Let G be an arbitrary cyclic group with generator g and order |G| with known factorization. G could be the subgroup generated by g within a larger group H. Based on an assumption about the existence of smooth numbers in short intervals, we prove that breaking the Diffie-Hellman protocol for G and base g is equivalent to computing discrete logarithms in G to the base g when a certain side information string S of length 2 log |G| is given, where S depends only on |G| but not on the definition of G and appears to be of no help for computing discrete logarithms in G. If every prime factor p of |G| is such that one of a list of expressions in p, including p − 1 and p + 1, is smooth for an appropriate smoothness bound, then S can efficiently be constructed and therefore breaking the Diffie-Hellman protocol is equivalent to computing discrete logarithms.
CITATION STYLE
Maurer, U. M. (1994). Towards the equivalence of breaking the diffie-hellman protocol and computing discrete logarithms. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 839 LNCS, pp. 271–281). Springer Verlag. https://doi.org/10.1007/3-540-48658-5_26
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