Kim型APN函数与Gold函数是仿射等价的
Kim-type APN functions are affine equivalent to Gold functions
摘要：
寻找${\mathbb F}_{2^n}$的APN排列的问题，其中$n$是偶数，$n>6$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函数的特征，其形式为$f(x)=x^{3q}+a_1x^{2q+1}+a_2x^{q+2}+a_3x^3$，其中$q=2^m$和$m\ge 4$。我们将这种形式的函数称为Kim型函数，因为它们推广了用于构造${\mathbb F}_{2^6}$的APN置换的Kim函数的形式。我们推广了Li，Li，Helleseth和Qu的结果，证明了如果Kim型函数$f$是APN和$m\ge 4$，则$f$仿射等价于两个Gold函数$G_1(x)=x^3$或$G_2(x)=x^{2^{m-1}+1}$中的一个.结合最近的结果Gölophillu和Langevin谁证明，即使$n$，黄金APN功能从来没有CCZ等价的排列，它遵循的是$m\ge 4$金型APN功能${\mathbb F}_{2^{2m}}$从来没有CCZ等价的排列。
英文摘要：
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 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.
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