Absolute Maximum Nonlinear Functions on Finite Nonabelian Groups

Abstract

Highly nonlinear functions (perfect nonlinear, maximum nonlinear, etc.) between finite fields have been studied in numerous papers. These concepts have been generalized first to finite abelian groups and then to finite nonabelian groups. In 2011, Poinsot and Pott introduced a new class of highly nonlinear functions between finite nonabelian groups, which are closely related to maximum nonlinear functions. They found such a function by a computer search, and proposed to find more such functions by theoretical constructions. Since then there is no progress in this topic. In this paper we continue the research in the direction proposed by Poinsot and Pott. Our purpose is to study the properties and constructions of this new class of highly nonlinear functions, called the absolute maximum nonlinear functions in this paper. In particular, for arbitrary finite (abelian or nonabelian) groups $K$ and $N$ with an absolute maximum nonlinear function $f: K \to N$ , we will prove that $N$ has to be abelian, and $f$ cannot be perfect nonlinear if $K$ is not abelian. Then we investigate the numerical constraints on the groups that have absolute maximum nonlinear functions. These constraints will eliminate the existence of absolute maximum nonlinear functions for many finite nonabelian groups. More importantly, using these constraints we will develop a method to construct absolute maximum nonlinear functions for some groups. In particular, we will determine all absolute maximum nonlinear functions on $A_4$ (the alternating group of degree 4). Furthermore, a general method to construct new absolute maximum nonlinear functions from the old ones will be given.