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Algebraic Properties of Subquasigroups and Construction of Finite Quasigroups

Algebraic Properties of Subquasigroups and Construction of Finite Quasigroups Many important properties are identified and criteria are developed for the existence of subquasigroups in finite quasigroups. Based on these results, we propose an effective method that concludes the nonexistence of proper subquasigroups in a given finite quasigroup, or finds all its proper subquasigroups. This has an important application in checking the cryptographic suitability of a quasigroup. Using arithmetic of finite fields, we introduce a binary operation to construct quasigroups of order pr. Criteria are developed under which the quasigroups mentioned have desirable cryptographic properties, such as polynomial completeness and absence of proper subquasigroups. Effective methods are given for constructing cryptographically suitable quasigroups. The efficiency of the methods is illustrated by some academic examples and implementation of all proposed algorithms in the computer algebra system Singular. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Algebra and Logic Springer Journals

Algebraic Properties of Subquasigroups and Construction of Finite Quasigroups

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References (23)

Publisher
Springer Journals
Copyright
Copyright © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
ISSN
0002-5232
eISSN
1573-8302
DOI
10.1007/s10469-023-09695-1
Publisher site
See Article on Publisher Site

Abstract

Many important properties are identified and criteria are developed for the existence of subquasigroups in finite quasigroups. Based on these results, we propose an effective method that concludes the nonexistence of proper subquasigroups in a given finite quasigroup, or finds all its proper subquasigroups. This has an important application in checking the cryptographic suitability of a quasigroup. Using arithmetic of finite fields, we introduce a binary operation to construct quasigroups of order pr. Criteria are developed under which the quasigroups mentioned have desirable cryptographic properties, such as polynomial completeness and absence of proper subquasigroups. Effective methods are given for constructing cryptographically suitable quasigroups. The efficiency of the methods is illustrated by some academic examples and implementation of all proposed algorithms in the computer algebra system Singular.

Journal

Algebra and LogicSpringer Journals

Published: Sep 1, 2022

Keywords: quasigroup; subquasigroup; polynomial completeness

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