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 $f(x)=x^{3q}+a_1{x}^{2q+1}+a_2{x}^{q+2}+a_3{x}^3$, where $q=2^m$ and $m\ge 4$. 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 extend the result of Li, Li, Helleseth and Qu by proving that if a Kim-type function $f$ is APN and $m\ge 4$, then $f$ is affine equivalent to one of two Gold functions $G_1(x)=x^3$ 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 $m\ge 4$ Kim-type APN functions on ${\mathbb{F}}_{2^{2m}}$ are never CCZ equivalent to permutations.

17 pages