Dispersers for Affine Sources with Sub-polynomial Entropy

Abstract

We construct an explicit disperser for affine sources over $\F_2^n$ with entropy $k=2^{\log^{0.9} n}=n^{o(1)}$. This is a polynomial time computable function $D:\F_2^n \ar \B$ such that for every affine space $V$ of $\F_2^n$ that has dimension at least $k$, $D(V)=\set{0,1}$. This improves the best previous construction of Ben-Sasson and Kop party (STOC 2009) that achieved $k = \Omega(n^{4/5})$.Our technique follows a high level approach that was developed in Barak, Kindler, Shaltiel, Sudakov and Wigderson (J. ACM 2010) and Barak, Rao, Shaltiel and Wigderson (STOC 2006) in the context of dispersers for two independent general sources. The main steps are:\begin{itemize}\item Adjust the high level approach to make it suitable for affine sources. \item Implement a ``challenge-response game'' for affine sources (in the spirit of the two aforementioned papers that introduced such games for two independent general sources).\item In order to implement the game, we construct extractors for affine block-wise sources. For this we use ideas and components by Rao (CCC 2009). \item Combining the three items above, we obtain dispersers for affine sources with entropy larger than $\sqrt{n}$.We use a recursive win-win analysis in the spirit of Rein gold, Shaltiel and Wigderson (SICOMP 2006) and Barak, Rao, Shaltiel and Wigderson (STOC 2006) to get affine dispersers with entropy less than $\sqrt{n}$.\end{itemize}