Improved Extractors for Recognizable and Algebraic Sources

We study the task of seedless randomness extraction from recognizable sources, which are uniform distributions over sets of the form {x : f(x) = 1} for functions f in some specified class C. We give two simple methods for constructing seedless extractors for C-recognizable sources. Our first method shows that if C admits XOR amplification, then we can construct a seedless extractor for C-recognizable sources by using a mildly hard function for C as a black box. By exploiting this reduction, we give polynomial-time, seedless randomness extractors for three natural recognizable sources: (1) constant-degree algebraic sources over any prime field, where constant-degree algebraic sources are uniform distributions over the set of zeros of a system of constant degree polynomials; (2) sources recognizable by randomized multiparty communication protocols of cn bits, where c>0 is a small enough constant; (3) halfspace sources, or sources recognizable by linear threshold functions. In particular, the new extractor for each of these three sources has linear output length and exponentially small error for min-entropy k >= (1-alpha)n, where alpha>0 is a small enough constant. Our second method shows that a seed-extending pseudorandom generator with exponentially small error for C yields an extractor with exponentially small error for C-recognizable sources, improving a reduction by Kinne, Melkebeek, and Shaltiel [Kinne et al., 2012]. Using the hardness of the parity function against AC^0 [Håstad, 1987], we significantly improve Shaltiel’s extractor [Shaltiel, 2011] for AC^0-recognizable sources. Finally, assuming sufficiently strong one-way permutations, we construct seedless extractors for sources recognizable by BPP algorithms, and these extractors run in quasi-polynomial time.