A new framework for identifying absolute maximum nonlinear functions

Abstract

Highly nonlinear functions (perfect nonlinear, maximum nonlinear, etc.) on finite fields and finite (abelian or nonabelian) groups have been studied in numerous papers. Among them are absolute maximum nonlinear functions on finite nonabelian groups introduced by Poinsot and Pott [15] in 2011. Recently, properties and constructions of absolute maximum nonlinear functions were studied in [19]. In this paper we study the characterizations of absolute maximum nonlinear functions on arbitrary finite groups. Then as an application of these characterizations, we discuss the existence of absolute maximum nonlinear functions on dihedral groups. We will prove that for a dihedral group D2n of order 2n, where n≥3, if there is an absolute maximum nonlinear function on D2n, then n∈{3,12,15,18,30,33,42,66,138}. In particular, if there exists an absolute maximum nonlinear function from D2n to Z2, where Z2 is the group of order 2, then we show that n=12. All absolute maximum nonlinear functions from D24 to Z2 will be determined.