Paper 2025/2130

Weightwise (almost) perfectly balanced functions: $t$-concatenation and the general Maiorana-McFarland class

Abstract

The study of cryptographic criteria for Boolean functions with restricted domains has been an important topic over the last 20 years. A revived interest has sparked after the work of Carlet, Méaux and Rotella in 2017, where the authors studied cryptographic properties of restricted-domain functions and introduced the concept of weightwise perfectly balanced functions as part of the analysis of the FLIP stream cipher. Weightwise (almost) perfectly balanced functions are defined as Boolean functions that are (almost) balanced on each of the sets of vectors of the same Hamming weight. Several approaches have been considered to build new families of such functions. In this article, we present some new constructions of weightwise (almost) perfectly balanced functions via two approaches, the first class is constructed using the $t$-concatenation of Boolean functions, whereas the second one draws certain functions from the so-called general Maiorana-McFarland class. A generic analysis of these two classes is given, as well as explicit examples in both classes. Namely, we provide instances of functions in both classes attaining high overall nonlinearities, as well as slice nonlinearities. Notably, we present examples in 16 variables that attain some of the best overall nonlinearities, and more importantly, the highest slice nonlinearities among all of the constructions presented in the literature.