Equivalences of power APN functions with power or quadratic APN functions (Q345143)

This paper gives two main theorems. Theorem 1: On a finite field \(F=\mathbb{F}_{2^n}\) with \(n\geq 3\), 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 \(0\leq a\leq n-1\) such that either (A) \(e\equiv d2^a \mod 2^n-1\) or (B) \(de\equiv 2^a \mod 2^a-1\), where case (B) occurs only when \(n\) is odd. Theorem 2: 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. Using Theorem 1, a complete answer is given for the question exactly when two known power APN functions are CCZ-equivalent. The key result to establish Theorem 1 is the conjugacy of some cyclic subgroups in the automorphism group of a power APN function. Theorem 2 characterizes the Gold functions as unique quadratic APN functions which are CCZ-equivalent to power functions.