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**2020/1113**

**Title:**

On combinatorial approaches to search for quadratic APN functions

**Authors:**
Konstantin Kalgin, 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.

**ePrint:**
https://eprint.iacr.org/2020/1113

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