# On constructions and properties of $(n,m)$-functions with maximal number of bent components

## TL;DR

This paper determines the maximum number of bent components in $(n,m)$-functions, characterizes functions achieving this maximum, and provides new constructions with spectral analysis, advancing understanding of bent functions in cryptography.

## Contribution

It establishes the maximal number of bent components for $(n,m)$-functions, characterizes trivial and nontrivial functions attaining this maximum, and introduces new constructions with spectral properties.

## Key findings

- Maximum number of bent components is $2^{m}-2^{m-k}$ for $(n,m)$-functions.
- All functions with maximum bent components are either trivial or of specific form $F^{i}$.
- New constructions yield functions with maximal bent components and diverse spectral properties.

## Abstract

For any positive integers $n=2k$ and $m$ such that $m\geq k$, in this paper we show the maximal number of bent components of any $(n,m)$-functions is equal to $2^{m}-2^{m-k}$, and for those attaining the equality, their algebraic degree is at most $k$. It is easily seen that all $(n,m)$-functions of the form $G(x)=(F(x),0)$ with $F(x)$ being any vectorial bent $(n,k)$-function, have the maximum number of bent components. Those simple functions $G$ are called trivial in this paper. We show that for a power $(n,n)$-function, it has such large number of bent components if and only if it is trivial under a mild condition. We also consider the $(n,n)$-function of the form $F^{i}(x)=x^{2^{i}}h({\rm Tr}^{n}_{e}(x))$, where $h: \mathbb{F}_{2^{e}} \rightarrow \mathbb{F}_{2^{e}}$, and show that $F^{i}$ has such large number if and only if $e=k$, and $h$ is a permutation over $\mathbb{F}_{2^{k}}$. It proves that all the previously known nontrivial such functions are subclasses of the functions $F^{i}$. Based on the Maiorana-McFarland class, we present constructions of large numbers of $(n,m)$-functions with maximal number of bent components for any integer $m$ in bivariate representation. We also determine the differential spectrum and Walsh spectrum of the constructed functions. It is found that our constructions can also provide new plateaued vectorial functions.