Oriented Steiner quasigroups.
type: Item
instance of: scholarly article
zbMATH Open document ID: 1315.20061
DOI: 10.1142/S0219498814500728
author: Izabella Stuhl
publication date: 2014-09-05T00:00:00Z
full work available at URL: https://arxiv.org/abs/1402.4644
Mathematics Subject Classification ID: 20N05
Mathematics Subject Classification ID: 05B07
Mathematics Subject Classification ID: 20E22
Mathematics Subject Classification ID: 94A60
zbMATH DE Number: 6340463
zbMATH Keywords: extensions of quasigroups
zbMATH Keywords: oriented Steiner quasigroups
zbMATH Keywords: oriented Steiner triple systems
zbMATH Keywords: cryptography
MaRDI profile type: Publication
arXiv ID: 1402.4644
cites work: Q3411976
cites work: Q4256358
cites work: Q4004080
cites work: Moufang quasigroups
cites work: Schreier loops
cites work: Right Nuclei of Quasigroup Extensions
cites work: Oriented Steiner loops.
published in: Journal of Algebra and its Applications
title: Oriented Steiner quasigroups.
review text: Given the quasigroups \(Q\), \(K\) and a function \(f\colon Q\times Q\to K\) a new structure of quasigroup, denoted by \(\mathcal Q_f\), can be defined on \(Q\times K\) and it is called the \(f\)-extension of the quasigroup \(K\) by the quasigroup \(Q\) (the function \(f\) is also called factor system of \(\mathcal Q_f\)). A first result of the paper is that if \(K\) is a commutative quasigroup of order 3, the quasigroup \(f\)-extensions of \(K\) with the same factor system are isomorphic for any quasigroup \(Q\).NEWLINENEWLINE Then the author investigates \(f\)-extensions where \(K\) is the group \(\mathbb{Z}_2\) or \(\mathbb{Z}_3\) and \(Q\) is a Steiner quasigroup associated to an oriented Steiner Triple System (briefly STS) \((\mathcal S,T)\) namely an STS where in each block is assigned a cyclic order. This implies that the following orientation function can be defined: \(f^*\colon\mathcal S\times\mathcal S\setminus\{(x,x)\mid x\in\mathcal S\}\to\{\pm 1\}\) with \(f^*(a_1, a_2)=+1\) or \(f^*(a_1,a_2)=-1\) according with the cyclic order of the points of the block determined by \(a_1\) and \(a_2\). When \(K=\mathbb{Z}_2\), the author considers two different \(f\)-extensions obtained by means of the orientation function \(f^*\) and such \(f\)-extensions are called oriented Steiner quasigroups. When \(K=\mathbb{Z}_3\), the factor system is defined in the following way: for all \(x,y\in\mathcal S\), \(f(x,y):=f^*(x,y)\) if \(x\neq y\) and \(f(x,x):=0\) and such \(f\)-extension is called canonical oriented Steiner quasigroup. Some algebraic properties of these classes of quasigroups are investigated such as flexible laws, cross inverse properties, etc. -- In the final section an encryption algorithm based on quasigroup extensions is proposed.
reviewed by: Elena Zizioli
description: scientific article; zbMATH DE number 6340463