Extractors for Low-Weight Affine Sources

We give polynomial time computable extractors for low-weight affine sources. A distribution is affine if it samples a random points from some unknown low dimensional subspace of F₂ 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,isin, our extractors can extract almost all the entropy from weight k^isin 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 condensers for affine somewhere random sources.