Equivalences of power APN functions with power or quadratic APN functions

Abstract

On $F={\mathbb {F}}_{2^{n}}$ ($n\ge 3$), the power APN function $f_{d}$ with exponent d is CCZ-equivalent to the power APN function $f_{e}$ with exponent e if and only if there is an integer a with $0\le a\le n-1$ such that either (A) $e\equiv d 2^{a}$ mod $2^{n}-1$ or (B) $de\equiv 2^{a}$ mod $2^{n}-1$, 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.