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. 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. 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.



Linear Span Full Version Dimension Expansion Support Size Individual Degree 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Eli Ben-Sasson
    • 1
    • 2
  • Ariel Gabizon
    • 1
  1. 1.Department of Computer ScienceTechnionHaifaIsrael
  2. 2.Microsoft Research New-EnglandCambridgeUSA

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