Computational and Statistical Indistinguishabilities

We prove that a pair of polynomially samplable distributions are statistically indistinguishable if and only if no polynomial size circuits relative to NP sets (P NP nu -distinguishers) can tell them apart. As one application of this observation, we classify "zero-knowledge" notions that are used for interactive protocols. 1. Introduction For any pair of probability distributions, we say that they are computationally indistinguishable [GM84, Yao82] if no polynomial size circuits (which are called P nu -distinguishers) can tell them apart, and we say that they are statistically indistinguishable [GMR89] if no distinguishers (that could be infinitely powerful) can tell them apart. (See Section 2.1 for the precise definition.) Intuitively, a pair of statistically indistinguishable distributions are "statistically" so close to each other that no one can find their difference, while a pair of computationally indistinguishable distributions may be statistically different, but such a dif..