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Varieties of superalgebras
Lišková, Adéla ; Žemlička, Jan (advisor) ; Barto, Libor (referee)
The goal of the thesis is to introduce the basics of the theory of superalgebras, that is Z2-graded algebras over a field of characteristic different from two, as well as to present necessary basics of universal and multilinear algebra, especially the tensor product and the terms variety of algebra and ideal of identities. We present the definitions of algebra and superalgebra including examples, we then look into the tensor product of superalgebras and its properties, Clifford and Grassmann superalgebras. A part of the thesis is dedicated to the construction of the free nonassociative algebra and the clarification of the relationship between varieties of algebras and ideals of identities including the specification of said relationship for superalgebras. The thesis also deals with varieties of superalgebras. 1
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Applications of Gröbner bases in cryptography
Fuchs, Aleš ; Šťovíček, Jan (advisor) ; Žemlička, Jan (referee)
Title: Applications of Gröbner bases in cryptography Author: Aleš Fuchs Department: Department of Algebra Supervisor: Mgr. Jan Št'ovíček Ph.D., Department of Algebra Abstract: In the present paper we study admissible orders and techniques of multivariate polynomial division in the setting of polynomial rings over finite fields. The Gröbner bases of some ideal play a key role here, as they allow to solve the ideal membership problem thanks to their properties. We also explore features of so called reduced Gröbner bases, which are unique for a particular ideal and in some way also minimal. Further we will discuss the main facts about Gröbner bases also in the setting of free algebras over finite fields, where the variables are non-commuting. Contrary to the first case, Gröbner bases can be infinite here, even for some finitely generated two- sided ideals. In the last chapter we introduce an asymmetric cryptosystem Polly Cracker, based on the ideal membership problem in both commutative and noncommutative theory. We analyze some known cryptanalytic methods applied to these systems and in several cases also precautions dealing with them. Finally we summarize these precautions and introduce a blueprint of Polly Cracker reliable construction. Keywords: noncommutative Gröbner bases, Polly Cracker, security,...
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Factorization of polynomials over finite fields
Straka, Milan ; Žemlička, Jan (advisor) ; Stanovský, David (referee)
Nazcv prace: Faktorizace polynoinu nad konccnynii telesy Autor: Milan Straka Katcdra (ustav): Katcdra algebry Vedouci bakalarske prace: Mgr. Jan Zcmlicka, Ph.D. E-mail vedouciho: Jan.Zemlicka((hnff. cuni.cz Abstrakt: Cilem prace je prozkoumat problem rozkladu polynomn nad konecnym telc- scm na soucin ircducibilnich polynoinu. PopHanim nekolika algoritmu hledaji- cich tento rozklad se ukaze, ze tento problem je vzdy fcsitclny v polynornialnim case vzhleclem kc stupni polynomu a poctu prvku konecneho telcsa. U jeduoho z algoritnm je po])sana implenientace s vclnii clobrou asymptotic- kou casovou slozito.sti O(nLylD log c/}, kdc i\. jc stupen rozkladaneho polynuinn nad telesem « q prvky. Program pouzivajiei jcdnodnssi, ale prakticky rychlcjsi variantu tohoto algoritnm jc soucasti ])racc. Klicova slova: faktorizace, kouecna telesa, polynoniy, algoritmns Title: Factoring polynomials over finite fields Author: Milan Straka Department: Department of Algebra Supervisor: Mgr. Jan Zemlicka, Ph.D. Supervisor's e-mail address: Jan. Zcirilicka@mJJ.cum.cz Abstract: The goal of this work is to present the problem of the decomposition of a polyno- mial over a finite field into a product of irreducible polynomials. By describing algorithms solving this problem, we show that the decomposition can always be found in...
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Multivariate cryptography
Jančaříková, Irena ; Žemlička, Jan (advisor) ; Růžička, Pavel (referee)
This thesis deals with multivariate cryptography. It includes specifically a description of the MQ problem and the proof of it's NP-completness. In the part of the MQ problem there is a description of a general pattern for the creation of the public part of asymetric cryptosystems based on the MQ problem. It this part the thesis describes the QMLE problem, which is important for the figure of the cryptosystem private key based on the MQ problem. Further, the thesis includes a description of the influence of the structure display, which appears in the QMLE problem, on time solution complexity of QMLE problem. The influence of time complexity has been detected by means of experimental measurement with programed algorithm. At the end of the thesis there is specified description of selected multivariety cryptosystems based on the MQ problem. Selected cryptosystems are provided with detailed description of encryption and decryption by means of selected cryptosystems and time estimations of these operations. The thesis includes estimations of memory requirements on saving of private and public key of the selected cryptosystems. Powered by TCPDF (www.tcpdf.org)
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Structure of self-small groups and modules
Dvořák, Josef ; Žemlička, Jan (advisor) ; Šaroch, Jan (referee)
Title: Structure of self-small groups and modules Author: Josef Dvořák Department: Department of Algebra Supervisor: Mgr. Žemlička Jan, Ph.D. Supervisor's e-mail address: zemlicka@karlin.mff.cuni.cz Abstract: The thesis sums up the basic properties of self-small groups. Furthermore it thoroughly builds the theory od quotient categories by Serre classes, with focus on quotient category modulo the class B of boun- ded groups, which, as demonstrated, is equivalent to the quasicategory, i.e. category of abelian groups with Hom-sets being Q⊗Z HomA (A, B). This approach is developed into the theory of generalized quasi-categories. The dualities between quasi-caterogories od torsion-free and quotient-divisible categories of finite rank, resp. between categories of finite-rank self-small groups are studied and they are emloyed to the partial solution of Fuchs' problem no. 34. Keywords: self-small group, quotient divisible group, quasicategory, quo- tient category 1
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Comparison of public key cryptography algorithms
Mareš, Jiří ; Stanovský, David (advisor) ; Žemlička, Jan (referee)
In the present work we study comparison of basic public key encryption algorithms - RSA, Rabin and ElGamel method. We derive theoretic complexity of encrypting / decrypting of one block and we derive an expected model of its behavior with the key of double size. We also take practical measurements of speed of each algorithm using keys sized 64 - 4096 bits and we statistically analyze the results. We also mention special cases of some algorithms and discuss the advantages and disadvantages of their practical usage. At the end of this thesis we make a comparison of the speed of algorithms and we also compare the measured data with theoretical hypothesis.
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Vychylující moduly nad Gorensteinovými okruhy
Pospíšil, David ; Trlifaj, Jan (advisor) ; Žemlička, Jan (referee)
Let R be a commutative 1-Gorenstein ring. Our main result characterizes all tilting and cotilting R-modules: up to equivalence: they are parametrized by subsets of the set of all prime ideals of height one. More precisely, every tilting (cotilting) R-module is equivalent to some Bass tilting (cotilting) module. This characterization was known in the particular case of Dedekind domains: Chapter 4 contains a new and simpler proof of this fact. Our main result is proved in Chapter 5, while Chapter 6 deals with the cotilting case. In Chapter 4, there is also a proof of the less well-known fact that all finitely generated tilting modules over commutative rings are projective.
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