Extractors for Polynomial Sources over Fields of Constant Order and Small Characteristic

Abstract

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

DOI: 10.4086/toc.2013.v009a021

Extracted Key Phrases

Cite this paper

@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} }