Hard Functions for Low-degree Polynomials over Prime Fields

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 lowdegree 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