A New Approach to Affine Extractors and Dispersers

We study the problem of constructing affine extractors over GF(2). Previously the only known construction that can handle sources with arbitrarily linear entropy is due to Bourgain (and a slight modification by Yehudayoff), which makes extensive use of complicated inequality manipulations and relies on a careful choice of a polynomial. In this paper we give a new and conceptually much cleaner construction of affine extractors for linear entropy sources that outputs a constant fractionof the entropy with exponentially small error. This matches theprevious best result of Bourgain. The extractor can be pushed tohandle affine sources with entropy n/√(log n log n). This slightly improves Bourgain's result andmatches the recent result of Yehudayoff. We also give a zero-error disperser for affine sources with entropy n/√(log n) that outputs n
^Ω(1)
bits. This improves previousconstructions of affine dispersers that output more than 1 bit. In contrast to Bourgain's construction, our construction mainly uses extractor machinery and basic properties of polynomials. Some of our techniques may be of independent interest.