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Alternative methods in cryptology
Hampl, Dalibor ; Doležel, Radek (referee) ; Hajný, Jan (advisor)
In the first part of this thesis explains, what is generally cryptography, generally analyze the problem of the current encryption methods. These include asymmetric, symmetric and hybrid encryption methods. The second part focuses on the theory of lattices and their main mathematical problems, then the shortest vector problem, the closest vector problem, the approximate shortest vector problem and the approximate closest vector problem. Mathematically difficult problems of lattice are used in an encryption method GGH for asymmetrically encrypted communications method and GGH for digital signatures. It explains mathematical methods, which use these encryption methods. Another part of this thesis deals with the encryption method and the establishment of XTR on Diffie-Hellman protocol, thus XTR-DH. It explains mathematical methods, which use methods XTR, XTR-DH and Diffie-Hellman protocol. There are shown the main mathematical problems for this part of thesis DHP (Diffie-Hellman problem) and DLP (discrete logarithm problem). The fourth part focuses on a program designed to demonstrate the GGH encryption method for asymmetrically encrypted communications. Describes the structure, functionality, graphical part of the program and shows to a user interface. Students can test how the GGH encryption method behave when entering different input values. During the work with the program, the user should obtain sufficient information to understand the principles of this encryption method.
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PWR fuel assembly model
Cekl, Jakub ; Novotný, Filip (referee) ; Vojáčková, Jitka (advisor)
PWR, fuel assembly, benchmark, burnup, lattice, SCALE, Polaris, validation, reactivity
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Static Value Analysis over C Programs
Ďuričeková, Daniela ; Peringer, Petr (referee) ; Vojnar, Tomáš (advisor)
Value-range analysis is a static analysis technique based on arguing about the values that a variable may take on a given program point. It can be used to prove absence of run-time errors such as out-of-bound array accesses. Since value-range analysis collects information on each program point, data-flow analysis can be used in association with it. The main goal of this work is designing and implementing such a value-range analysis tool. The work begins with an introduction into the topic, an explanation of data-flow and value-range analyses and a description of abstract interpretation, which provides the formal basis of the analyser. The core of this work is the design, implementation, testing and evaluation of the analyser. In the conclusion, our personal experience obtained in the area of the thesis is mentioned, along with a discussion of a possible future development of the designed tool.
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Lattice constructions and Priestley duality
Hartman, Juraj ; Růžička, Pavel (advisor) ; Tůma, Jiří (referee)
In this thesis after recalling some basic definitions and theorems in category theory, lattice theory and topology we first introduce the so called Stone duality of the category of boolean lattices and the category of boolean topological spaces. Then we introduce its generalization, the so called Priestley duality of the category of bounded distributive lattices and the category of total order disconnected topological spaces. Then we introduce the M3[.] lattice construction and prove that for every bounded distributive lattice L there is an isomorphism from the lattice M3[L] to the lattice of all continuous monotone maps from the Priestley space of L to the lattice M3 with discrete topology. Finally we introduce the so called boolean power, which we generalize to the so called priestley power and we prove that for every natural number n ≥ 3 and every bounded distributive lattice L there is an isomorphism from the lattice Mn to the priestley power of the lattice Mn by the lattice L. 1
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Short invertible elements in cyclotomic rings
Kroutil, Jaroslav ; Žemlička, Jan (advisor) ; Příhoda, Pavel (referee)
This bachelor's thesis is based on an article about the criterion for invertibility of elements in special- chosen cyclotomic rings. In this thesis, we start with defining important terms and statements from algebra that we need. Then we will deal with the existence of infinitely many prime numbers which satisfy conditions that are used for irreducible decomposition of cyclotomic polynomials. Based on this polynomials we define cyclotomic rings and at the end of this thesis we prove invertibility of elements from this rings depending on the size of their norm.
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LWE and provably secure key exchange schemes
Václavek, Jan ; Příhoda, Pavel (advisor) ; Žemlička, Jan (referee)
The threat of large-scale quantum computers motivates cryptographers to base cryptosystems on problems believed to be resistant against quantum computers. In this thesis, we focus on the LWE problem which is believed to be resistant against quantum computers. First, we describe lattices which are closely related to the LWE problem. We introduce basic notions, describe lattice problems and solve exercises related to the covering radius of lattice. After that, we introduce the LWE problem and its variants. We prove reductions from two lattice problems to certain variant of the LWE problem. We define the notion of statistical distance and prove some lemmata about it which we need within reductions. Moreover, we show concrete application of the LWE problem. We describe a scheme for key exchange and briefly prove its security under the assumption that the LWE problem is hard. 1
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PWR fuel assembly model
Cekl, Jakub ; Novotný, Filip (referee) ; Vojáčková, Jitka (advisor)
PWR, fuel assembly, benchmark, burnup, lattice, SCALE, Polaris, validation, reactivity
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Data Visualisation Methods in the Statistical Computational Programme R
Hanková, Monika ; Procházka, Jiří (advisor) ; Malá, Ivana (referee)
Visualisation is a key element for analysis in statistics. This thesis describes data visualisation methods in the statistical computational programme R which, thanks to add-on packages, contains several tools for plotting. Data visualisation may be much more complicated than it seems to be, the focus is aimed at the general connection between packages as well as the particular visualisation of base graphics, ggplot2 and lattice package. These packages are built based on the traditional graphical model in R, grammar of graphics and trellis plotting. The interactive shiny app provides an overall view about how the data are visualised by each package.
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Maximal Orders
Tlustá, Stanislava ; Příhoda, Pavel (advisor) ; Růžička, Pavel (referee)
Maximal Orders Stanislava Tlustá Abstract This thesis summarizes basic properties of lattices and orders over Dedekind domain in separable algebras. The concepts of reduced norm and reduced trace are introduced and applied to few examples of rational algebras. By that the maximal orders are found. The properties of maximal orders are stated and used to explore new types of ideals: normal ideals and Λ-ideals. At the end of this thesis the isomorphisms of lattices are examined and the Jordan-Zassenhaus theorem is proved. 1
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