Given a cyclic group G and a generator g, the Diffie-Hellman function (DH) maps two group elements (ga, gb) to gab. 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 (ga, gb) to gP(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.
CITATION STYLE
Kiltz, E. (2001). A tool box of cryptographic functions related to the Diffie-Hellman function. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 2247, pp. 339–349). Springer Verlag. https://doi.org/10.1007/3-540-45311-3_32
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