Computational Indistinguishability: A Sample Hierarchy

Abstract

We consider the existence of pairs of probability ensembles that may be efficiently distinguished given k samples, but cannot be efficiently distinguished given k′<k samples. It is well known that in any such pair of ensembles it cannot be that both are efficiently computable (and that such phenomena cannot exist for nonuniform classes of distinguishers, say, polynomial-size circuits). It was also known that there exist pairs of ensembles that may be efficiently distinguished based on two samples, but cannot be efficiently distinguished, based on a single sample. In contrast, it was not known whether the distinguishing power increases when one moves from two samples to polynomially-many samples. We show the existence of pairs of ensembles which may be efficiently distinguished given k+1 samples but cannot be efficiently distinguished given k samples, where k can be any function bounded above by a polynomial in the security parameter. In the course of establishing the above result, we prove several technical lemmas regarding polynomials and graphs. We believe that these may be of independent interest.