Distinguishing SAT from polynomial-size circuits, through black-box queries

We may believe SAT does not have small Boolean circuits. But is it possible that some language with small circuits looks indistinguishable from SAT to every polynomial-time bounded adversary? We rule out this possibility. More precisely, assuming SAT does not have small circuits, we show that for every language A with small circuits, there exists a probabilistic polynomial-time algorithm that makes black-box queries to A, and produces, for a given input length, a Boolean formula on which A differs from SAT. A key step for obtaining this result is a new proof of the main result by Gutfreund, Shaltiel, and Ta-Shma reducing average-case hardness to worst-case hardness via uniform adversaries that know the algorithm they fool. The new adversary we construct has the feature of being black-box on the algorithm it fools, so it makes sense in the non-uniform setting as well. Our proof makes use of a refined analysis of the learning algorithm of Bshouty et al