Deterministic extractors for affine sources over large fields

An (n, k)-affine source over a finite field F is a random variable X = (X₁, ..., X_n) ε F_n, which is uniformly distributed over an (unknown) k-dimensional affine subspace of F_n. We show how to (deterministically) extract practically all the randomness from affine 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 n₂₀, we give an explicit construction for a function D : F_n → F_k-1, such that for any (n, k)-affine source X over F, the distribution of D(X) is ε-close to uniform, where ε is polynomially small in |F|. 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 : Fⁿ → {0,1}^(1-σ)log₂|F|, such that for any (n, 1)-affine source X over F, the distribution of D(X) is ε-close to uniform, where ε is polynomially small in |F|. Here, δ > 0 is an arbitrary small constant, and c is a constant depending on δ.