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Revision #1 to TR18-110 | 11th September 2018 22:12

#### Improved Extractors for Recognizable and Algebraic Sources

Revision #1
Authors: Fu Li, David Zuckerman
Accepted on: 11th September 2018 22:12
Keywords:

Abstract:

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-c)n, where c > 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 $AC^0$ [Has87], we significantly improve Shaltiel’s extractor [Sha11] 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.

Changes to previous version:

Added new extractors for sources recognizable by communication protocols and LTFs, and corrected minor typos.

### Paper:

TR18-110 | 4th June 2018 21:30

#### Improved Extractors for Recognizable and Algebraic Sources

TR18-110
Authors: Fu Li, David Zuckerman
Publication: 4th June 2018 21:34
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 algebraic sources over any prime field, where algebraic sources are uniform distributions over the set of solutions of a system of low degree polynomials. In particular, the new extractor has linear output length and exponentially small error for min-entropy $k \geq (1 - \alpha)n$, 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 [KvMS12]. Using the hardness of the parity function against $AC^0$ [Has87], we significantly improve Shaltiel’s extractor [Sha11] 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.