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) = v} 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 solutions to 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 − α)n, where α > 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 [KvMS12]. Using the hardness of the parity function against AC0 [H̊as87], we significantly improve Shaltiel’s extractor [Sha11] for AC0-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. ∗Supported by NSF Grant CCF-1526952, NSF Grant CCF-1705028, and a Simons Investigator Award (#409864, David Zuckerman). ISSN 1433-8092 Electronic Colloquium on Computational Complexity, Revision 1 of Report No. 110 (2018)