Requirements for Group Independent Linear Threshold Secret Sharing Schemes
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, 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.
Keywordssecret sharing linear secret sharing threshold cryptography group independent linear threshold schemes monotone span programs and bounds on share size
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- 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
- 6.C. Boyd, Digital Multisignatures, Cryptography and coding, Clarendon Press, 1989, pp 241–246.Google Scholar
- 8.R. Cramer. Personal communication Google Scholar
- 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.Y. Desmedt. Society and group oriented cryptography: a new concept. In Advances of Cryptology-Crypto’ 87 Google Scholar
- 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.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.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.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
- 21.M. Karchmer and A. Wigderson. On span programs In Proc. of 8 th annual Complexity Theory Conference, pp 102–111, 1993.Google Scholar
- 24.B. King. “Randomness Required for Linear Threshold Sharing Schemes Defined over Any Finite Abelian Group”. In ACISP 2001. pp. 376–391.Google Scholar