Paper 2023/460

A unified construction of weightwise perfectly balanced Boolean functions

Abstract

At Eurocrypt 2016, Méaux et al. presented FLIP, a new family of stream ciphers {that aimed to enhance the efficiency of homomorphic encryption frameworks. Motivated by FLIP, recent research has focused on the study of Boolean functions with good cryptographic properties when restricted to subsets of the space $\mathbb{F}_2^n$. If an $n$-variable Boolean function has the property of balancedness when restricted to each set of vectors with fixed Hamming weight between $1$ and $n-1$, it is a weightwise perfectly balanced (WPB) Boolean function. In the literature, a few algebraic constructions of WPB functions are known, in which there are some constructions that use iterative method based on functions with low degrees of 1, 2, or 4. In this paper, we generalize the iterative method and contribute a unified construction of WPB functions based on functions with algebraic degrees that can} be any power of 2. For any given positive integer $d$ not larger than $m$, we first provide a class of $2^m$-variable Boolean functions with a degree of $2^{d-1}$. Utilizing these functions, we then present a construction of $2^m$-variable WPB functions $g_{m;d}$. In particular, $g_{m;d}$ includes four former classes of WPB functions as special cases when $d=1,2,3,m$. When $d$ takes other integer values, $g_{m;d}$ has never appeared before. In addition, we prove the algebraic degree of the constructed WPB functions and compare the weightwise nonlinearity of WPB functions known so far in 8 and 16 variables.