Extractors for Low-Weight Affine Sources

We give polynomial time computable extractors for \emph{low-weight affince sources}. A distribution is affine if it samples a random points from some unknown low dimensional subspace of $\mathbb{F}_2^n$. A distribution is low weight affine if the corresponding linear space has a basis of low-weight vectors. Low-weight affine sources are thus a generalization of the well studied models of bit-fixing sources (which are just weight $1$ affine sources). For universal constants $c,\epsilon$, our extractors can extract almost all the entropy from weight $k^{\epsilon}$ affine sources of dimension $k$, as long as $k ≫ \log ^c n$, with error $2^{-k^{\Omega(1)}}$. In particular, our results give new extractors for low entropy bit-fixing sources, with exponentially small error, a parameter that is important for the application of these extractors to cryptography. Our techniques involve constructing new \emph{condensers} for \emph{affine somewhere random sources}.