On Generic Groups and Related Bilinear Problems

Groups with pairing are now considered as standard building blocks for cryptographic primitives. The security of schemes based on su ch groups relies on hypotheses related to the discrete logarithm problem. As the e ypotheses are not proved, one would like to have some positive security argument for them. It is usual to assess their security in the so called generic group model i ntroduced by Nechaev and Shoup. Over the time, this model has been extended in differ ent directions to cover new features. The relevance of this model is nevertheless subject to critic isms: in particular, the fact that the answer to any fresh query is a random bit stri ng is not what one expects from a usual group law. In this paper, we develop a generic group model with pairing wh ich generalizes all the models seen so far in the literature. We provide a gener al framework in order to prove difficulty assumptions in this setting. In order to imp rove the realism of this model, we introduce the notion of pseudo-random families of groups. We show how to reduce the security of a problem in such a family to the se curity of the same problem in the generic group model and to the security of an und erlying strong pseudo-random family of permutations.