Equivalences of power APN functions with power or quadratic APN functions

Abstract

On $$F={\mathbb {F}}_{2^{n}}$$F=F2n ($$n\ge 3$$nź3), the power APN function $$f_{d}$$fd with exponent d is CCZ-equivalent to the power APN function $$f_{e}$$fe with exponent e if and only if there is an integer a with $$0\le a\le n-1$$0≤a≤n-1 such that either (A) $$e\equiv d 2^{a}$$eźd2a mod $$2^{n}-1$$2n-1 or (B) $$de\equiv 2^{a}$$deź2a mod $$2^{n}-1$$2n-1, where case (B) occurs only when n is odd (Theorem 1). A quadratic APN function f is CCZ-equivalent to a power APN function if and only if f is EA-equivalent to one of the Gold functions (Theorem 2). Using Theorem 1, a complete answer is given for the question exactly when two known power APN functions are CCZ-equivalent (Proposition 2). The key result to establish Theorem 1 is the conjugacy of some cyclic subgroups in the automorphism group of a power APN function (Corollary 3). Theorem 2 characterizes the Gold functions as unique quadratic APN functions which are CCZ-equivalent to power functions.