Hard functions for low-degree polynomials over prime fields

  • Authors:

  • Andrej Bogdanov;Akinori Kawachi;Hidetoki Tanaka

  • Affiliations:

  • Department of Computer Science and Engineering, The Chinese University of Hong Kong;Department of Mathematical and Computing Sciences, Graduate School of Information Science and Engineering, Tokyo Institute of Technology;Department of Mathematical and Computing Sciences, Graduate School of Information Science and Engineering, Tokyo Institute of Technology

  • Venue:
  • MFCS'11 Proceedings of the 36th international conference on Mathematical foundations of computer science
  • Year:
  • 2011

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Abstract

In this paper, 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 [22]. 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, Kaufman, Krivelevich, Litsyn and Ron [2]. 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 [17]. 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.