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Quasigroups with few associative triples
Valent, Viliam ; Drápal, Aleš (advisor) ; Kepka, Tomáš (referee)
This bachelor thesis deals with quasigroups with a small number of associative triples. They were studied from the algebraic point of view by Drápal, Ježek and Kepka, Kotzig, and recently by Grošek and Horák. The aim of this thesis is to build on the research of Grošek and Horák, replicate and improve their findings concerning the minimum number of associative triples in small quasigroups. Another important part is an establishment of a new upper bound on the minimum number of associative triples among all quasigroups of the same order. We provide an algorithm that can produce quasigroups with the number of associative triples less or equal to the second power of their order. We also present the applications of such quasigroups in cryptography, namely in hash functions and zero-knowledge protocols. Powered by TCPDF (www.tcpdf.org)
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Small order quasigroups with minimum number of associative triples
Valent, Viliam ; Drápal, Aleš (advisor) ; Lisoněk, Petr (referee)
This thesis is concerned with quasigroups with a small number of associative triples. The minimum number of associative triples among quasigroups of orders up to seven has already been determined. The goal of this thesis is to determine the minimum for orders eight and nine. This thesis reports that the minimum number of associative triples among quasigroups of order eight is sixteen and among quasigroups of order nine is nine. The latter finding is rather significant and we present a construction of an infinite series of quasigroups with the number of associative triples equal to their order. Findings of this thesis have been a result of a computer search which used improved algorithm presented in this thesis. The first part of the thesis is devoted to the theory that shows how to reduce the search space. The second part deals with the development of the algorithm and the last part analyzes the findings and shows a comparison of the new algorithm to the previous work. It shows that new search program is up to four orders of magnitude faster than the one used to determine the minimum number of associative triples among quasigroups of order seven.
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Quasigroups with few associative triples
Valent, Viliam ; Drápal, Aleš (advisor) ; Kepka, Tomáš (referee)
This bachelor thesis deals with quasigroups with a small number of associative triples. They were studied from the algebraic point of view by Drápal, Ježek and Kepka, Kotzig, and recently by Grošek and Horák. The aim of this thesis is to build on the research of Grošek and Horák, replicate and improve their findings concerning the minimum number of associative triples in small quasigroups. Another important part is an establishment of a new upper bound on the minimum number of associative triples among all quasigroups of the same order. We provide an algorithm that can produce quasigroups with the number of associative triples less or equal to the second power of their order. We also present the applications of such quasigroups in cryptography, namely in hash functions and zero-knowledge protocols. Powered by TCPDF (www.tcpdf.org)
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Magic squares
SUCHÁ, Lucie
This diploma thesis deals with basic features of magic squares and analyses these features with regard to usability during the teaching at elementary schools. Magic squares are known for hundreds years and since then they have changed due to various modifications, from which other kinds were derived. The first part of the thesis is therefore dedicated to the history. Next chapter deals with the construction of magic squares. The following chapters study similar games as Sudoku, Kakuro and Latin squares. The final part of the thesis is dedicated to the usability of magic squares in teaching mathematics. To practice the given topic, the worksheets which are divided according to their difficulty, were created.
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