Balanced Boolean functions with few-valued Walsh spectra parameterized by $P(x^2+x)$

TL;DR

This paper introduces a parametric construction method for balanced Boolean functions with few-valued Walsh spectra, producing new families with strong cryptographic properties and several classes of plateaued functions.

Contribution

It presents a novel family of four-valued spectrum Boolean functions and seven classes of plateaued functions, enhancing cryptographic design options.

Findings

New family of four-valued spectrum Boolean functions with high nonlinearity

Seven classes of plateaued functions, including semi-bent and near-bent functions

Functions achieve maximal algebraic degree and optimal algebraic immunity for certain dimensions

Abstract

Boolean functions with few-valued spectra have wide applications in cryptography, coding theory, sequence designs, etc. In this paper, we further study the parametric construction approach to obtain balanced Boolean functions using $2$-to-$1$ mappings of the form $P(x^2+x)$, where $P$ denotes carefully selected permutation polynomials. The key contributions of this work are twofold: (1) We establish a new family of four-valued spectrum Boolean functions. This family includes Boolean functions with good cryptographic properties, e.g., the same nonlinearity as semi-bent functions, the maximal algebraic degree, and the optimal algebraic immunity for dimensions $n \leq 14$. (2) We derive seven distinct classes of plateaued functions, including four infinite families of semi-bent functions and a class of near-bent functions.