Trims and Extensions of Quadratic APN Functions

TL;DR

This paper investigates restrictions of quadratic APN functions to affine hyperplanes, introduces an EA-invariant based on these restrictions, and constructs numerous new quadratic APN functions, especially in dimension eight, with a focus on maximum linearity.

Contribution

It introduces a new EA-invariant for vectorial Boolean functions based on hyperplane restrictions and constructs many new quadratic APN functions in dimension eight.

Findings

A multiset of restrictions defines an EA-invariant for vectorial Boolean functions.

Identified which restrictions of known quadratic APN functions remain APN in dimensions less than 10.

Constructed 6,368 new quadratic APN functions in dimension eight via extensions from dimension seven.

Abstract

In this work, we study functions that can be obtained by restricting a vectorial Boolean function $F \colon \mathbb{F}_2^n \rightarrow \mathbb{F}_2^n$ to an affine hyperplane of dimension $n-1$ and then projecting the output to an $n-1$-dimensional space. We show that a multiset of $2 \cdot (2^n-1)^2$ EA-equivalence classes of such restrictions defines an EA-invariant for vectorial Boolean functions on $\mathbb{F}_2^n$. Further, for all of the known quadratic APN functions in dimension $n < 10$, we determine the restrictions that are also APN. Moreover, we construct 6,368 new quadratic APN functions in dimension eight up to EA-equivalence by extending a quadratic APN function in dimension seven. A special focus of this work is on quadratic APN functions with maximum linearity. In particular, we characterize a quadratic APN function $F \colon \mathbb{F}_2^n \rightarrow \mathbb{F}_2^n$ with linearity of $2^{n-1}$ by a property of the ortho-derivative of its restriction to a linear hyperplane. Using the fact that all quadratic APN functions in dimension seven are classified, we are able to obtain a classification of all quadratic 8-bit APN functions with linearity $2^7$ up to EA-equivalence.