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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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Selfdistributive quasigroups of size 2^k
Nagy, Tomáš ; Stanovský, David (advisor) ; Kepka, Tomáš (referee)
We present the theory of selfdistributive quasigroups and the construction of non-affine selfdistributive quasigroup of size 216 that was presented by Onoi in 1970 and which was the least known example of such structure of size 2k . Based on this construction, we introduce the notion of Onoi structures and Onoi mappings between them which generalizes Onoi's construction and which allows us to construct non-affine selfdistributive quasigroups of size 22k for k ≥ 3. We present and implement algorithm for finding central extensions of self- distributive quasigroups which enables us to classify non-affine selfdistributive quasigroups of size 2k and prove that those quasigroup exists exactly for k ≥ 6, k ̸= 7. We use this algorithm also in order to better understand the structure of non-affine selfdistributive quasigroups of size 26 . 1
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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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