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

This article introduces the concept of \(\mathcal {S}_0\)-equivalence class, i.e. , n-variable Boolean functions up to the addition of a symmetric function null in \(0_n\) and \(1_n\), and investigates its application 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 equivalence classes. On the other hand, we analyze the variation of global parameters inside the same class, and prove, for example, that there is always a function with high degree, algebraic immunity, or nonlinearity in the \(\mathcal {S}_0\)-equivalence class of a function. Finally, we discuss how these results can be extended to other equivalence relations and their applications in cryptography.