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Braid Group Cryptography
Frnka, Jan ; Středa, Adolf (advisor) ; Stanovský, David (referee)
Braid groups involve certain problems that enable the construction of trapdoor func- tions for the purposes of asymmetric cryptography. Specifically, the conjugacy problem has shown potential in this direction, leading to the development of several schemes. However, it was soon revealed that instances of this problem used in designed schemes are vulnerable to attacks. The aim of this thesis is to formally describe braid groups and construct a theoretical framework to study this problem, selected derived cryptosystems, and attacks on these cryptosystems. In the conclusion, we will explore further potential problems that could be utilized to construct a new asymmetric cryptosystem.
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Coloring invariants of knots
Chwiedziuk, Ondřej ; Stanovský, David (advisor) ; Vojtěchovský, Petr (referee)
We can color knots by various finite quandles and check if they have non-trivial coloring. If so, we can say that the knot is not an unknot. However, we will focus on quandles that always give trivial coloring. It will turn out that they have interesting algebraic properties. In this work, we will show that a quandle gives a trivial coloring for each knot if and only if the quandle is reductive, which is exactly when the coloring invariant is Vassiliev's. We will make a similar characterization for links. That is, a quandle gives a trivial coloring for each link if and only if it is a trivial quandle. 1
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Cycles in translations in connected quandles
Filipi, Filip ; Stanovský, David (advisor) ; Vojtěchovský, Petr (referee)
In the thesis, we are dealing with Hayashi's conjecture in the context of conjugation quandles. We analyze their connectedness and, by using ideas presented by David Stanovský and Petr Vojtěchovský in the proof of the claim that every quandle of this type, derived from symmetric groups, satisfies this conjecture, we derive a characterization of Hayashi's conjecture for a narrow class of quandles using purely group-theoretic concepts. This characterization states, among other things, that if we find a finite non-abelian simple group containing an element that is not the identity and that commutes with every element of its conjugacy class in at least one of its non-trivial powers, then Hayashi's conjecture does not hold. Furthermore, we follow up on the aforementioned proof and prove that the conjecture also holds for conjugation quandles derived from alternating and dihedral groups. In conclusion, we formulate attractive possibilities for further research on these quandles. 1
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Isomophism problem for quandles derived from groups
Pudich, Ondřej ; Stanovský, David (advisor) ; Vojtěchovský, Petr (referee)
In this bachelor thesis, we focus on the mathematical structure called quandle. The point of interest shall be to provide the solution to the isomorphism problem, i.e., to determine exactly when two quandles are isomorphic. We address this problem in the case of principal quandles. Firstly, we prove the abstract characterization of when two principal quandles are isomorphic, and secondly, we imply the results on dihedral groups and obtain a partial classification. 1
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Quasigroups, one-way functions and hash mappings
Machek, Ivo ; Drápal, Aleš (advisor) ; Stanovský, David (referee)
In the rst part of this work we study the complexity of solving nonlinear quasigroup equations for di erent classes of quasigroups. In particular we study the application of principle of central quasigroups on the blocks of congruence. We show that these quasigroups can be shapeless and therefore we gain counterexample to the hypothesis which was stated by D. Gligoroski. In the second part of this work we apply previous results on the concrete quasigroups of the type Edon-R-I,II and we deduce the complexity of the corresponding algorithm for inverting the hash function Edon-R.
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Some questions of definability
Lechner, Jiří ; Stanovský, David (advisor) ; Kepka, Tomáš (referee)
We focus on first-order definability in the quasiordered class of finite digraphs ordered by embeddability. At first we will prove definability of each digraph up to size three. We will need to add to the quasiorder structure some digraphs as constants, so we try to find the needed set of constants as small as possible with small digraph as well. Gradually we make instruments that we can use to express the inner structure of each digraphs in the language of embeddability. At the end we investigate definability in the closely related lattice of universal classes of digraphs. We show that the set of finitely generated and also the set of finitely axiomatizable universal classes are definable subsets of the lattice.
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