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Requirements for Group Independent Linear Threshold Secret Sharing Schemes

  • Brian King
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2384)

Abstract

In a t out of n threshold scheme, any subset of t or more participants can compute the secret key k, while subsets of t − 1 or less participants cannot compute k. Some schemes are designed for specific algebraic structures, for example finite fields. Whereas other schemes can be used with any finite abelian group. In[24], the definition of group independent sharing schemes was introduced. In this paper, we develop bounds for group independent t out of n threshold schemes. The bounds will be lower bounds which discuss how many subshares are required to achieve a group independent linear threshold scheme. In particular, we will show that our bounds for the n − 1 out of n threshold schemes are tight for infinitely many n.

Keywords

secret sharing linear secret sharing threshold cryptography group independent linear threshold schemes monotone span programs and bounds on share size 

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References

  1. 1.
    W. Adkins and S. Weintrab. Algebra, an approach via module theory. Springer-Verlag, NY, 1992.zbMATHGoogle Scholar
  2. 2.
    A. Beimel, A. G’al, and M. Paterson. Lower bounds for monotone span programs. Computational Complexity, 6(1):29–45, 1997.CrossRefMathSciNetGoogle Scholar
  3. 3.
    S. Blackburn, M. Burmester, Y. Desmedt, and P. Wild. “Efficient Multiplicative Sharing schemes”. In Advances in Cryptology-Eurocrypt’ 96, LNCS 1070, pp. 107–118, Springer-Verlag, 1996.Google Scholar
  4. 4.
    C. Blundo, A. De Santis, A.G. Gaggia, and U. Vaccaro. “New Bounds on the Information rate of Secret Sharing Schemes”. In IEEE Trans. on Inform. Theory, 41, no. 2, pp. 549–554, 1995.zbMATHCrossRefGoogle Scholar
  5. 5.
    C. Blundo, A. De Santis, R. De Simone,, and U. Vaccaro. “Tight Bounds on the Information rate of secret Sharing Schemes”. In Design, Codes and Cryptography, 11, pp. 107–122, 1997.zbMATHCrossRefGoogle Scholar
  6. 6.
    C. Boyd, Digital Multisignatures, Cryptography and coding, Clarendon Press, 1989, pp 241–246.Google Scholar
  7. 7.
    R.M. Capocelli, A. De Santis, L. Gargano, and U. Vaccaro, “On the Size of Shares for secret Sharing Schemes” In Journal of Cryptology, 6, pp. 157–167, 1993.zbMATHCrossRefGoogle Scholar
  8. 8.
    R. Cramer. Personal communication Google Scholar
  9. 9.
    L. Csirmaz. “The Size of a Share Must Be large”. In Journal of Cryptology, 10, pp. 223–231, 1997.zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    N. De Lillo. Advanced Calculus with applications. MacMillan, NY, 1982.zbMATHGoogle Scholar
  11. 11.
    A. De Santis, Y. Desmedt, Y. Frankel, and M. Yung. “How to share a function”. In Proceedings of the twenty-sixth annual ACM Symp. Theory of Computing (STOC), pp. 522–533, 1994.Google Scholar
  12. 12.
    Y. Desmedt. Society and group oriented cryptography: a new concept. In Advances of Cryptology-Crypto’ 87 Google Scholar
  13. 13.
    Y. Desmedt, G. Di Crescenzo, and M. Burmester. “Multiplicative non-Abelian sharing schemes and their application to threshold cryptography”. In Advances in Cryptology-Asiacrypt’ 94, LNCS 917. pp. 21–32, Springer-Verlag, 1995.CrossRefGoogle Scholar
  14. 14.
    Y. Desmedt and Y. Frankel. “Homomorphic zero-knowledge threshold schemes over any finite Abelian group”. In Siam J. Disc. Math. vol 7, no. 4 pp. 667–679, SIAM, 1994.zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Y. Desmedt and S. Jajodia. Redistributing secret shares to new access structures and its applications. Tech. Report ISSE-TR-97-01, George Mason University, July 1997 ftp://isse.gmu.edu/pub/techrep/97.01.jajodia.ps.gz
  16. 16.
    Y. Desmedt, B. King, W. Kishimoto, and K. Kurosawa, “A comment on the efficiency of secret sharing scheme over any finite Abelian group”, In Information Security and Privacy, ACISP’98 (Third Australasian Conference on Information Security and Privacy), LNCS 1438, 1998, 391–402.Google Scholar
  17. 17.
    Y. Frankel, Y. Desmedt, and M. Burmester. “ Non-existence of homomorphic general sharing schemes for some key spaces”, in Advances of Cryptology-Crypto’ 92, 740, 1992 pp 549–557Google Scholar
  18. 18.
    Y. Frankel, P. Gemmel, P. Mackenzie, and M. Yung. “Optimal-Resilience Proactive Public-key Cryptosystems”. In Proc. 38th FOCS, IEEE, 1997, p. 384–393.Google Scholar
  19. 19.
    R. Gennaro, S. Jarecki, H. Krawczyk, and T. Rabin. “Robust and efficient sharing of RSA functions”. In Advances of Cryptology-Crypto’ 96, LNCS 1109, Springer Verlag, 1996, p. 157–172.CrossRefGoogle Scholar
  20. 20.
    T. Hungerford. Algebra. Springer-Verlag, NY, 1974.zbMATHGoogle Scholar
  21. 21.
    M. Karchmer and A. Wigderson. On span programs In Proc. of 8 th annual Complexity Theory Conference, pp 102–111, 1993.Google Scholar
  22. 22.
    E. Karnin, J. Greene, and M. Hellman. “On secret sharing systems.” In IEEE Trans. Inform. Theory, 29(1), pp. 35–41, 1983.zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    H.L. Keng. Introduction to Number Theory. Springer Verlag, NY 1982zbMATHGoogle Scholar
  24. 24.
    B. King. “Randomness Required for Linear Threshold Sharing Schemes Defined over Any Finite Abelian Group”. In ACISP 2001. pp. 376–391.Google Scholar
  25. 25.
    R. Lidl and G. Pilz. Applied Abstract Algebra. Springer Verlag, NY 1984zbMATHGoogle Scholar
  26. 26.
    R. Rivest, A. Shamir, and L. Adelman, A method for obtaining digital signatures and public key cryptosystems, Comm. ACM, 21(1978), pp 294–299.CrossRefGoogle Scholar
  27. 27.
    A. Shamir, How to share a secret, Comm. ACM, 22(1979), pp 612–613.zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    M. van Dijk. “A Linear Construction of Secret Sharing Schemes”. In Design, Codes and Cryptography 12, pp. 161–201, 1997.zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Brian King
    • 1
  1. 1.IUPUI campusPurdue School of Engineering and TechnologyUSA

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