On the image of an affine subspace under the inverse function within a finite field

Abstract

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.