Paper 2021/361

Some New Constructions of Generalized Plateaued Functions

Jiaxin Wang and Fang-Wei Fu

Abstract

Plateaued functions as an extension of bent functions play a significant role in cryptography, coding theory, sequences and combinatorics. In 2019, Hod\v{z}i\'{c} et al. designed Boolean plateaued functions in spectral domain and provided some construction methods in spectral domain. However, in their constructions, the Walsh support of Boolean $s$-plateaued functions in $n$ variables, when written as a matrix of order $2^{n-s} \times n$, contains at least $n-s$ columns corresponding to affine functions on $\mathbb{F}_{2}^{n-s}$. They proposed an open problem to provide constructions of Boolean $s$-plateaued functions in $n$ variables whose Walsh support, when written as a matrix, contains strictly less than $n-s$ columns corresponding to affine functions. In this paper, we focus on the constructions of generalized $s$-plateaued functions from $V_{n}$ to $\mathbb{Z}_{p^k}$, where $V_{n}$ is an $n$-dimensional vector space over $\mathbb{F}_{p}$, $p$ is a prime, $k\geq 1$ and $n+s$ is even when $p=2$. Firstly, inspired by the work of Hod\v{z}i\'{c} et al., we give a complete characterization of generalized plateaued functions with affine Walsh support in spectral domain and provide some construction methods of generalized plateaued functions with (non)-affine Walsh support in spectral domain. In our constructions of generalized $s$-plateaued functions with non-affine Walsh support, the Walsh support, when written as a matrix, can contain strictly less than $n-s$ columns corresponding to affine functions. When $p=2, k=1$, these constructions provide an answer to the open problem in \cite{Hodzic2}. Secondly, we provide a generalized indirect sum construction method of generalized plateaued functions, which can also be used to construct (non)-weakly regular generalized bent functions. In particular, we show that the canonical way to construct Generalized Maiorana-McFarland bent functions can be obtained by the generalized indirect sum construction method and we illustrate that the generalized indirect sum construction method can be used to construct bent functions not in the completed Generalized Maiorana-McFarland class. Furthermore, based on this construction method, we give constructions of plateaued functions in the subclass \emph{WRP} of the class of weakly regular plateaued functions and vectorial plateaued functions.