Hard Functions for Low-Degree Polynomials over Prime Fields

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.