An Open Problem About Monomial Bent Functions

In 2018, Pott et al. investigated vectorial functions with maximal number of bent components. They found one class of binomial functions attaining the upper bound. They also proposed an open problem regarding monomial functions that have the maximal number of bent components. In this paper, we solve this open problem. Specifically, we prove that if k = 2, then xs(2(k+ 1)) are the only monomial functions over F22k that have the maximal number of bent components, where s. {1, 2, 22,..., 2(k-1)}. As a consequence, we also solve an open problem of Ness and Helleseth about the cross-correlation function between two sequences in 2006.