Accelerating the Distributed Multiplication Protocol with Applications to the Distributed Miller-Rabin Primality Test

In the light of information security it is highly desirable to avoid a “single point of failure” because this would be an attractive target for attackers. Cryptographic protocols for distributed computations are important techniques in pursuing this goal. An essential module in this context is the secure multiparty multiplication of two polynomially shared values over Zq with a public prime number q. The multiplication protocol of Gennaro, Rabin and Rabin (1998) is considered as the best protocol for this purpose. It requires a complexity of O(nk log n + nk) bit operations per player, where k is the bit size of the prime q and n is the number of players. The present paper reduces this complexity to O(nk + nk) with unaltered communication and round complexities. This improvement is possible by a loan from the field of numerical analysis, namely by the use of Newton’s classical interpolation formula. The distributed version of the famous probabilistic primality test of Miller and Rabin is built of several modules, which depend on distributed multiplications. Applications of the new method to these modules is studied and its importance for distributed signatures is outlined.