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

TL;DR

This paper introduces a linear matrix-based framework for representing nonlinear maps over finite fields, enabling analysis of their invertibility, cycle structure, and group properties.

Contribution

It develops a novel linear representation for nonlinear maps over finite fields, facilitating inverse computation and structural analysis.

Findings

Linear representation associates maps with matrices over finite fields.

Inverse maps are represented by matrix inverses for permutation maps.

The framework extends to parameterized maps and group structures.

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 $F_{\lambda}(x): \mathbb{F} \to \mathbb{F}$, defined in terms of a parameter $\lambda \in \mathbb{F}$, leading to the definition of an analogous linear complexity of the map $F_{\lambda}(x)$, and a parameter-dependent matrix representation $M_\lambda$ defined over the univariate polynomial ring $\mathbb{F}[\lambda]$. 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 $M_\lambda$. 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 $\mathbb{F}$.