# Constructing vectorial bent functions via second-order derivatives

## TL;DR

This paper introduces a new method for constructing vectorial bent and plateaued functions using second-order derivatives, leading to multiple infinite families with high algebraic degrees and maximal bent components.

## Contribution

It provides a generic construction approach linking second-order derivatives to known conditions, resulting in new classes of vectorial bent functions with high algebraic degrees.

## Key findings

- Three new infinite families of vectorial bent functions
- Construction of vectorial plateaued functions with maximal bent components
- Efficient generic method based on second-order derivatives

## Abstract

Let $n$ be an even positive integer, and $m<n$ be one of its positive divisors. In this paper, inspired by a nice work of Tang et al. on constructing large classes of bent functions from known bent functions [27, IEEE TIT, 63(10): 6149-6157, 2017], we consider the construction of vectorial bent and vectorial plateaued $(n,m)$-functions of the form $H(x)=G(x)+g(x)$, where $G(x)$ is a vectorial bent $(n,m)$-function, and $g(x)$ is a Boolean function over $\mathbb{F}_{2^{n}}$. We find an efficient generic method to construct vectorial bent and vectorial plateaued functions of this form by establishing a link between the condition on the second-order derivatives and the key condition given by [27]. This allows us to provide (at least) three new infinite families of vectorial bent functions with high algebraic degrees. New vectorial plateaued $(n,m+t)$-functions are also obtained ($t\geq 0$ depending on $n$ can be taken as a very large number), two classes of which have the maximal number of bent components.