Deterministic extractors for affine sources over large fields

An (n,k)-affine source over a finite field F is a random variable X=(X-1,..,X-n)is an element of F-n which is uniformly distributed over an (unknown) k-dimensional affine subspace of Fn. We show how to (deterministically) extract practically all the randomness from sources, for any field of size larger than n(c) (where c is a large enough constant). Our main results are as follows: 1. (For arbitrary k): For any n,k and any F of size larger than n(20), we give an explicit construction for a function D:Fn -> Fk(-1), such that for any (n,k)-affine source X over F, the distribution of D(X) is -close to uniform, where is polynomially small in F, the distribution D(X) is epsilon-close to uniform, where epsilon is polynomially small in vertical bar F vertical bar. 2. (For k = 1): For any n and any F of size larger than n(c), we give an explicit construction for a function D:F-n ->+{0,1}((1-delta)log2 vertical bar F vertical bar), such that for any (n,1)-affine source X over F. the distribution of D(X) is epsilon-close to uniform, where epsilon is polynomially small in vertical bar F vertical bar Here, delta > 0 is an arbitrary small constant, and c is a constant depending on delta.