**Image sets of perfectly nonlinear maps**

*Lukas Kölsch and Björn Kriepke and Gohar Kyureghyan*

**Abstract: **We consider image sets of $d$-uniform maps of finite fields. We present a lower bound on the image size of such maps and study their preimage distribution, by extending methods used for planar maps.
We apply the results to study $d$-uniform Dembowsi-Ostrom polynomials.
Further, we focus on a particularly interesting case of APN maps on binary fields $\F_{2^n}$. For these maps our lower
bound coincides with previous bounds.
We show that APN maps fulfilling the lower bound have a very special preimage distribution. We observe that for an even $n$ the image sets of several well-studied families of APN maps are minimal. In particular, for $n$ even, a Dembowski-Ostrom polynomial of form $f(x) =f'(x^3)$ is APN if and only if $f$ is almost-3-to-1, that is when its image set is minimal.
Also, any almost-3-to-1 component-wise plateaued map is necessarily APN, if $n$ is even.
For $n$ odd, we believe that the lower bound is not sharp. For $n$ odd, we present APN Dembowski-Ostrom polynomials
of form $f'(x^3)$ on $\F_{2^n}$ with image sizes $ 2^{n-1}$ and $5\cdot 2^{n-3}$.

We present results connecting the image sets of special APN maps with their Walsh spectrum. Especially, we show that a large class of APN maps has the classical Walsh spectrum. Finally, we present upper bounds on the image size of APN maps. In particular, we show that the image set of a non-bijective almost bent map contains at most $2^n-2^{(n-1)/2}$ elements.

**Category / Keywords: **secret-key cryptography / Image set, APN map, differential uniformity, Walsh spectrum, quadratic map, Dembowski-Ostrom polynomial, plateaued function

**Date: **received 26 Jan 2021

**Contact author: **lukas koelsch at uni-rostock de, bjoern kriepke@uni-rostock de, gohar kyureghyan@uni-rostock de

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

**Version: **20210127:133816 (All versions of this report)

**Short URL: **ia.cr/2021/098

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