Extractors for Samplable Distributions from the Two-Source Extractor Recipe

Seedless extractors for samplable distributions were first constructed under a very strong complexity-theoretic hardness assumption: that E=DTIME(2O(n)) is hard for exponential-size circuits with oracle access to a fixed level of the polynomial hierarchy. That construction applies to sources with min-entropy k=(1-Y)n for an arbitrarily small constant Y>0. Subsequent works weakened the hardness assumption and improved the min-entropy threshold to k=n1-Y and then to k=nω(1), though these improvements again relied on hardness against circuits with oracle access to the polynomial hierarchy. We introduce a new approach to constructing extractors for samplable distributions, inspired by constructions of two-source extractors. Our approach relies on a new, incomparable hardness assumption involving only deterministic circuits, and reduces the task of constructing extractors for samplable distributions to that of constructing explicit non-malleable extractors with short seed length. The new assumption has the same flavor as the classic assumption that E is hard for exponential-size circuits. Specifically, we assume that there exists a constant 0<α<1 such that for every constant Chard≥ 1, there exist a constant Ceasy and a problem in DTIME(2Ceasyn) that is not in DTIME(2Chardn)/2α n. A notable feature of this assumption is that the adversary is allowed to run in time exceeding 2n, while still being restricted to fewer than 2n bits of nonuniformity. Under this assumption, we construct an explicit extractor for samplable distributions with min-entropy k=O(logn), matching the threshold achieved by the probabilistic method. More precisely, for every constant c≥ 1 and every constant ϵ >0, there exists a constant c′ and an explicit extractor Ext:{0,1}n→{0,1} with error ϵ for distributions of min-entropy at least c′logn that are samplable by circuits of size nc. The key observation underlying our construction is that for a samplable source, the set of bad seeds for a non-malleable extractor is efficiently recognizable. We use this observation to show that, in the relevant two-source extractor constructions, the second source can be replaced by the truth table of a sufficiently hard function. This yields an unexpected connection between two-source extractors and extractors for samplable distributions, paralleling the classical connection between extractors and pseudorandom generators in the opposite direction.