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Affine extractors over large fields with exponential error

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Abstract

We describe a construction of explicit affine extractors over large finite fields with exponentially small error and linear output length. Our construction relies on a deep theorem of Deligne giving tight estimates for exponential sums over smooth varieties in high dimensions.

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References

  • B. Barak, A. Rao, R. Shaltiel & A. Wigderson (2006). 2-source dispersers for sub-polynomial entropy and Ramsey graphs beating the Frankl-Wilson construction. In Proceedings of the thirty-eighth annual ACM symposium on Theory of computing, 671–680. ACM Press, New York, NY, USA. ISBN 1-59593-134-1.

  • E. Ben-Sasson & A. Gabizon (2012). Extractors for Polynomials Sources over Constant-Size Fields of Small Characteristic. In APPROX-RANDOM, Lecture Notes in Computer Science, 399–410. Springer.

  • Bourgain J. (2007) On the construction of affine extractors. Geometric And Functional Analysis 17(1): 33–57

    Article  MathSciNet  MATH  Google Scholar 

  • J. Bourgain (2010). On Exponential Sums in Finite Fields. In An Irregular Mind, I. Bárány, J. Solymosi & G. Sági, editors, volume 21 of Bolyai Society Mathematical Studies, 219–242. Springer Berlin Heidelberg. ISBN 978-3-642-14443-1.

  • Deligne P. (1974) La conjecture de Weil. I, Inst. Hautes -Etudes Sci. Publ. Math. 43: 273–307

    Article  MathSciNet  MATH  Google Scholar 

  • Z. Dvir (2012). Extractors for varieties. Comput. Complex. 21, 515–572. ISSN 1016-3328.

  • Z. Dvir, A. Gabizon & A. Wigderson (2009). Extractors And Rank Extractors For Polynomial Sources. Comput. Complex. 18(1), 1–58. ISSN 1016-3328.

  • P. Erdös (1935). On the normal number of prime factors of p−1 and some related problems concerning Euler’s \({\varphi}\)-function. Quart. J. Math., Oxford Ser. 205–213.

  • Gabizon A., Raz R. (2008) Deterministic extractors for affine sources over large fields. Combinatorica 28(4): 415–440

    Article  MathSciNet  MATH  Google Scholar 

  • G.H. Hardy & E.M Wright (1979). An Introduction to the Theory of Numbers. Oxford Science Pub.

  • X. Li (2011). A New Approach to Affine Extractors and Dispersers. In IEEE Conference on Computational Complexity, 137–147. IEEE Computer Society.

  • A. Lubotzky, R. Phillips & P. Sarnak (1988). Ramanujan graphs. Combinatorica 8(3), 261–277.

  • Moreno O., Kumar P. (1993) Minimum distance bounds for cyclic codes and Deligne’s theorem. IEEE Transactions on Information Theory 39(5): 1524–1534

    Article  MathSciNet  MATH  Google Scholar 

  • A. Rao (2007). An Exposition of Bourgain+s 2-Source Extractor. In Electronic Colloquium on Computational Complexity (ECCC), volume 14, 034.

  • Reingold O., Vadhan S., Wigderson A. (2002) Entropy waves, the zig-zag graph product, and new constant-degree expanders. Ann. Math 155(1): 157–187

    Article  MathSciNet  MATH  Google Scholar 

  • A. Yehudayoff (2011). Affine extractors over prime fields. Combinatorica 31(2), 245–256. ISSN 0209-9683. URL http://dx.doi.org/10.1007/s00493-011-2604-9.

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Authors and Affiliations

  1. School of Mathematics, Institute for Advanced Study, Princeton, NJ, USA

    Jean Bourgain

  2. Department of Computer Science, Princeton University, Princeton, NJ, USA

    Zeev Dvir

  3. Department of Mathematics, Princeton University, Princeton, NJ, USA

    Zeev Dvir

  4. Department of Mathematics, The University of Texas at Austin, Austin, TX, USA

    Ethan Leeman

Authors
  1. Jean Bourgain
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  2. Zeev Dvir
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  3. Ethan Leeman
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Correspondence to Zeev Dvir.

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Bourgain, J., Dvir, Z. & Leeman, E. Affine extractors over large fields with exponential error. comput. complex. 25, 921–931 (2016). https://doi.org/10.1007/s00037-015-0108-5

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  • DOI: https://doi.org/10.1007/s00037-015-0108-5

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