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Modification of electroinsulating varnish with ground mica
Drápal, Aleš ; Frk, Martin (referee) ; Rozsívalová, Zdenka (advisor)
This master's thesis focuses on composite systems based on varnish combined with different filler ratios of micronized mica. The aim is to analyse impact of the filler on dielectric properties of the varnish, i.e. relative permittivity and loss factor as functions of frequency as well as charging and discharging currents as functions of time. Dielectric mixture formulas are applied on relative permittivity values. Calculated and measured values are compared.
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Edwards curves and elliptic function fields
Beran, Adam ; Drápal, Aleš (advisor) ; Žemlička, Jan (referee)
In this thesis, we study twisted Edwards curves using the theory of algebraic function fields. After summarizing the basic theory, we focus on the structure of the function field for curves that are given by an equation of the form x2 2 = f(x1), where f is a monic polynomial of degree four. We show that twisted Edwards curves correspond to a special case when f(x1) = g(x2 1), where g is a quadratic polynomial with two distinct nonzero roots. We describe the basic properties of twisted Edwards curves, with special attention given to possible places at infinity. Next, we derive formulas for the point addition, which is achieved by using the relation between points on the curve, places of degree one and elements of the Picard group. Furthermore, we summarize how the point addition can be interpreted geometrically, and outline several alternative coordinate systems based on projective coordinates. Finally, we present two examples of twisted Edwards curves that are nowadays being used in cryptographic applications. 1
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Schoof's algorithm for Weierstrass curves
Zvoníček, Václav ; Drápal, Aleš (advisor) ; Mareš, Martin (referee)
Schoof's algorithm is the starting point for the most efficient methods for determining the number of rational points on an elliptic curve defined over a finite field. The goal of this thesis is to introduce the subject of elliptic curves, with the emphasis on Weierstrass curves over a finite field, to describe Schoof's algorithm and its time complexity, and, finally, to implement it in C++ with the support of NTL. The implementation provides a user with a reasonably fast utility for determining the order of Weierstrass curves over finite fields of size up to 128 bits. 1
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Nonassociativity in two operations
Lehká, Martina ; Drápal, Aleš (advisor) ; Patáková, Zuzana (referee)
This thesis follows up mainly on the research of Drápal and Valent, who studied the nonassociativity of one quasigroup operation. Its central objective is to examine the number of triples (x, y, z) ∈ Q3 such that (x ∗ y) ◦ z = x ∗ (y ◦ z), where (Q, ∗) and (Q, ◦) are two quasigroups, |Q| = n. Let a2(C) be the number of such triples in a quasigroup couple C. Call it the associativity index. Denote by a2(n) the minimal a2(C), where C is a couple of order n. By averaging the associativity index over all the principal isotopes of a quasigroup couple, we prove that a2(n) ≤ n2 (1+1/(n−1)), n > 2. We then characterize the couples C that, on average, attain a2(C) = n2 and we prove that this value is an improved upper bound on a2(n), n > 2. Furthermore, we begin research on couples of quasigroups isotopic to groups. Lastly, we present computational results with examples, including a2(4) = 8 and a2(5) = 9. 1
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Elementary theory for groups of linear fractional transformations
Tomášková, Sára ; Drápal, Aleš (advisor) ; Růžička, Pavel (referee)
The thesis focuses on the properties of general projective linear group PGL2(F) and its action on the projective line P1 (F), both for a finite and an infinite field F. Only the basic knowledge from the Bachelor studies is used to prove these properties. Sharp 3- transitivity of the said group is discussed. Then, we deal with the subgroups consisting of identity and all elements whose sets of fixed points coincide. Furthermore, we show under which conditions all these subgroups have the property that all their finite subgroups are cyclic. We deduce that for a finite field F, it holds that all of these groups are cyclic if and only if F is equal to Zp for a prime number p. The thesis then focuses on the action of PGL2(F) by conjugation on the set of these sungroups. Finally, it is shown that projective special linear group PSL2(F) is simple. 1
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Sudé triangulace a Abelovy grupy
Hrbek, Michal ; Drápal, Aleš (advisor) ; Kepka, Tomáš (referee)
Title: Even triangulations and Abelian groups Author: Michal Hrbek Department: Department of Algebra Supervisor: prof. RNDr. Aleš Drápal CSc., DSc. Abstract: This thesis takes interest in spherical Eulerian triangulations and the algebraic structure defined on its vertices corresponding with the latin bitrade equivalent to the triangulation. First, we introduce needed results about the properties of the triangulations and their embeddings into Abelian groups. Then we get concerned with a particular kind of almost 6-homogenous triangulations. The text presents several examples, then the groups of the simplest sequence of triangulations are explicitly described. In order to investigate more complicated cases, we introduce a recursive formula for defining relations of the groups and we show an example of its usage with modular arithmetic. The thesis is completed by discussing computed data. Keywords: latin bitrade, eulerian triangulation, Abelian group 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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Quasigroup based cryptography
Frisová, Andrea ; Stanovský, David (advisor) ; Drápal, Aleš (referee)
In this work, we study some properties of an in nite matrix, which consists of quasigroup elements. This matrix is generated from a certain sequence X using left iterated translations. We suppose that the sequence X is periodic and we examine how the periods of the rows of our matrix behave for various types of quasigroups. We show that for central quasigroups the periods increase at most linearly. Further, we try to apply our result to the stream cipher Edon-80.
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On DSA
Čadová, Veronika ; Drápal, Aleš (advisor) ; Jedlička, Přemysl (referee)
This thesis deals with problems of comparing the safety and running time of digital signatures DSA and Schnorr. Digital signature is almost full, legally recognized alternative to physical sign, intended for use in a digital environment. Digital signature uses asymmetric codes and hash functions which are easily described, as well as other basic concepts such as discrete logarithm and cyclic groups. The thesis deals with the analysis of possible attacks on DSA and compares DSA and Schnorr algorithm. Digital signature history and its implementation is part of the thesis.
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