Computational Indistinguishability: Algorithms vs. Circuits

We present a simple proof to the existence of a probability ensemble with tiny support which cannot be distinguished from the uniform ensemble by any recursive computation. Since the support is tiny (i.e, sub-polynomial), this ensemble can be distinguished from the uniform ensemble by a (non-uniform) family of small circuits. It also provides an example of an ensemble which cannot be (recursively) distinguished from the uniform by one sample, but can be so distinguished by two samples. In case we only wish to fool probabilistic polynomial-time algorithms the ensemble can be constructed in slightly super-exponential time.