New Instances of Quadratic APN Functions

TL;DR

This paper adapts a recursive search method to discover numerous new quadratic APN functions in small dimensions, including permutations and functions with unique spectral properties, expanding the known landscape of APN functions.

Contribution

It introduces an adapted search technique that uncovers many previously unknown quadratic APN functions in dimensions 8 to 10, including permutations and functions with novel spectral features.

Findings

Found 12,921 new quadratic APN functions in dimension 8.

Discovered 35 new quadratic APN functions in dimension 9, including 2 permutations.

Identified 5 new quadratic APN functions in dimension 10.

Abstract

In a recent work, Beierle, Brinkmann and Leander presented a recursive tree search for finding APN permutations with linear self-equivalences in small dimensions. In this paper, we describe how this search can be adapted to find many new instances of quadratic APN functions. In particular, we found 12,921 new quadratic APN functions in dimension eight, 35 new quadratic APN functions in dimension nine and five new quadratic APN functions in dimension ten up to CCZ-equivalence. Remarkably, two of the 35 new APN functions in dimension nine are APN permutations. Among the 8-bit APN functions, there are three extended Walsh spectra that do not correspond to any of the previously-known quadratic 8-bit APN functions and, surprisingly, there exist at least four CCZ-inequivalent 8-bit APN functions with linearity $2^7$, i.e., the highest possible non-trivial linearity for quadratic functions in dimension eight.