Computational indistinguishability: a sample hierarchy

We consider the existence of pairs of probability ensembles which may be efficiently distinguished from each other given k samples but cannot be efficiently distinguished given k'<k samples. If 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 non-uniform classes of distinguishers, say, polynomial-size circuits). It was also known that there exist pairs of ensembles which 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 course of establishing the above result, we prove several technical lemmas regarding polynomials and graphs. We believe that these may be of independent interest.