**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.

**Category / Keywords: **boolean functions, APN functions

**Original Publication**** (with minor differences): **SETA-2020 Proceedings

**Date: **received 14 Sep 2020

**Contact author: **vvitkup at yandex ru

**Available format(s): **PDF | BibTeX Citation

**Version: **20200915:113218 (All versions of this report)

**Short URL: **ia.cr/2020/1113

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