On Linear Representation, Complexity and Inversion of maps over finite fields

TL;DR

The paper introduces a Koopman-operator–driven linear representation for nonlinear maps over finite fields, mapping compositions to matrix powers via a minimal invariant subspace and a companion matrix $M$ with size $N$ (the map’s linear complexity). For permutation maps, inverses are obtained as $M^{-1}$-based linear combinations of iterates, and the framework extends to parametric families where a matrix $M_{\lambda}$ over a rational-function ring governs invertibility and inverse construction. The authors further generalize to multivariate maps on $\mathbb{F}^n$ by constructing a common $\mathbf{K}$-invariant subspace $W$ and a linear representation $F(x)=V M \hat{\psi}(x)$, with $F^{-1}$ given by $G(x)=V M^{-1} \hat{\psi}(x)$ when invertible, and show how to represent the group generated by several maps via a shared linear action. The approach yields concrete methods for computing cycle structures, parametric inverses, and group representations, with illustrative examples on Dickson polynomials and parameterized polynomials, and connections to dynamical-systems viewpoints through the Koopman framework.

Abstract

This paper defines a linear representation for nonlinear maps $F:\mathbb{F}^n\rightarrow\mathbb{F}^n$ where $\mathbb{F}$ is a finite field, in terms of matrices over $\mathbb{F}$. This linear representation of the map $F$ associates a unique number $N$ and a unique matrix $M$ in $\mathbb{F}^{N\times N}$, called the Linear Complexity and the Linear Representation of $F$ respectively, and shows that the compositional powers $F^{(k)}$ are represented by matrix powers $M^k$. It is shown that for a permutation map $F$ with representation $M$, the inverse map has the linear representation $M^{-1}$. This framework of representation is extended to a parameterized family of maps $F_λ(x): \mathbb{F} \to \mathbb{F}$, defined in terms of a parameter $λ\in \mathbb{F}$, leading to the definition of an analogous linear complexity of the map $F_λ(x)$, and a parameter-dependent matrix representation $M_λ$ defined over the univariate polynomial ring $\mathbb{F}[λ]$. Such a representation leads to the construction of a parametric inverse of such maps where the condition for invertibility is expressed through the unimodularity of this matrix representation $M_λ$. Apart from computing the compositional inverses of permutation polynomials, this linear representation is also used to compute the cycle structures of the permutation map. Lastly, this representation is extended to a representation of the cyclic group generated by a permutation map $F$, and to the group generated by a finite number of permutation maps over $\mathbb{F}$.

Theorems & Definitions (22)

• Lemma 1
• proof
• Theorem 1
• proof
• Remark 1
• Theorem 2
• proof
• Theorem 3
• Theorem 4
• Proposition 1