## From Affine to Two-Source Extractors via Approximate Duality

by Eli Ben-Sasson and Noga Zewi

Letting $E:\{0,1\}^n\to\bitset^m$ denote the extractor for affine sources and $IP$ denote the inner produce modulo 2, two constructions that have the form $E_2(x,y)=IP(h(x),h(y))$ are presented: (1) $h(z)=z E(z)$, and (2) $h$ is a 1-1 mapping to $\bitset^{n-m}$ to $E^{-1}(v_o)$, where $v_0$ is an arbitrary value such that $|E^{-1}(v_o)| \geq 2^{n-m}$. The analysis first shows that, for these choices of $h$, the mapping $E_2$ is a two-source disperser; and next that, for any $h$, if $E_2$ is a two-source disperser then it is a two-source extractor. The exact parameters in the latter claim depend on a (new) Approximate Duality Conjecture.