skip to main content
research-article

Hard Functions for Low-Degree Polynomials over Prime Fields

Published:01 July 2013Publication History
Skip Abstract Section

Abstract

In this article, we present a new hardness amplification for low-degree polynomials over prime fields, namely, we prove that if some function is mildly hard to approximate by any low-degree polynomials then the sum of independent copies of the function is very hard to approximate by them. This result generalizes the XOR lemma for low-degree polynomials over the binary field given by Viola and Wigderson [2008]. The main technical contribution is the analysis of the Gowers norm over prime fields. For the analysis, we discuss a generalized low-degree test, which we call the Gowers test, for polynomials over prime fields, which is a natural generalization of that over the binary field given by Alon et al. [2003]. This Gowers test provides a new technique to analyze the Gowers norm over prime fields. Actually, the rejection probability of the Gowers test can be analyzed in the framework of Kaufman and Sudan [2008]. However, our analysis is self-contained and quantitatively better. By using our argument, we also prove the hardness of modulo functions for low-degree polynomials over prime fields.

References

  1. Noga Alon and Richard Beigel. 2001. Lower bounds for approximations by low degree polynomials over Zm. In Proceedings of the 16th IEEE Conference on Computational Complexity. 184--187. Google ScholarGoogle ScholarDigital LibraryDigital Library
  2. Noga Alon, Tali Kaufman, Michael Krivelevich, Simon Litsyn, and Dana Ron. 2003. Testing low-degree polynomials over GF(2). In Proceedings of the 6th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX’03), and the 7th International Workshop on Randomization and Approximation Techniques in Computer Science (RANDOM’03). 188--199.Google ScholarGoogle ScholarCross RefCross Ref
  3. Laszlo Babai, Lance Fortnow, and Carsten Lund. 1991. Non-deterministic exponential time has two-prover interactive protocols. Comput. Complex. 1, 1, 3--40.Google ScholarGoogle ScholarCross RefCross Ref
  4. Arnab Bhattacharyya, Swastik Kopparty, Grant Schoenebeck, Madhu Sudan, and David Zuckerman. 2010. Optimal testing of Reed-Muller codes. In Proceedings of the 51st Annual IEEE Symposium on Foundations of Computer Science. Google ScholarGoogle ScholarDigital LibraryDigital Library
  5. Manuel Blum, Michael Luby, and Ronitt Rubinfeld. 1993. Self-testing/correcting with applications to numerical problems. J. Comput. System Sci. 3, 549--595. Google ScholarGoogle ScholarDigital LibraryDigital Library
  6. Jean Bourgain. 2005. Estimation of certain exponential sums arising in complexity theory. Comptes Rendus Mathematique 340, 9, 627--631.Google ScholarGoogle ScholarCross RefCross Ref
  7. Arkadev Chattopadhyay. 2006. An improved bound on correlation between polynomials over Zm and MODq. Tech. rep. TR06-107, Electronic Colloquium on Computational Complexity.Google ScholarGoogle Scholar
  8. Uriel Feige, Shafi Goldwasser, Laszlo Lovasz, Shmuel Safra, and Mario Szegedy. 1996. Interactive proofs and the hardness of approximating cliques. J. ACM 43, 2, 268--292. Google ScholarGoogle ScholarDigital LibraryDigital Library
  9. Peter Gemmell, Richard J. Lipton, Ronitt Rubinfeld, Madhu Sudan, and Avi Wigderson. 1991. Self-testing/correcting for polynomials and for approximate functions. In Proceedings of the 23rd Annual ACM Symposium on Theory of Computing. 32--42. Google ScholarGoogle ScholarDigital LibraryDigital Library
  10. Timothy Gowers. 1998. A new proof of Szemeredi’s theorem for arithmetic progressions of length four. Geom. Functional Anal. 8, 3, 529--551.Google ScholarGoogle ScholarCross RefCross Ref
  11. Timothy Gowers. 2001. A new proof of Szemeredi’s theorem. Geom. Functional Anal. 11, 3, 465--588.Google ScholarGoogle ScholarCross RefCross Ref
  12. Ben Green and Terence Tao. 2008. An inverse theorem for the gowers U3(G) norm. Proc. Edinburgh Math. Soc. 51, 1, 73--153.Google ScholarGoogle ScholarCross RefCross Ref
  13. Frederic Green, Amitabha Roy, and Howard Straubing. 2005. Bounds on an exponential sum arising in boolean circuit complexity. Comptes Rendus Mathematique 341, 5, 279--282.Google ScholarGoogle ScholarCross RefCross Ref
  14. Elad Haramaty, Amir Shpilka, and Madhu Sudan. 2011. Optimal testing of multivariate polynomials over small prime fields. In Proceedings of the 52nd Annual Symposium on Foundations of Computer Science. 629--637. Google ScholarGoogle ScholarDigital LibraryDigital Library
  15. Charanjit S. Jutla, Anindya C. Patthak, Atri Rudra, and David Zuckerman. 2009. Testing low-degree polynomials over prime fields. Random Struct. Algor. 35, 2, 163--193. Google ScholarGoogle ScholarDigital LibraryDigital Library
  16. Tali Kaufman and Dana Ron. 2006. Testing polynomials over general fields. SIAM J. Comput. 36, 3, 779--802. Google ScholarGoogle ScholarDigital LibraryDigital Library
  17. Tali Kaufman and Madhu Sudan. 2008. Algebraic property testing: The role of invariance. In Proceedings of the 40th Annual ACM Symposium on Theory of Computing. 403--412. Google ScholarGoogle ScholarDigital LibraryDigital Library
  18. Alexander Razborov. 1987. Lower bounds on the size of bounded depth circuits over a complete basis with logical addition. Math. Notes Acad. Sci. USSR 41, 4, 333--338.Google ScholarGoogle ScholarCross RefCross Ref
  19. Alex Samorodnitsky. 2007. Low degree tests at large distances. In Proceedings of the 39th Annual ACM Symposium on Theory of Computing. 506--515. Google ScholarGoogle ScholarDigital LibraryDigital Library
  20. Roman Smolensky. 1987. Algebraic methods in the theory of lower bounds for boolean circuit complexity. In Proceedings of the 19th Annual ACM Symposium on Theory of Computing. 77--82. Google ScholarGoogle ScholarDigital LibraryDigital Library
  21. Terence Tao. 2008. Some notes on non classical polynomials in finite characteristic. http://terrytao.wordpress.com/2008/11/13/some-notes-on-non-classical-polynomials-in-finite-characteristic/.Google ScholarGoogle Scholar
  22. Terence Tao and Tamar Ziegler. 2008. The inverse conjecture for the Gowers norm over finite fields via the correspondence principle. http://arxiv.org/abs/0810.5527.Google ScholarGoogle Scholar
  23. Emanuele Viola and Avi Wigderson. 2008. Norms, XOR lemmas, and lower bounds for polynomials and protocols. Theory Comput. 4, 1, 137--168.Google ScholarGoogle ScholarCross RefCross Ref
  24. Andrew C.-C. Yao. 1982. Theory and applications of trapdoor functions (extended abstract). In Proceedings of the 23rd Annual IEEE Symposium on Foundations of Computer Science. 80--91. Google ScholarGoogle ScholarCross RefCross Ref

