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.
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Acknowledgements
We are grateful to Ms U. Jeya Santhi (Director SAG, DRDO) and Dr. Sudhir Kamath (DG, MED&CoS, DRDO) for their support and encouragement in carrying out this collaborative research work. Thanks also are due to all the team members of Indo-Russian joint project QGSEC for technical support and fruitful discussions.
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Translated from Algebra i Logika Vol. 61 No. 4 pp. 375-400 July-August 2022. Russian DOI: https://doi.org/10.33048/alglog.2022.61.401
Supported by Russian Science Foundation project No. 22-21-00745.
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Artamonov, V.A., Chakrabarti, S., Tiwari, S.K. et al. Algebraic Properties of Subquasigroups and Construction of Finite Quasigroups. Algebra Logic 61, 251–270 (2022). https://doi.org/10.1007/s10469-023-09695-1
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DOI: https://doi.org/10.1007/s10469-023-09695-1