Classification of $(q,q)$-biprojective APN functions

TL;DR

This paper classifies a specific class of APN functions over finite fields, revealing that only known Gold functions and a particular $ppa$-function satisfy certain properties, addressing an open problem in cryptography.

Contribution

It provides a complete classification of $(q,q)$-biprojective APN functions under group actions, identifying the only quadratic APN functions with the subfield property.

Findings

Gold functions are the only quadratic APN functions with subfield property.

The $ppa$-function is unique among known APN functions equivalent to permutations.

No other quadratic APN functions satisfying the subfield property are known or likely to exist.

Abstract

In this paper, we classify $(q,q)$-biprojective almost perfect nonlinear (APN) functions over $\mathbb{LL} \times \mathbb{LL}$ under the natural left and right action of $\mathrm{GL}(2,\mathbb{LL})$ where $\mathbb{LL}$ is a finite field of characteristic $2$. This shows in particular that the only quadratic APN functions (up to CCZ-equivalence) over $\mathbb{LL} \times \mathbb{LL}$ that satisfy the so-called subfield property are the Gold functions and the function $\kappa : \mathbb{F}_{64} \to \mathbb{F}_{64}$ which is the only known APN function that is equivalent to a permutation over $\mathbb{LL} \times \mathbb{LL}$ up to CCZ-equivalence. The $\kappa$-function was introduced in (Browning, Dillon, McQuistan, and Wolfe, 2010). Deciding whether there exist other quadratic APN functions (possibly CCZ-equivalent to permutations) that satisfy subfield property or equivalently, generalizing $\kappa$ to higher dimensions was an open problem listed for instance in (Carlet, 2015) as one of the interesting open problems on cryptographic functions.