Vectorial Boolean functions map n bits to m. These functions and their properties are of particular interest to cryptographers since they correspond to the S-boxes used in block ciphers, to the filter functions of stream ciphers, etc.

Quadratic functions are especially interesting for two reasons. From the implementation perspective, they ease the use of some counter-measures against side-channel attacks. From a theoretical stand-point, they play a big role in the ongoing investigation of the big APN problem (an APN function is one with optimal differential properties). In this talk, I will present some tools that allow an efficient investigation of the equivalence class in which a given function lives. The focus will be on CCZ-equivalence, including the particular case of extended-affine equivalence.

Two approaches will be presented.
1. The one relies on the Jacobian matrix of the function, and allows recovering the specifics of the extended-affine relation between two functions (if it exists). It is a joint work with Anne Canteaut and Alain Couvreur.
2. The other can very efficiently « label » the extended-affine equivalence class of a quadratic APN function, and thus its CCZ-class. As an illustration of the latter, I will present a significantly large number of new quadratic APN functions found by a team from Bochum, and then quickly mention some fun functions in this set that I investigated with them.