Index Terms

  1. Hard Functions for Low-Degree Polynomials over Prime Fields

    Recommendations

    Comments

    Login options

    Check if you have access through your login credentials or your institution to get full access on this article.

    Sign in

    Full Access

    • Published in

      cover image ACM Transactions on Computation Theory
      ACM Transactions on Computation Theory  Volume 5, Issue 2
      July 2013
      104 pages
      ISSN:1942-3454
      EISSN:1942-3462
      DOI:10.1145/2493246
      Issue’s Table of Contents

      Copyright © 2013 ACM

      Publisher

      Association for Computing Machinery

      New York, NY, United States

      Publication History

      • Published: 1 July 2013
      • Accepted: 1 March 2013
      • Revised: 1 December 2012
      • Received: 1 March 2012
      Published in toct Volume 5, Issue 2

      Permissions

      Request permissions about this article.

      Request Permissions

      Check for updates

      Qualifiers

      • research-article
      • Research
      • Refereed
    • Article Metrics

      • Downloads (Last 12 months)3
      • Downloads (Last 6 weeks)1

      Other Metrics

    PDF Format

    View or Download as a PDF file.

    PDF

    eReader

    View online with eReader.

    eReader
    About Cookies On This Site

    We use cookies to ensure that we give you the best experience on our website.

    Learn more

    Got it!