Computing inverses over a shared secret modulus

Reference: Advances in Cryptology - EUROCRYPT 2000. Lecture Notes in Computer Science, vol. 1807, Pages 190-206, Springer-Verlag, 2000.

Abstract: We discuss the following problem: Given an integer phi, shared secretly among n players and a prime number e, how can the players efficiently compute a sharing of e^{-1} mod phi. The most interesting case is when phi is the Euler function of a known RSA modulus N, phi = phi(N). The problem has several applications, among which the construction of threshold variants for two recent signature schemes proposed by Gennaro-Halevi-Rabin and Cramer-Shoup.

We present new and efficient protocols to solve this problem, improving over previous solutions by Boneh-Franklin and Frankel et al. Our basic protocol (secure against honest but curious players) requires only two rounds of communication and a single GCD computation. The robust protocol (secure against malicious players) adds only a couple of rounds and a few modular exponentiations to the computation.

Keywords: Mudular inversion, Threshold computation, RSA Signatures.