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Geometric and Computational Approach to Classical and Quantum Secret Sharing

  • Ryutaroh Matsumoto
  • Diego Ruano
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 198)

Abstract

Secret sharing is a cryptographic scheme to encode a secret to multiple shares being distributed to participants, so that only qualified (or authorized) sets of participants can reconstruct the original secret from their shares. It is also known that every linear ramp secret sharing can be expressed by a nested pair of linear codes \(C_2 \subset C_1 \subset \mathbf {F}_q^n\). On the other hand, a nest code pair \(C_2 \subset C_1 \subset \mathbf {F}_q^n\) can also give a quantum secret sharing. Since \(C_1\) and \(C_2\) are linear codes, it is natural to use algebraic geometry codes to construct \(C_1\) and \(C_2\). The purpose of this work is to find sufficient conditions for qualified or forbidden sets by using geometric properties of the set of points.

Keywords

Algebraic geometry codes Quantum secret sharing Access structure 

Notes

Acknowledgements

The authors gratefully acknowledge the support from Japan Society for the Promotion of Science (Grant Nos. 23246071 and 26289116), from the Spanish MINECO/FEDER (Grant No. MTM2012-36917-C03-03 and No. MTM2015-65764-C3-2-P), the Danish Council for Independent Research (Grant No. DFF-4002-00367) and from the “Program for Promoting the Enhancement of Research Universities” at Tokyo Institute of Technology.

References

  1. 1.
    Bains, T.: Generalized Hamming weights and their applications to secret sharing schemes. Master’s Thesis, University of Amsterdam (2008). Supervised by R. Cramer, G. van der Geer, and R. de HaanGoogle Scholar
  2. 2.
    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
  3. 3.
    Chen, H., Cramer, R., Goldwasser, S., de Haan, R., Vaikuntanathan, V.: Secure computation from random error correccting codes. In: Advances in Cryptology—EUROCRYPT 2007. Lecture Notes in Computer Science, vol. 4515, pp. 291–310. Springer (2007). doi: 10.1007/978-3-540-72540-4_17
  4. 4.
    Chen, H., Cramer, R., de Haan, R., Cascudo Pueyo, I.: Strongly multiplicative ramp schemes from high degree rational points on curves. In: Smart, N. (ed.) Advances in Cryptology—EUROCRYPT 2008. Lecture Notes in Computer Science, vol. 4965, pp. 451–470. Springer (2008). doi: 10.1007/978-3-540-78967-3_26
  5. 5.
    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
  6. 6.
    Geil, O., Pellikaan, R.: On the structure of order domains. Finite Fields Appl. 8, 369–396 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Gottesman, D.: Theory of quantum secret sharing. Phys. Rev. A 61(4), 042311 (2000). doi: 10.1103/PhysRevA.61.042311 MathSciNetCrossRefGoogle Scholar
  8. 8.
    Heegard, C., Little, J., Saints, K.: Systematic encoding via Gröbner bases for a class of algebraic-geometric Goppa codes. IEEE Trans. Inf. Theory 41(6), 1752–1761 (1995). doi: 10.1109/18.476247 CrossRefzbMATHGoogle Scholar
  9. 9.
    Kurihara, J., Uyematsu, T., Matsumoto, R.: Secret sharing schemes based on linear codes can be precisely characterized by the relative generalized Hamming weight. IEICE Trans. Fundam. E95-A(11), 2067–2075 (2012). doi: 10.1587/transfun.E95.A.2067
  10. 10.
    Matsumoto, R.: Coding theoretic construction of quantum ramp secret sharing, Version 4 or later (2014)Google Scholar
  11. 11.
    Matsumoto, R., Miura, S.: Finding a basis of a linear system with pairwise distinct discrete valuations on an algebraic curve. J. Symb. Comput. 30(3), 309–323 (2000). doi: 10.1006/jsco.2000.0372 MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Matsumoto, R., Miura, S.: On construction and generalization of algebraic geometry codes. In: Katsura, T. et al. (eds.) Proceedings of Algebraic Geometry, Number Theory, Coding Theory, and Cryptography, pp. 3–15. University of Tokyo, Japan (2000). http://www.rmatsumoto.org/repository/weight-construct.pdf
  13. 13.
    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
  14. 14.
    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 CrossRefGoogle Scholar
  15. 15.
    Shamir, A.: How to share a secret. Commun. ACM 22(11), 612–613 (1979). doi: 10.1145/359168.359176 MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Smith, A.D.: Quantum secret sharing for general access structures. arXiv:quant-ph/0001087 (2000)
  17. 17.
    Yamamoto, H.: Secret sharing system using \((k, l, n)\) threshold scheme. Electron. Commun. Jpn. (Part I: Communications) 69(9), 46–54 (1986). doi: 10.1002/ecja.4410690906 (The original Japanese version published in 1985)
  18. 18.
    Zhang, P., Matsumoto, R.: Quantum strongly secure ramp secret sharing. Quantum Inf. Process. 14(2), 715–729 (2015). doi: 10.1007/s11128-014-0863-2 MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

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

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