Deterministic extractors for affine sources over large fields

Abstract

An (n,k)-affine source over a finite field $\mathbb{F}$ is a random variable X = (X ₁,..., X _n ) ∈ $\mathbb{F}^n$, 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 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 $\mathbb{F}$ of size larger than n ²⁰, 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 |$\mathbb{F}$|.

• 2. (For k=1): For any n and any $\mathbb{F}$ of size larger than n ^c, 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, δ>0 is an arbitrary small constant, and c is a constant depending on δ.