# Attività del Dipartimento

### Colloquium di Matematica

Optimal non-linear Boolean functions as multivariable polynomials: the even case.

Massimiliano Sala

11-04-2018 - 16:00
Aula F, primo piano, edificio Aule - Largo San Leonardo Murialdo,1

To guarantee security w.r.t. known attacks (especially differential cryptanalysis) it is necessary to design a block cypher very carefully. One type of component which is often used is the so-called S-Box (Substitution Box). It turns out that the ideal situation would be to have an APN (Almost Perfect Nonlinear) permutation of dimension d even, possible 4, 8 or another power of 2. The experimental results show that: there is no APN permutation for d=4, there is one APN permutation for d=6 (but there could be more) and none has been found so far for d=8, being any higher dimension intractable with a computer nowadays. In recent papers, we have investigated the situation by considering APN permutations as multivariable polynomials (vectorial Boolean functions). In other words, they are polynomial maps from a binary space of dimension d to itself. We have proved several theoretical results (that partially explain the computational findings): no component can have degree less than three (for any d even), no APN permutations exist for d=4, no pure cubic APN permutations exist for d=6. This is joint work with M. Calderini, I. Villa and M. Zaninelli.
org: SCOPPOLA Elisabetta