The Generic Complexity of Index - Search

This paper is concerned with the complexity of generic algorithms solving so-called index-search problems such as the discrete logarithm problem in a cyclic group. The index-search problem was deened as follows: given a list (a i) i2I of elements of some set S, and given an element b of S, nd an i 2 I such that b = a i. The notion of a generic algorithm on the other hand was introduced by Shoup. Intuitively, a generic algorithm is a general-purpose algorithm which does not exploit any particular property of the representation of the objects, for instance of the group elements. By using purely information-theoretic arguments , a general lower bound on the complexity of generic index-search problems is proved. Some implications of this result are the optimality of the baby-step giant-step algorithm in many situations, Shoup's result concerning the hardness of the generic discrete logarithm problem, and a complete characterization of groups for which breaking the Diie-Hellman key-distribution protocol is computationally equivalent in a generic sense to computing discrete logarithms in the same group: this holds exactly for groups whose order is not divisible by a large multiple prime factor.