, 31:245 | Cite as

Affine extractors over prime fields



An affine extractor is a map that is balanced on every affine subspace of large enough dimension. We construct an explicit affine extractor AE from \(\mathbb{F}^n \) to \(\mathbb{F}\), \(\mathbb{F}\) a prime field, so that AE(x) is exponentially close to uniform when x is chosen uniformly at random from an arbitrary affine subspace of \(\mathbb{F}^n \) of dimension at least δn, 0<δ≤1 a constant. Previously, Bourgain constructed such affine extractors when the size of \(\mathbb{F}\) is two. Our construction is in the spirit of but different than Bourgain’s construction. This allows for simpler analysis and better quantitative results.

Mathematics Subject Classification (2000)



  1. [1]
    J. Bourgain: On the construction of affine extractors, Geometric and Functional Analysis 17(1) (2007), 33–57.MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    J. Bourgain: Multilinear exponential sum bounds with optimal entropy assignments, Geometric and Functional Analysis 18(5) (2009), 1477–1502.MathSciNetCrossRefGoogle Scholar
  3. [3]
    J. Bourgain: On exponential sums in finite fields, manuscript, 2009.Google Scholar
  4. [4]
    J. Bourgain, A. A. Glibichuk and S. V. Konyagin: Estimates for the number of sums and products and for exponential sums in fields of prime order, Journal of the London Mathematical Society 73(2) (2006), 380–398.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    A. Gabizon and R. Raz: Deterministic extractors for affine sources over large fields, in: Proceedings of the 46th FOCS, pages 407–418, 2005.Google Scholar
  6. [6]
    O. Goldreich: Three XOR-lemmas — an exposition, manuscript, 1997.Google Scholar
  7. [7]
    A. Rao: Extractors for low-weight affine sources, in: Proceedings of CCC, pages 95–101, 2009.Google Scholar
  8. [8]
    R. Raz and A. Yehudayoff: Multilinear formulas, maximal-partition discrepancy and mixed-sources extractors; in: proceedings of FOCS, pages 273–282, 2008.Google Scholar

Copyright information

© János Bolyai Mathematical Society and Springer Verlag 2011

Authors and Affiliations

  1. 1.Department of MathematicsTechnion — Israel Institute of TechnologyHaifaIsrael

Personalised recommendations