Deterministic Extractors for Affine Sources over Large Fields

Abstract

An (n, k)-affine source over a finite field F is a random variable X = (X/sub 1/, ..., X/sub n/) /spl epsi/ F/sub n/, which is uniformly distributed over an (unknown) k-dimensional affine subspace of F/sub n/. We show how to (deterministically) extract practically all the randomness from affine sources, for any field of size larger than n/sup 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/sub 20/, we give an explicit construction for a function D : F/sub n/ /spl rarr/ F/sub k-1/, such that for any (n, k)-affine source X over F, the distribution of D(X) is /spl epsiv/-close to uniform, where /spl epsiv/ is polynomially small in |F|. 2. (For k = 1): For any n and any F of size larger than n/sup c/, we give an explicit construction for a function D : F/sup n/ /spl rarr/ {0,1}/sup (1-/spl sigma/)log//sub 2/|F|, such that for any (n, 1)-affine source X over F, the distribution of D(X) is /spl epsiv/-close to uniform, where /spl epsiv/ is polynomially small in |F|. Here, /spl delta/ > 0 is an arbitrary small constant, and c is a constant depending on /spl delta/.