Extractors for Polynomials Sources over Constant-Size Fields of Small Characteristic

Abstract

A polynomial source of randomness over $\mathbb F_q^n$ is a random variable X = f(Z) where f is a polynomial map and Z is a random variable distributed uniformly on $\mathbb F_q^r$ for some integer r. The three main parameters of interest associated with a polynomial source are the field size q, the (total) degree D of the map f, and the “rate” k which specifies how many different values does the random variable X take, where rate k means X is supported on at least q ^k different values. For simplicity we call X a (q,D,k)-source.

Informally, an extractor for (q,D,k)-sources is a deterministic function $E:\mathbb F_q^n\to \left \{{0,1} \right \}^m$ such that the distribution of the random variable E(X) is close to uniform on $\left \{{0,1} \right \}^m$ for any (q,D,k)-source X. Generally speaking, the problem of constructing deterministic extractors for such sources becomes harder as q and k decrease and as D grows larger.

The only previous work of [Dvir et al., FOCS 2007] construct extractors for such sources when q ≫ n. In particular, even for D = 2 no constructions were known for any fixed finite field.

In this work we construct for the first time extractors for (q,D,k)-sources for constant-size fields. Our proof builds on the work of DeVos and Gabizon [CCC 2010] on extractors for affine sources, with two notable additions (described below). Like [DG10], our result makes crucial use of a theorem of Hou, Leung and Xiang [J. Number Theory 2002] giving a lower bound on the dimension of products of subspaces. The key insights that enable us to extend these results to the case of polynomial sources of degree D greater than 1 are

• 1 A source with support size q ^k must have a linear span of dimension at least k, and in the setting of low-degree polynomial sources it suffices to increase the dimension of this linear span.

• 2 Distinct Frobenius automorphisms of a (single) low-degree polynomial source are ‘pseudo-independent’ in the following sense: Taking the product of distinct automorphisms (of the very same source) increases the dimension of the linear span of the source.