Paper 2026/652

Counting and recovering the quadratic relations of a vectorial function

Abstract

A recent paper by Calderini et al. investigates the use of a CCZ transformation to mask the quadratic central map in a multivariate scheme, providing an instance leading to a system of degree four. A following paper by Caminata et al. presents two methods to reduce the masked system back to a quadratic system. In this work we further study the method based on the quadratic relations between input and output of the masked function, generalizing it and applying to any CCZ transformation (of any quadratic map). Moreover, we study how the existence of these quadratic relations can be used to study whether a function can be CCZ equivalent to a quadratic map and, more generally, to study whether two functions can be CCZ equivalent. In fact, this analysis gives us necessary conditions that can be checked also in relatively large dimensions. Particularly, we completely analyse the cases of vectorial Boolean functions with 1 and 2 output bits.