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 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 $q\in {\mathbb{F}}_{p^n}^{\ast }$ 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 $b\in {\mathbb{F}}_{p^n}^{\ast }$. As an example, it is shown that the condition works for S-box of AES. Also, we demonstrate that some functions of the form $\hat{I}\pm x^{-1}+b$ have no invariant affine subspaces except for ${\mathbb{F}}_{p^n}$, where $\hat{I}\pm ,b\in {\mathbb{F}}_{p^n}^{\ast }$ and $n$ is arbitrary.