Paper 2021/098

Image sets of perfectly nonlinear maps

Lukas Kölsch, 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.