Paper 2023/1101

${\mathcal{S}}_0$-equivalent classes, a new direction to find better weightwise perfectly balanced functions, and more

Abstract

We investigate the concept of ${\mathcal{S}}_0$-equivalent class, $n$-variable Boolean functions up to the addition of a symmetric function null in $0_n$ and $1_n$, as a tool to study weightwise perfectly balanced functions. On the one hand we show that weightwise properties, such as being weightwise perfectly balanced, the weightwise nonlinearity and weightwise algebraic immunity, are invariants of these classes. On the other hand we analyze the variation of global parameters inside the same class, showing for example that there is always a function with high degree, algebraic immunity, or nonlinearity in the $$-equivalent class of a function. Finally, we discuss how these results extend to other equivalence relations and their applications in cryptography.