By Dario Catalano, Ronald Cramer, Ivan Damgard, Giovanni Di Crescenzo, David Pointcheval, Tsuyoshi Takagi

The target of this article is to regard chosen issues of the topic of latest cryptology, established in 5 fairly autonomous yet similar subject matters: effective allotted computation modulo a shared mystery, multiparty computation, sleek cryptography, provable safeguard for public key schemes, and effective and safe public-key cryptosystems.

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48] T. Rabin. A Simpliﬁed Approach to Threshold and Proactive RSA. 1462, Springer-Verlag, 1998. [49] R. Rivest, A. Shamir and L. Adelman. A Method for Obtaining Digital Signature and Public Key Cryptosystems. Comm. of ACM, 21 (1978), pp. 120–126. [50] A. Schrijver. Theory of Linear and Integer Programming. John Wiley & Sons. 1986. [51] A. Shamir. How to share a secret . In Communications of the ACM 22 (11) pages 612–613, 1979. [52] V. Shoup. Practical Threshold Signatures. 1807, Springer-Verlag, 2000.

Frankel. A Practical protocol for large group oriented networks. In Advances in Cryptology–Eurocrypt’89, Lecture Notes in Computer Science Vol. 434, pp. 56–61, Springer-Verlag, 1990. [25] Y. Frankel, P. Gemmell, P. Mackenzie, and M. Yung. Optimal Resilience Proactive Public-Key Cryptosystems. 384–393, IEEE Computer Society Press, 1997. [26] Y. Frankel, P. Gemmell, and M. Yung. Witness-based Cryptographic Program Checking and Robust Function Sharing. 499–508, ACM Press, 1996. [27] Y. Frankel, P.

Assume that a set of n players wants to establish whether a shared value a belongs to a set of (shared) integers b1 , . . , bm . A simple strategy to solve this problem is to check if there is a bj for which a ≡ bj mod p holds. To perform this check in a distributed way one may simply compute (for each bj ) the value a − bj mod p, multiply it with a jointly generated random element and check if the obtained result is zero or not. Unfortunately, however, this solution does not quite solve the problem in our setting.