

Abstract

Quantum secret sharing is a scheme for encoding a quantum state (the secret) into multiple shares and distributing them among several participants. If a sufficient number of shares are put together, then the secret can be fully reconstructed. If an insufficient number of shares are put together however, no information about the secret can be revealed. In quantum ramp secret sharing, partial information about the secret is allowed to leak to a set of participants, called an unqualified set, that cannot fully reconstruct the secret. By allowing this, the size of a share can be drastically reduced. This paper introduces a quantum analog of classical strong security in ramp secret sharing schemes. While the ramp secret sharing scheme still leaks partial information about the secret to unqualified sets of participants, the strong security condition ensures that qudits with critical information can no longer be leaked.

Keywords

Quantum secret sharing Non-perfect secret sharing   Ramp secret sharing Strong security 

Mathematics Subject Classification

81P94 94A62 

Notes

Acknowledgments

The authors would like to thank the reviewers for their careful reading that improved the presentation. This research was conducted as part of Tokyo Institute of Technology International Research Opportunities Program under Re-Inventing Japan Project funded by Ministry of Education, Culture, Sports, Science and Technology. This research is partly supported by the National Institute of Information and Communications Technology, Japan, and by the Japan Society for the Promotion of Science Grant Nos. 23246071 and 26289116, and the Villum Foundation through their VELUX Visiting Professor Programme 2013–2014.

References

  1. 1.
    Blakley, G.R., Meadows, C.: Security of ramp schemes. In: Advances in Cryptology-CRYPTO’84. Lecture Notes in Computer Science, vol. 196, pp. 242–269. Springer (1985). doi: 10.1007/3-540-39568-7_20
  2. 2.
    Capocelli, R.M., De Santis, A., Gargano, L., Vaccaro, U.: On the size of shares for secret sharing schemes. J. Cryptol. 6(3), 157–167 (1993). doi: 10.1007/BF00198463 CrossRefzbMATHGoogle Scholar
  3. 3.
    Cleve, R., Gottesman, D., Lo, H.K.: How to share a quantum secret. Phys. Rev. Lett. 83(3), 648–651 (1999). doi: 10.1103/PhysRevLett.83.648 ADSCrossRefGoogle Scholar
  4. 4.
    Gottesman, D.: Theory of quantum secret sharing. Phys. Rev. A 61(4), 042311 (2000). doi: 10.1103/PhysRevA.61.042311
  5. 5.
    Iwamoto, M., Yamamoto, H.: Strongly secure ramp secret sharing schemes for general access structures. Inf. Process. Lett. 97(2), 52–57 (2006). doi: 10.1016/j.ipl.2005.09.012 CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    McEliece, R.J., Sarwate, D.V.: On sharing secrets and Reed–Solomon codes. Commun. ACM 24(9), 583–584 (1981). doi: 10.1145/358746.358762 CrossRefMathSciNetGoogle Scholar
  7. 7.
    Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge, UK (2000)zbMATHGoogle Scholar
  8. 8.
    Nishiara, M., Takizawa, K.: Strongly secure secret sharing scheme with ramp threshold based on Shamir’s polynomial interpolation scheme. Trans. IEICE J92-A(12), 1009–1013 (2009). http://ci.nii.ac.jp/naid/110007483234/en
  9. 9.
    Ogata, W., Kurosawa, K., Tsujii, S.: Nonperfect secret sharing schemes. In: Advances in Cryptology—AUSCRYPT ’92. Lecture Notes in Computer Science, vol. 718, pp. 56–66. Springer (1993). doi: 10.1007/3-540-57220-1_52
  10. 10.
    Ogawa, T., Sasaki, A., Iwamoto, M., Yamamoto, H.: Quantum secret sharing schemes and reversibility of quantum operations. Phys. Rev. A 72(3), 032318 (2005). doi: 10.1103/PhysRevA.72.032318
  11. 11.
    Shamir, A.: How to share a secret. Commun. ACM 22(11), 612–613 (1979)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Smith, A.D.: Quantum secret sharing for general access structures (2000). URL arXiv:quant-ph/0001087
  13. 13.
    Stinson, D.R.: Cryptography Theory and Practice, 3rd edn. Chapman & Hall, London (2006)zbMATHGoogle Scholar
  14. 14.
    Yamamoto, H.: Secret sharing system using \((k, l, n)\) threshold scheme. Electron. Commun. Jpn. I Commun. 69(9), 46–54 (1986). doi:  10.1002/ecja.4410690906. (the original Japanese version published in 1985)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.California Institute of TechnologyPasadenaUSA
  2. 2.Department of Communications and Computer EngineeringTokyo Institute of TechnologyTokyoJapan
  3. 3.Department of Mathematical SciencesAalborg UniversityAalborgDenmark

Personalised recommendations