论文标题
在有限字段的逆函数下的仿射子空间的图像上
On the image of an affine subspace under the inverse function within a finite field
论文作者
论文摘要
我们考虑函数$ x^{ - 1} $，该函数会倒入有限的字段元素$ x \ in \ in \ mathbb {f} _ {p^n} $（$ p $是prime，$ 0^{ - 1} = 0 $）和affine $ \ $ \ \ \ \ \ m mathbb {f} _ {p} $ - {p} $ - 他们的图像也是仿射子空间。事实证明，仿射子空间$ l $，$ | l |的图像> 2 $，是一个仿射子空间，仅当$ l = q \ mathbb {f} _ {p^k} $中，其中$ q \ in \ mathbb {f} _ {p^n}^n}^{**} $ and $ k \ mid n $。换句话说，它要么是$ \ mathbb {f} _ {p^n} $的子场，要么是由子场的所有元素乘以$ q $的子空间。这概括了2006年线性不变子空间获得的结果。因此，我们提出了一个足够的条件，前提是函数$ a（x^{ - 1}） + b $没有不变的仿射子空间$ u $ u $ u $ u $ u $ u $ u $ 2 <| u | <p^n $用于可逆线性转换$ a：\ mathbb {f} _ {p^n} \ to \ m athbb {f} _ {p^n} $和$ b \ in \ in \ m athbb {f} _ {f} _ {p^n}^}^{*} $。例如，显示该条件适用于AES的S-Box。另外，我们证明了$αx^{ - 1} + b $的某些功能没有不变的仿射子空间，除了$ \ m athbb {f} _ {p^n} $，其中$α，b \ in \ mathbb {f} _ {f} _ {p^n} _ {p^n}^{*} $和$ n $是仲裁。
We consider the function $x^{-1}$ that inverses a finite field element $x \in \mathbb{F}_{p^n}$ ($p$ is prime, $0^{-1} = 0$) and affine $\mathbb{F}_{p}$-subspaces of $\mathbb{F}_{p^n}$ such that their images are affine subspaces as well. It is proven that the image of an affine subspace $L$, $|L| > 2$, is an affine subspace if and only if $L = q \mathbb{F}_{p^k}$, where $q \in \mathbb{F}_{p^n}^{*}$ and $k \mid n$. In other words, it is either a subfield of $\mathbb{F}_{p^n}$ or a subspace consisting of all elements of a subfield multiplied by $q$. This generalizes the results that were obtained for linear invariant subspaces in 2006. As a consequence, we propose a sufficient condition providing that a function $A(x^{-1}) + b$ has no invariant affine subspaces $U$ of cardinality $2 < |U| < p^n$ for an invertible linear transformation $A: \mathbb{F}_{p^n} \to \mathbb{F}_{p^n}$ and $b \in \mathbb{F}_{p^n}^{*}$. As an example, it is shown that the condition works for S-box of AES. Also, we demonstrate that some functions of the form $αx^{-1} + b$ have no invariant affine subspaces except for $\mathbb{F}_{p^n}$, where $α, b \in \mathbb{F}_{p^n}^{*}$ and $n$ is arbitrary.
