Summary: Extractors for Low-Weight Affine Sources
Institute for Advanced Study
January 22, 2008
We give polynomial time computable extractors for low-weight affince sources. A distribution
is affine if it samples a random points from some unknown low dimensional subspace of Fn
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, , our extractors can extract almost all the entropy from weight k
affine sources of dimension k, as long as k > logc
n, with error 2-k(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.
Keywords: Extractors, Affine Sources, Exposure Resilient Cryptography