## Extractors for Low-Weight Affine Sources

#### Anup RaoInstitute for Advanced Study

We give polynomial time computable extractors for low-weight affine sources. A distribution is affine if it samples a random point from some unknown low dimensional subspace of F^n_2 . 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,e , our extractors can extract almost all the entropy from weight k affine sources of dimension k, as long as k > log^c n, with error 2^{-k^Omega(1)} . This gives 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.

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