A Tool Box of Cryptographic Functions Related to the Diffie-Hellman Function

Abstract

Given a cyclic group G and a generator g, the Diffie-Hellman function (DH) maps two group elements (g ^a, g ^b) to g ^ab. For many groups G this function is assumed to be hard to compute. We generalize this function to the P-Diffie-Hellman function (P-DH) that maps two group elements (g ^a, g ^b) to g ^P(a,b) for a (non-linear) polynomial P in a and b. In this paper we show that computing DH is computationally equivalent to computing P-DH. In addition we study the corresponding decision problem. In sharp contrast to the computational case the decision problems for DH and P-DH can be shown to be not generically equivalent for most polynomials P. Furthermore we show that there is no generic algorithm that computes or decides the P-DH function in polynomial time.