On quasigroups with few associative triples
type: Item
instance of: scholarly article
title: On quasigroups with few associative triples
zbMATH Open document ID: 1250.94036
author: Otokar Grošek
author: Peter Horák
published in: Designs, Codes and Cryptography
publication date: 2012-05-22T00:00:00Z
review text: The paper focuses on the idea of employing quasigroups in designs of hash functions. It turns out that quasigroups with a relatively small number of associative triples are more suitable as building blocks in these designs. Given a quasigroup \(Q\) of order \(n\), one is interested in the total number \(a(Q)\) of associative triples in \(Q\). The main result of the paper is a lower bound for the number of associative triples: \(a(Q)\geq 2n-i(Q)\), where \(i(Q)\) is the cardinality of the set of all idempotent elements of the quasigroup \(Q\). An upper bound for \(a(Q)\) is investigated as well and constructions of quasigroup sequences \(\{Q_n\}_{n=1}^\infty\) are described for which \(a(Q_n)=|Q_n|^t\) for \(t=\frac{\ln 15}{\ln 5},\frac{\ln 19}{\ln 6}\) and \(\frac{\ln 19}{\ln 7}\).
reviewed by: Vladimír Lacko
Mathematics Subject Classification ID: 94A60
Mathematics Subject Classification ID: 05B15
Mathematics Subject Classification ID: 11T71
Mathematics Subject Classification ID: 14G50
Mathematics Subject Classification ID: 20N05
zbMATH DE Number: 6037532
zbMATH Keywords: Latin squares
zbMATH Keywords: associativity index
zbMATH Keywords: combinatorial designs
zbMATH Keywords: hash functions
zbMATH Keywords: quasigroups
zbMATH Keywords: associative triples
MaRDI profile type: Publication
full work available at URL: s10623-010-9482-9
OpenAlex ID: W2017271227
cites work: Q4529534
cites work: On quasigroups rich in associative triples
cites work: Q3135917
cites work: Q3887681
cites work: Q3310065
cites work: Q3369009
DOI: 10.1007/S10623-010-9482-9
description: scientific article; zbMATH DE number 6037532