Non Abelian Bent Functions

TL;DR

This paper extends the concept of bent functions, originally defined for Abelian groups, to non-Abelian groups using linear representation theory, providing new characterizations of perfect nonlinear functions in this broader context.

Contribution

It introduces a novel framework for non-Abelian bent functions, generalizing existing Abelian concepts through the use of linear representations.

Findings

Characterizations of bentness in non-Abelian groups

Extension of perfect nonlinear functions to non-Abelian groups

Application of linear representation theory to bent function analysis

Abstract

Perfect nonlinear functions from a finite group $G$ to another one $H$ are those functions $f: G \rightarrow H$ such that for all nonzero $\alpha \in G$, the derivative $d_{\alpha}f: x \mapsto f(\alpha x) f(x)^{-1}$ is balanced. In the case where both $G$ and $H$ are Abelian groups, $f: G \rightarrow H$ is perfect nonlinear if and only if $f$ is bent i.e for all nonprincipal character $\chi$ of $H$, the (discrete) Fourier transform of $\chi \circ f$ has a constant magnitude equals to $|G|$. In this paper, using the theory of linear representations, we exhibit similar bentness-like characterizations in the cases where $G$ and/or $H$ are (finite) non Abelian groups. Thus we extend the concept of bent functions to the framework of non Abelian groups.