Decomposing generalized bent and hyperbent functions

In this paper we introduce generalized hyperbent functions from $F_{2^n}$ to $Z_{2^k}$, and investigate decompositions of generalized (hyper)bent functions. We show that generalized (hyper)bent functions from $F_{2^n}$ to $Z_{2^k}$ consist of components which are generalized (hyper)bent functions from $F_{2^n}$ to $Z_{2^{k^\prime}}$ for some $k^\prime < k$. For odd $n$, we show that the Boolean functions associated to a generalized bent function form an affine space of semibent functions. This complements a recent result for even $n$, where the associated Boolean functions are bent.Comment: 24 page

On permutation polynomials EA-equivalent to the inverse function over GF(2n)GF(2^n)

It is proved that there does not exist a linearized polynomial $L(x)\in\mathbb{F}_{2^n}[x]$ such that $x^{-1}+L(x)$ is a permutation on $\mathbb{F}_{2^n}$ when $n\geq5$, which is proposed as a conjecture in \cite{li}. As a consequence, a permutation is EA-equivalent to the inverse function over $\mathbb{F}_{2^n}$ if and only if it is affine equivalent to it when $n\geq 5$

A note on constructions of bent functions from involutions

Bent functions are maximally nonlinear Boolean functions. They are important functions introduced by Rothaus and studied rstly by Dillon and next by many researchers for four decades. Since the complete classication of bent functions seems elusive, many researchers turn to design constructions of bent functions. In this note, we show that linear involutions (which are an important class of permutations) over nite elds give rise to bent functions in bivariate representations. In particular, we exhibit new constructions of bent functions involving binomial linear involutions whose dual functions are directly obtained without computation