Paper 2020/1113

On combinatorial approaches to search for quadratic APN functions

Konstantin Kalgin and Valeriya Idrisova

Abstract

Almost perfect nonlinear functions possess the optimal resistance to the differential cryptanalysis and are widely studied. Most known APN functions are obtained as functions over finite fields ${\mathbb{F}}_{2^n}$ and very little is known about combinatorial constructions in ${\mathbb{F}}_2^n$. In this work we proposed two approaches for obtaining quadratic APN functions in ${\mathbb{F}}_2^n$. The first approach exploits a secondary construction idea, it considers how to obtain quadratic APN function in $n+1$ variables from a given quadratic APN function in $n$ variables using special restrictions on new terms. The second approach is searching quadratic APN functions that have matrix form partially filled with standard basis vectors in a cyclic manner. This approach allowed us to find a new APN function in 7 variables. Also, we conjectured that a quadratic part of an arbitrary APN function has a low differential uniformity. This conjecture allowed us to introduce a new subclass of APN functions, so-called stacked APN functions. We found cubic examples of such functions for dimensions up to 6.