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.
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Kiltz, E. (2001). A Tool Box of Cryptographic Functions Related to the Diffie-Hellman Function. In: Rangan, C.P., Ding, C. (eds) Progress in Cryptology — INDOCRYPT 2001. INDOCRYPT 2001. Lecture Notes in Computer Science, vol 2247. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45311-3_32
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DOI: https://doi.org/10.1007/3-540-45311-3_32
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