Deterministic extractors for affine sources over large fields

• Authors: Ariel Gabizon;Ran Raz
• Venue: Combinatorica
• Year: 2008

An (n,k)-affine source over a finite field $$\mathbb{F}$$is a random variable X = (X1,..., Xn) ∈ $$\mathbb{F}^n$$, which is uniformly distributed over an (unknown) k-dimensional affine subspace of $$\mathbb{F}^n$$. We show how to (deterministically) extract practically all the randomness from affine sources, for any field of size larger than nc (where c is a large enough constant). Our main results are as follows: (For arbitraryk): For any n,k and any $$\mathbb{F}$$of size larger than n20, we give an explicit construction for a function D : $$\mathbb{F}^n $$→ $$\mathbb{F}^{k - 1}$$, such that for any (n,k)-affine source X over $$\mathbb{F}$$, the distribution of D(X) is ∊-close to uniform, where ∊ is polynomially small in |$$\mathbb{F}$$|. (Fork=1): For any n and any $$\mathbb{F}$$ of size larger than nc, we give an explicit construction for a function D: $$\mathbb{F}^n \to \{ 0,1\} ^{(1 - \delta )log_2 |\mathbb{F}|}$$, such that for any (n, 1)-affine source X over $$\mathbb{F}$$, the distribution of D(X) is ∊-close to uniform, where ∊ is polynomially small in |$$\mathbb{F}$$|. Here, δ0 is an arbitrary small constant, and c is a constant depending on δ.