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 proved that the image of an affine subspace L, \(|L |> 2\), is an affine subspace if and only if \(L = s\mathbb {F}_{p^k}\), where \(s\in \mathbb {F}_{p^n}^{*}\) 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 \(s\). This generalizes the results that were obtained for linear invariant subspaces in 2006. As a consequence, the function \(x^{-1}\) maps the minimum number of affine subspaces to affine subspaces among all invertible power functions. In addition, 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 \(A: \mathbb {F}_{p^n} \rightarrow \mathbb {F}_{p^n}\) and \(b \in \mathbb {F}_{p^n}^{*}\). As an example, it is shown that the S-box of the AES satisfies the condition. Also, we demonstrate that some functions of the form \(\alpha x^{-1} + b\) have no invariant affine subspaces except for \(\mathbb {F}_{p^n}\), where \(\alpha , b \in \mathbb {F}_{p^n}^{*}\) and n is arbitrary.

我们考虑函数\(x^{-1}\)来反转有限域元素\(x \in \mathbb {F}_{p^n}\)（p是素数，\(0^{-1 } = 0\) ) 和仿射\(\mathbb {F}_{p}\) -子空间\(\mathbb {F}_{p^n}\)使得它们的图像也是仿射子空间。证明了仿射子空间L的图像 \ ( |L |> 2\)是仿射子空间当且仅当\(L = s\mathbb {F}_{p^k}\)，其中\(s\in \mathbb {F}_{p^n}^{*}\)和\(k \mid n\)。换句话说，它要么是\(\mathbb {F}_{p^n}\)的子域，要么是由子域的所有元素乘以\(s\)组成的子空间。这概括了 2006 年针对线性不变子空间获得的结果。因此，函数\(x^{-1}\)将最小数量的仿射子空间映射到所有可逆幂函数中的仿射子空间。此外，我们提出了一个充分条件，假设函数\ (A(x^{-1}) + b\)不具有基数\(2< |U |< p^n\)的不变仿射子空间U可逆线性变换\(A: \mathbb {F}_{p^n} \rightarrow \mathbb {F}_{p^n}\) 和 \(b \in \mathbb { F }_{p^n} ^{*}\)。作为例子，表明AES的S盒满足条件。此外，我们证明了一些形式为\(\alpha x^{-1} + b\)的函数没有不变仿射子空间，除了\(\mathbb {F}_{p^n}\)，其中\( \alpha , b \in \mathbb {F}_{p^n}^{*}\)且n是任意的。