Two new families of bivariate APN functions

TL;DR

This paper introduces two new quadratic APN function families constructed via biprojective polynomials, expanding the known classes and providing CCZ-inequivalent examples over ^{12}.

Contribution

The paper presents two novel families of quadratic APN functions, including one that encompasses previously known families, and demonstrates CCZ-inequivalence over ^{12}.

Findings

Includes one of the two APN families by G;lo;glu (2022).

Constructs a second family by adding terms to the first, including one of Li et al.'s families.

Provides CCZ-inequivalent APN functions over ^{12}.

Abstract

In this work, we present two new families of quadratic APN functions. The first one (F1) is constructed via biprojective polynomials. This family includes one of the two APN families introduced by G\"olo\v{g}lu in 2022. Then, following a similar approach as in Li \emph{et al.} (2022), we give another family (F2) obtained by adding certain terms to F1. As a byproduct, this second family includes one of the two families introduced by Li \emph{et al.} (2022). Moreover, we show that for $n=12$, from our constructions, we can obtain APN functions that are CCZ-inequivalent to any other known APN function over $\mathbb{F}_{2^{12}}$.