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Mathematical security methods in digital data transfer
Bartušek, Petr ; Kureš, Miroslav (referee) ; Vašík, Petr (advisor)
This master’s thesis deals with an analysis of digital security with CRC. In the thesis there is described a principle of coding theory, especially then digital security with CRC, for which there is explained a mathematical principle of their encoding and decoding, software implementation and a description of frequently used generator polynomials. The main aim of the thesis is a testing of undetected errors and a finding of number of these errors. After that it is used for the computation of probability with which undetected errors can occur. The thesis is supplemented with several programs which are programmed in the software Matlab.
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Economic Curves
Hrubešová, Gabriela ; Doupovec, Miroslav (referee) ; Kureš, Miroslav (advisor)
The topic of this bachelor's thesis is to study some economic curves, a description of their attributes and a determining the mathematical expression. In the first part, it is explained the issues of general plane curves. Next part focuses on the characteristics of the most important economic curves and their use. The mathematical properties of the individual curves are described below. The last part is a software implementation in the program Wolfram Mathematica.
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Type-preserving Matrices and Block Cipher Security
Okediran, Tunmbi Olayemi ; Kureš, Miroslav (referee) ; Aragona, Riccardo (advisor)
Zavedli jsme novou vlastnost směšovacích vrstev blokových šifer. Tato vlastnost se nazývá non-type-preserving a zaručuje odolnost vůči algebraickým útokům na základě imprimitivity skupiny generované kruhovými funkcemi. Uvažovali jsme binární matici odpovídající směšovací vrstvě a dali jsme potřebné a dostatečné podmínky pro binární matici, která zajišťuje typ nezachovávající vlastnost. Pak jsme ukázali, že některé reálné šifry splňují tyto nezbytné a postačující podmínky, a proto jsou typ nezachovávající. Uvažované šifry byly GOST, PRESENT a AES. Nakonec jsme v kapitole 4 ukázali, že pokud míchací vrstva šifry SPN, která používá adiční modulo 2^n pro míchání klíčů, nezachovává typ, pak je skupina generovaná funkcí round primitivní.
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Fibonacci numbers with applications
Váňa, Viktor ; Kureš, Miroslav (referee) ; Klaška, Jiří (advisor)
This bachelor thesis deals with golden ratio, its appearance and possible use in technical practice. It is divided into two main parts. First part deals with golden ratio appearance in chemistry. Specificly inquire into distance between centers of atoms in a halide salts compounds, asymmetric features of chemical processes, stability of transuranic elements, energy levels of gas electrons and spin number of subatomic elements. Another part takes an interest in simulation of electrical power lines supplied by two sides and calculatinon of its parameters. Last section is included in order to present variety of Fibonacci numbers appearence. It shows us relations between insects and flower, where insect live.
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Measurement of Micro-Accelerations on Board of the Russian Spacecraft „Universat -2“
Fedosov, Viktor ; Kureš, Miroslav (referee) ; Bezděk, Aleš (referee) ; Daněk, Vladimír (advisor)
The Thesis of the dissertation work present results of the accelerometer test operation on board of the Russian small spacecraft Universat – 2. This technical experiment was performed in the frame of the project TEASER (Technological Experiment And Space Environmental Resistance) financed by Ministry of Industry and Trade of the Czech Republic. The principal aim of the project was development and application of end-to-end tests procedures for MAC04TS. Generally, End-to-end Testing is intended for verification of the new product comprehensive operational sequence. In the context of the accelerometer this process includes following aspects: On-Ground Qualification of the instrument flight model and its integration with space platform, operation in orbit to verify the device functionality and performances. The MAC04TS is the new modification of the triaxial electrostatic high sensitive microaccelerometer MAC (or MACEK) designed for measurement of non-gravitational accelerations acting to orbiting spacecraft. Mentioned non-conservative perturbations are primary constraint for theoretical predictions of the spacecrafts orbit evolution. Problem of the precise orbit determination is important not only for ballistic mission analysis and support but at planning and realization of the science researches in Geophysics, Geodesy or in Space Physics branches… Presented text to examination is devoted to partial task of the MAC04TS end-to-end testing, especially, measurement and analysis of the micro-gravitational accelerations in the instrument position on board of the spacecraft. Solution of the problem is based on two methods. The first method was based on the calculation of the accelerations by telemetry information about satellite attitude motion. The second method consisted in direct measuring the accelerations by the triaxial low frequency accelerometer MAC04TS and subsequent smoothing measurement data. Comparison of the acceleration values, obtained by different ways, was carried out as a result of constructing the approximation of the measured acceleration values by their calculated values. The approximation was constructed by the least squares method. Both methods gave similar results. The received estimations of the accelerometer measurements can be used for the analysis of the accelerometer verification.
