论文标题
Kim型APN功能是相当于黄金功能的元素
Kim-type APN functions are affine equivalent to Gold functions
论文作者
论文摘要
查找$ {\ mathbb f} _ {2^n} $的APN排列的问题，其中$ n $均匀，$ n> 6 $被称为大apn问题。 Li，Li，Heleseth和Qu最近表征了$ {\ Mathbb f} _ {q^2} $在$ f（x）= x^{3Q}+a_1x+a_1x^{2q+1}+a_1x+a_3x^{q+2}+a_3x^3 $，$ ge 4 $，q = 2^m = 2^m $ ge 4我们将其称为此表单的函数Kim型函数，因为它们概括了用于构建$ {\ Mathbb f} _ {2^6} $的APN置换的KIM函数的形式。我们通过证明Kim型函数$ f $为apn和$ m \ ge 4 $，扩展了Li，Li，Helleseth和Qu的结果，那么$ f $的仿射等同于两个金函数之一$ g_1（x）= x^3 $或$ g_2或$ g_2（x）= x^^^{2^{2^{2^{2^{2^{2^{M-1} $ 1} +1} +1} +1} +1} +1}+。再加上Göloğlu和Langevin的最新结果，他们证明，即使对于$ n $，黄金APN功能都不相当于排列，因此，对于$ M \ ge 4 $ kim type type apn在$ {\ mathbb f} _ {2m {2m {2m {2M {2M} $ equect上都从来都不是$ m \ ge ge 4 $ \ ge type apn函数。
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.
