## Circuit size lower bounds and #SAT upper bounds through a general framework

- Alexander Golovnev, Alexander S. Kulikov, Alexander Smal, Suguru Tamaki
- Electronic Colloquium on Computational Complexity
- 2016

2 Excerpts

- Published 2013 in Theory of Computing

A polynomial source of randomness over Fq is a random variable X = f (Z) where f is a polynomial map and Z is a random variable distributed uniformly over Fq for some integer r. The three main parameters of interest associated with a polynomial source are the order q of the field, the (total) degree D of the map f , and the base-q logarithm of the size of the range of f over inputs in Fq, denoted by k. For simplicity we call X a (q,D,k)-source. Informally, an extractor for (q,D,k)-sources is a function E : Fq→{0,1} m such that the distribution of the random variable E(X) is close to uniform over {0,1} for any (q,D,k)source X . Generally speaking, the problem of constructing extractors for such sources becomes harder as q and k decrease and as D increases. A rather large number of recent ∗A conference version of this paper appeared in the Proceedings of RANDOM 2012 [1]. †Supported by funding from the European Community’s Seventh Framework Programme (FP7/2007-2013) under grant agreement number 240258. ‡Supported by funding from the European Community’s Seventh Framework Programme (FP7/2007-2013) under grant agreement number 240258. ACM Classification: F.0, G.2.0, G.3 AMS Classification: 11T06, 11T23, 12F05

@article{BenSasson2013ExtractorsFP,
title={Extractors for Polynomial Sources over Fields of Constant Order and Small Characteristic},
author={Eli Ben-Sasson and Ariel Gabizon},
journal={Theory of Computing},
year={2013},
volume={9},
pages={665-683}
}