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

• Eli Ben-Sasson
• Ariel Gabizon
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7408)

## 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. 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.

## Keywords

Linear Span Full Version Dimension Expansion Support Size Individual Degree
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Preview

Unable to display preview. Download preview PDF.

## References

1. 1.
Ben-Sasson, E., Hoory, S., Rozenman, E., Vadhan, S., Wigderson, A.: Extractors for affine sources (2001) (unpublished Manuscript)Google Scholar
2. 2.
Ben-Sasson, E., Kopparty, S.: Affine dispersers from subspace polynomials. In: Proceedings of the 41st Annual ACM Symposium on Theory of Computing, pp. 65–74 (2009)Google Scholar
3. 3.
Ben-Sasson, E., Zewi, N.: From affine to two-source extractors via approximate duality. In: Fortnow, L., Vadhan, S.P. (eds.) STOC, pp. 177–186. ACM (2011)Google Scholar
4. 4.
Blum, N.: A boolean function requiring 3n network size. Theor. Comput. Sci. 28, 337–345 (1984)
5. 5.
Bourgain, J.: On the construction of affine extractors. Geometric & Functional Analysis 17(1), 33–57 (2007)
6. 6.
Chor, B., Goldreich, O.: Unbiased bits from sources of weak randomness and probabilistic communication complexity. SIAM Journal on Computing 17(2), 230–261 (1988); Special issue on cryptography
7. 7.
De, A., Watson, T.: Extractors and Lower Bounds for Locally Samplable Sources. In: Goldberg, L.A., Jansen, K., Ravi, R., Rolim, J.D.P. (eds.) APPROX/RANDOM 2011. LNCS, vol. 6845, pp. 483–494. Springer, Heidelberg (2011)
8. 8.
Demenkov, E., Kulikov, A.S.: An Elementary Proof of a 3no(n) Lower Bound on the Circuit Complexity of Affine Dispersers. In: Murlak, F., Sankowski, P. (eds.) MFCS 2011. LNCS, vol. 6907, pp. 256–265. Springer, Heidelberg (2011)
9. 9.
DeVos, M., Gabizon, A.: Simple affine extractors using dimension expansion. In: Proceedings of the 25th Annual IEEE Conference on Computational Complexity, p. 63 (2010)Google Scholar
10. 10.
Dvir, Z.: Extractors for varieties (2009)Google Scholar
11. 11.
Dvir, Z., Gabizon, A., Wigderson, A.: Extractors and rank extractors for polynomial sources. Computational Complexity 18(1), 1–58 (2009)
12. 12.
Dvir, Z., Lovett, S.: Subspace evasive sets. Electronic Colloquium on Computational Complexity (ECCC) 18, 139 (2011)Google Scholar
13. 13.
Gabizon, A., Raz, R.: Deterministic extractors for affine sources over large fields. Combinatorica 28(4), 415–440 (2008)
14. 14.
Guruswami, V.: Linear-algebraic list decoding of folded reed-solomon codes. In: IEEE Conference on Computational Complexity, pp. 77–85. IEEE Computer Society (2011)Google Scholar
15. 15.
Hou, X., Leung, K.H., Xiang, Q.: A generalization of an addition theorem of kneser. Journal of Number Theory 97, 1–9 (2002)
16. 16.
Li, X.: A new approach to affine extractors and dispersers (2011)Google Scholar
17. 17.
Lidl, R., Niederreiter, H.: Introduction to finite fields and their applications. Cambridge University Press, Cambridge (1994)
18. 18.
Shaltiel, R.: Dispersers for affine sources with sub-polynomial entropy. In: Ostrovsky, R. (ed.) FOCS, pp. 247–256. IEEE (2011)Google Scholar
19. 19.
Viola, E.: Extractors for circuit sources. Electronic Colloquium on Computational Complexity (ECCC) 18, 56 (2011)Google Scholar
20. 20.
von Neumann, J.: Various techniques used in connection with random digits. Applied Math Series 12, 36–38 (1951)Google Scholar
21. 21.
Weil, A.: On some exponential sums. Proc. Nat. Acad. Sci. USA 34, 204–207 (1948)
22. 22.
Yehudayoff, A.: Affine extractors over prime fields (2009) (manuscript)Google Scholar

© 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