On the Equivalence of Quadratic APN Functions

TL;DR

This paper proves that quadratic APN functions are CCZ-equivalent to Gold functions if and only if they are EA-equivalent, and applies this to show certain APN trinomials are CCZ-inequivalent to Gold functions.

Contribution

It establishes a precise equivalence condition between CCZ- and EA-equivalence for quadratic APN functions, simplifying their classification.

Findings

Quadratic APN functions CCZ-equivalent to Gold functions are exactly EA-equivalent.

A family of APN functions on fields with n ≡ 2 mod 4 are CCZ-inequivalent to Gold functions.

The method involves automorphism groups of associated codes.

Abstract

Establishing the CCZ-equivalence of a pair of APN functions is generally quite difficult. In some cases, when seeking to show that a putative new infinite family of APN functions is CCZ inequivalent to an already known family, we rely on computer calculation for small values of n. In this paper we present a method to prove the inequivalence of quadratic APN functions with the Gold functions. Our main result is that a quadratic function is CCZ-equivalent to an APN Gold function if and only if it is EA-equivalent to that Gold function. As an application of this result, we prove that a trinomial family of APN functions that exist on finite fields of order 2^n where n = 2 mod 4 are CCZ inequivalent to the Gold functions. The proof relies on some knowledge of the automorphism group of a code associated with such a function.