Changing Almost Perfect Nonlinear Functions on Affine Subspaces of Small Codimensions

Abstract

A function $F\colon \mathbb{F}_2^n \to \mathbb{F}_2^m$ with $m\ge n$ is called almost perfect nonlinear (APN) if, for every nonzero $a \in \mathbb{F}_2^n$ and every $b \in \mathbb{F}_2^m$, the equation $F(x + a) + F(x) = b$ has at most two solutions $x\in\mathbb{F}_2^n$. One of the central problems in the research on APN functions lies in discovering new constructions of these mappings. In this paper, we introduce secondary construction methods for APN functions by modifying given ones on affine subspaces of small codimensions. We provide explicit criteria for determining when such modifications preserve the APN property and show that that some of the newly constructed functions are inequivalent to the original ones.