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 $f(x)=x^{3q}+a_1x^{2q+1}+a_2x^{q+2}+a_3x^3$, 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 G\"{o}lo\u{g}lu 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.

中文翻译：

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Kim 型 APN 函数仿射等效于 Gold 函数

找到 ${\mathbb F}_{2^n}$ 的 APN 排列的问题，其中 $n$ 是偶数且 $n>6$ 被称为大 APN 问题。Li、Li、Helleseth 和 Qu 最近描述了定义在 ${\mathbb F}_{q^2}$ 上的 APN 函数，形式为 $f(x)=x^{3q}+a_1x^{2q+1}+a_2x ^{q+2}+a_3x^3$，其中 $q=2^m$ 和 $m\ge 4$。我们将这种形式的函数称为 Kim 类型函数，因为它们概括了用于构造 ${\mathbb F}_{2^6}$ 的 APN 置换的 Kim 函数的形式。我们扩展了 Li、Li、Helleseth 和 Qu 的结果，证明如果 Kim 型函数 $f$ 是 APN 且 $m\ge 4$，则 $f$ 仿射等价于两个 Gold 函数 $G_1( x)=x^3$ 或 $G_2(x)=x^{2^{m-1}+1}$。结合最近 G\"{o}lo\u{g}lu 和 Langevin 的结果，他们证明，即使对于 $n$，Gold APN 函数也永远不会等同于排列的 CCZ，