Image sets of perfectly nonlinear maps

Abstract

We consider image sets of differentially d-uniform maps of finite fields. We present a lower bound on the image size of such maps and study their preimage distribution. Further, we focus on a particularly interesting case of APN maps on binary fields $$\mathbb {F}_{2^n}$$ F 2 n . We show that APN maps with the minimal image size are very close to being 3-to-1. We prove that for n even the image sets of several important families of APN maps are minimal, and as a consequence they have the classical Walsh spectrum. Finally, we present upper bounds on the image size of APN maps. For a non-bijective almost bent map f, these results imply $$\frac{2^n+1}{3}+1 \le |{\text {Im}}(f)| \le 2^n-2^{(n-1)/2}$$ 2 n + 1 3 + 1 ≤ | Im ( f ) | ≤ 2 n - 2 ( n - 1 ) / 2 .