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Lower Bounds on Generic Algorithms in Groups

Ueli Maurer and Stefan Wolf

In this paper we consider generic algorithms for computational problems in cyclic groups. The model of a generic algorithm was proposed by Shoup at Eurocrypt '97. A generic algorithm is a general-purpose algorithm that does not make use of any particular property of the representation of the group elements. Shoup proved the hardness of the discrete logarithm problem and the Diffie-Hellman problem with respect to such algorithms for groups whose order contains a large prime factor. By extending Shoup's technique we prove lower bounds on the complexity of generic algorithms solving different problems in cyclic groups, and in particular of a generic reduction of the discrete logarithm problem to the Diffie-Hellmanproblem. It is shown that the two problems are not computationally equivalent in a generic sense for groups whose orders contain a multiple large prime factor. This complements earlier results which stated this equivalence for all other groups. Furthermore, it is shown that no generic algorithm exists that computes $p$-th roots efficiently in a group whose order is divisible by $p^2$ if $p$ is a large prime.