Two-to-one functions from Galois extensions
type: Item
instance of: scholarly article
title: Two-to-one functions from Galois extensions
zbMATH Open document ID: 1504.11130
author: Daniele Bartoli
author: Marco Timpanella
author: Massimo Giulietti
published in: Discrete Applied Mathematics
publication date: 2022-01-13T00:00:00Z
review text: An \(m\)-1 function over a finite field \(\mathbb{F}_q\) is a function \(f:\mathbb{F}_q\rightarrow\mathbb{F}_q\) such that for each \(b\in\mathbb{F}_q, |f^{-1}(b)|\in\{0, m\}\), except for at most a single value \(\in\mathbb{F}_q\) for which \(|f^{-1}(b)|\notin\{0, m\}\). The main achievement of the paper is the construction of new 2-1 rational functions in odd characteristic. The construction is relatively straightforward, related to algebraic curves over finite fields and Galois groups of rational functions. It is based on rational function \(f_{a,b}(X)=N(X)/D(X)\) associated to the group \(C_{2d}=\{ g\left(\begin{smallmatrix}1&0\\ 0&\delta^i\end{smallmatrix}\right)g^{-1}\mid i\in\{0,\dots,2d-1\}\}\). The authors also extend the notion of 2-1 mapping to that of \(m\)-1 mapping and provide explicit descriptions of 3-1 rational functions.
reviewed by: Vladimír Lacko
Mathematics Subject Classification ID: 11T71
Mathematics Subject Classification ID: 94A60
zbMATH DE Number: 7456391
zbMATH Keywords: two-to-one mappings
zbMATH Keywords: Galois groups
zbMATH Keywords: symmetric cryptography
zbMATH Keywords: algebraic curves
zbMATH Keywords: rational functions
describes a project that uses: Magma
MaRDI profile type: Publication
full work available at URL: j.dam.2021.12.008
OpenAlex ID: W4200021230
cites work: Rational functions with small value set
cites work: The Magma algebra system. I: The user language
cites work: On Two-to-One Mappings Over Finite Fields
DOI: 10.1016/J.DAM.2021.12.008
description: scientific article; zbMATH DE number 7456391