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Quaternion algebras
Bečka, Pavel ; Klaška, Jiří (referee) ; Kureš, Miroslav (advisor)
This thesis deals with quaternion algebras. A quaternion algebra is a four dimensional vector space with basis 1, i, j, k and multiplication defined as i2 = a, j2 = b, ij = -ji = k. The thesis deals with the basic attributes of quaternion algebras, quaternion orders and maximal orders. Lastly the thesis deals with the concept of discriminant of algebras and connected terms like Hilbert symbol and Legendre symbol. Throughout the thesis we show solved problems using mathematical software SAGE.
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Tensors and their geometrical and mechanical applications
Kunz, Daniel ; Kureš, Miroslav (referee) ; Tomáš, Jiří (advisor)
In this thesis I describe construction of tensor algebra and its following usage of this magnificent structure for description of curved surfaces. This structure can be used in geometry or for example in mechanic. The thesis is focused on clear construction and if it is possible than to sustain visual aspect of current problem. Main task of this thesis was got the feel of tensor algebra and its construction and then use it on tasks in physic and mechanic.
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Drozd rings
Nytra, Jan ; Doupovec, Miroslav (referee) ; Kureš, Miroslav (advisor)
This thesis focuses on Drozd rings. In the beginning, we mention important parts of algebraic theory for the definition of these rings. In the next chapter we describe an example of Drozd ring. In the following, we concentrate on Weil algebras - it shows up, that Drozd algebras over field of real numbers are specific examples of Weil algebras. We also construct groups of algebra automorphisms for these algebras. In the last part of the thesis, we mention Lie groups, because groups of algebra automorphisms of Weil algebras are examples of Lie groups.
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Rings of endomorphisms of elliptic curves and Mestre's theorem
Szásziová, Lenka ; Hrdina, Jaroslav (referee) ; Kureš, Miroslav (advisor)
Eliptické křivky jsou mocným nástrojem dnešní doby. Jednak přispěly k vyřešení mnoha matematických problémů, ale také nalezly četná uplatnění v aplikacích, jako je například kryptografie založená na eliptických křivkách (ECC). Tato metoda veřejného klíče má velkou budoucnost, neboť v mnohém doplňuje nedostatky známé RSA metody. Jedním z hlavních problémů kryptografie založené na eliptických křivkách je určení řádu eliptické křivky, tedy výpočet počtu bodů eliptické křivky nad prvočíselným polem. Tomuto zásadnímu problému je věnována tato práce. Na určení řádu eliptické křivky existuje řada algoritmů. Pro menší prvočísla (čili pro charakteristiku prvočíselného pole) se užívá metoda založená na přímém výpočtu, tzv. naivní algoritmus. Velkou pomocí v této problematice je Hasseho teorém, který omezuje řád eliptické křivky intervalem. Pro větší prvočísla se s úspěchem používají Shanksův algoritmus a jeho vylepšení Mestreho algoritmus. Oba algoritmy mají dvě části - Baby Step a Giant Step. Shanksův algoritmus je však v určitých případech nepoužitelný a tento problém řeší Mestreho algoritmus, který používá pojem twist eliptické křivky. Díky Mestreho teorému bylo dokázáno, že řád eliptické křivky nad prvočíselným polem muže být spočten pro každé prvočíslo vetší než 457. Důkaz, který spočívá především v isomorfismu okruhu endomorfismů nad eliptickými křivkami a imaginárního kvadratického řádu, je uveden na závěr této práce.
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The transfer of elliptic curves onto the torus
Bajko, Jaroslav ; Hrdina, Jaroslav (referee) ; Kureš, Miroslav (advisor)
Elliptic curves are an essential part of modern mathematics and play an important role especially in cryptography. The bachelor work focuses on the visualization elliptic curves and group operation in real plane and torus. In the first chapter we will introduce elliptic curves over field of real numbers and above all over prime fields. In order to describe the problematics rigorously the graphical outputs and also the experimental results in the field of discrete elliptic curves will be mentioned. In the next section we will pay a particular attention to topology, functions between topological spaces and to the introduction of the concept of smooth manifold. We will search the suitable functions which can transfer geometrical objects from the real plane onto torus. A software specifically developed for transfering the elliptic curves onto the torus works on the basis of aforementioned functions.
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