Kim-type APN functions are affine equivalent to Gold functions

Abstract

The problem of finding APN permutations of ${\mathbb {F}}_{2^{n}}$ where n is even and n > 6 has been called the Big APN Problem. Li, Li, Helleseth and Qu recently characterized APN functions defined on ${\mathbb {F}}_{q^{2}}$ of the form f(x) = x^3q + a₁x^2q+ 1 + a₂x^q+ 2 + a₃x³, where q = 2^m and m ≥ 4. We will call functions of this form Kim-type functions because they generalize the form of the Kim function that was used to construct an APN permutation of ${\mathbb {F}}_{2^{6}}$. We prove that Kim-type APN functions with m ≥ 4 (previously characterized by Li, Li, Helleseth, and Qu) are affine equivalent to one of two Gold functions G₁(x) = x³ or $G_{2}(x)=x^{2^{m-1}+1}$. Combined with the recent result of Göloğlu and Langevin who proved that, for even n, Gold APN functions are never CCZ equivalent to permutations, it follows that for m ≥ 4 Kim-type APN functions on ${\mathbb {F}}_{2^{2m}}$ are never CCZ equivalent to permutations.