In this paper, we discuss secure protocols for shared computation of algorithms associated with digital signature schemes based on discrete logarithms. Generic solutions to the problem of cooperatively computing arbitraty functions, though formally provable according to strict security notions, are inefficient in terms of communication--bits and rounds of interaction--; practical protocols for shared computation of particular functions, on the other hand, are often shown secure according to weaker notions of security. We propose efficient secure protocols to share the generation of keys and signatures in the digital signature schemes introduced by Schnorr (1989) and ElGamal (1985). The protocols are built on a protocol for non-interactive verifiable secret sharing (Feldman, 1987) and a novel construction for non-interactively multiplying secretly shared values. Together with the non-interactive protocols for shared generation of RSA signatures introduced by Desmedt and Frankel (1991), the results presented here show that practical signature schemes can be efficiently shared.

An extension of the notion of cryptographically strong pseudo-random generator to a distributed setting is proposed in this paper. Instead of a deterministic function to generate a pseudo-random bit string from a truly random shorter string, we have a deterministic secure protocol for a group of separate entities to compute a secretly shared pseudo-random string from a secretly shared and truly random shorter string. We propose a precise definition of this notion in terms of Yao's computational entropy and describe a concrete construction using Shamir's pseudo-random number generator. Several practical applications are also discussed.