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Key dependent message security
Hostáková, Kristina ; Hojsík, Michal (advisor) ; Kazda, Alexandr (referee)
In this work, we deal with cryptosystems which are provably secure even if we encrypt a key-dependent message. These cryptosystems are called KDM-secure. First, we define KDM-security and discuss its relationship with other kinds of security, especially IND-CPA-security. Thereafter, we construct the public- key and the symmetric-key encryption scheme of Applebaum et al. (CRYPTO 2009) and we prove KDM-security of these cryprosystems with respect to the set of affine functions. The security of our cryptosystems is based on the LWE problem and the LPN problem as its special case. We study these problems and their variants. Moreover, we give a brief introduction to lattices and hard lattice problems because there exist reductions from hard lattice problems to LWE. 1
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Model building using CSP
Peterová, Alena ; Stanovský, David (advisor) ; Kazda, Alexandr (referee)
In the present work, we study algorithms for building finite models of sets of first-order axioms with the aim of proposing and implementing a new method, based on translation onto constraint satisfaction problem (CSP). In the theoretical part, we describe the standard MACE-style method, based on translating problems onto SAT, and advanced techniques that improve the effectiveness of this method: clause splitting, term definitions and static symmetry reduction. Next, we propose an alternative method, which translates problems onto CSP in a similar way. In addition, we have newly proposed a static symmetry reduction technique for binary functions. Next, we describe an implementation of the alternative method using a CSP-modelling language MiniZinc and a CSP-solver Gecode. Finally, we compare performance of our model finder against state-of-the-art representatives of standard methods, systems Paradox and Mace4.
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Topological properties of algebraic curves
Hudec, Pavel ; Šťovíček, Jan (advisor) ; Kazda, Alexandr (referee)
The thesis aims to present a theory about algebraic curves over complex numbers from the topological perspective. The main result proved in the thesis is the classical degree-genus formula which states that in the projective setting, non-singular algebraic curves are compact surfaces whose genus depends only on the degree of the curve itself. The presented proof relies heavily on algebraic topology; it is shown that the curve acts as a covering space for the projective line (without a finite set of images of ramified points), then a suitable triangulation of a projective line is lifted to the curve. Later, we discuss how our result relates to the popular definition of genus as the number of handles attached to the sphere. Finally, we briefly go through singular curves showing that the degree-genus formula cannot, in general, be applied to them. 1
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Symbolické reprezentace kompaktních prostorů
Kazda, Alexandr
Title: Symbolic representations of compact spaces Author: Alexandr Kazda Department: Department of Algebra Supervisor: Prof. RNDr. Petr Kůrka, CSc. Supervisor's e-mail address: kurka@cts.cuni.cz Abstract: The thesis concerns itself with Möbius number systems. These systems represent points using sequences of Möbius transformations. We are mainly inter- ested in representing the unit circle (which is equivalent to representing R ∪ {∞}). The main aim of the thesis is to improve already known tools for proving that a given subshift-iterative system pair is in fact a Möbius number system. We also study the existence problem: How to describe iterative systems resp. subshifts for which there exists a subshift resp. iterative system such that the resulting pair forms a Möbius number system. While we were unable to provide a complete answer to this question, we present both positive and negative partial results. As Möbius number systems are also subshifts, we can ask when a given Möbius number system is sofic. We give this problem a short treatment at the end of our thesis. Keywords: Möbius transformation, numeral system, subshift
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Complex algebraic curves
Zvěřina, Adam ; Šťovíček, Jan (advisor) ; Kazda, Alexandr (referee)
The thesis describes the relationship between algebraic curves and Riemann surfaces. We define Weierstrass ℘-function and prove some of its properties. We further prove that every complex algebraic curve can be regarded as a Riemann surface. Finally, we demonstrate that an elliptic curve can be parametrised with Weierstrass ℘-function. 1
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Logic circuits as models of computation
Naumenko, Mykhailo ; Kazda, Alexandr (advisor) ; Kompatscher, Michael (referee)
This work focuses on the study of logic circuits. We investigated the basics of the theory of logic circuits following the textbook "Models of Computation" by John E. Savage and we used this knowledge to solve some of the examples and problems suggested in the textbook. In this work, you can find key concepts related to logical circuits. Our main topic is the estimation of the lower bounds of the circuit size and formula size of general Boolean function. We constructed simple examples of some known circuits and showed how the circuit designs may be offered. 1
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