Paper 2021/361

Some Generic Constructions of Generalized Plateaued Functions

Jiaxin Wang 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, Hodžić et al. [IEEE TIT 65(9): 5865-5879, 2019] designed Boolean plateaued functions in spectral domain and provided some efficient 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 $\mathbb{F}_{2}^{n-s}$. In this paper, we study generalized $s$-plateaued functions from $V_{n}$ to $\mathbb{Z}_{p^k}$ where $p$ is an odd prime and $k\geq 1$ or $p=2, k\geq 2$ and $n+s$ is even. Firstly, inspired by the work of Hodžić \emph{et al}., we give a complete characterization of generalized plateaued functions with affine Walsh support 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 can contain strictly less than $n-s$ columns corresponding to affine functions and our construction methods are also applicable to Boolean plateaued functions. 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 is a special case of 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 complete Generalized Maiorana-McFarland class. Furthermore, based on this construction method, we give constructions of plateaued functions in the subclass WRP of the class of weakly regular plateaued functions and vectorial plateaued functions.