Two Problems about Monomial Bent Functions

TL;DR

This paper advances understanding of monomial bent functions by refining conditions for their duals and identifying the unique monomial with maximal bent components, solving open problems in the field.

Contribution

It simplifies the condition for monomial bent functions and proves the uniqueness of the monomial with maximal bent components, resolving open questions.

Findings

First part of Langevin and Leander's condition implies the whole condition.

The monomial $x^{2^k+1}$ uniquely has maximal bent components over $F_{2^{2k}}$.

The results solve open problems posed by Pott et al. and Ness and Helleseth.

Abstract

In 2008, Langevin and Leander determined the dual function of three classes of monomial bent functions with the help of Stickelberger's theorem: Dillon, Gold and Kasami. In their paper, they proposed one very strong condition such that their method works, and showed that both Gold exponent and Kasami exponent satisfy this condition. In 2018, Pott {\em et al.} investigated the issue of vectorial functions with maximal number of bent components. They found one class of binomial functions which attains the upper bound. They also proposed an open problem regarding monomial function with maximal number of bent components. In this paper, we obtain an interesting result about the condition of Langevin and Leander, and solve the open problem of Pott {\em et al.}. Specifically, we show that: 1) for a monomial bent function over $\mathbb{F}_{2^{2k}}$, if the exponent satisfies the first part of the condition of Langevin and Leander, then it satisfies the entire condition; 2) $x^{2^k+1}$ is the only monomial function over $\mathbb{F}_{2^{2k}}$ which has maximal number of bent components. Fortunately, as a consequence, we also solve an open problem of Ness and Helleseth in 2006.