### Deterministic Extractors for Affine Sources over Large Fields [chapter]

Ariel Gabizon
2010 Monographs in Theoretical Computer Science. An EATCS Series
An (n, k)-affine source over a finite field F is a random variable X = (X 1 , ..., X n ) ∈ 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 n 20 , we give an explicit construction for a
more » ... function D : F n → F k−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|. 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−δ) log 2 |F| , such that for any (n, 1)-affine source X over F, the distribution of D(X) is -close to uniform, where is polynomially small in |F|. Here, δ > 0 is an arbitrary small constant, and c is a constant depending on δ.