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Lineární kódy nad okruhy
Kobrle, Tomáš ; Šťovíček, Jan (advisor) ; Příhoda, Pavel (referee)
This master thesis focus on special type of rings called path algebras with a goal to define and describe codes over these rings. The path algebras are defined by graphic structures called quivers which is transferred also on the modules of the path algebra. Codes themselves are defined over indecomposible injective modules of path algebra considering the latest result in ring-coding theory. So defined codes allow us to study the parameters and the versions of elementary theorems from theory of linear codes over fields for codes over rings. These are about duals codes especially, the MacWilliams identity theorem and about code equivalency. Finally we get back to path algebras and describe a way to make them applicable in theory of codes over rings.
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Description of the cryptosystem HFE
Jančaříková, Irena ; Žemlička, Jan (advisor) ; Příhoda, Pavel (referee)
This bachelor thesis deals with the description of the assymetric HFE cryptosystem. This thesis contains encryption and decryption using this cryptosystem, estimations of the time complexity of private and public transformation and the memory requirements to store secret and public keys. Thesis also contains a basic description of the predecessor HFE cryptosystem, cryptosystem C *. The work includes a short passage about MQ problem, which are cryptosystems based on and a short treatise on finite fields over which cryptosystems are both defined. The paper deals with the attack, which proves possibility of breaking C * for the majority of encrypted messages and contains a variant of this attack for HFE cryptosystem.
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Kompaktní objekty v kategoriích modulů
Kálnai, Peter ; Žemlička, Jan (advisor) ; Příhoda, Pavel (referee)
Title: Compact objects in categories of modules Author: Peter Kálnai Department: Department of Algebra Supervisor: Mgr. Jan Žemlička, Ph.D., Department of Algebra Abstract: In the thesis we state baic properties of compact objects in various appropriate categories like categories of modules, stable factor category over a perfect ring and Grothendieck categories. We find a ring R such that the class of dually slender R-modules is closed under direct products under some set-theoretic assumption. Finally, we characterize the conditions, when countably generat- ed projective modules are finitely generated, expressed by their Grothendieck monoid. Keywords: compact, dually slender module, stable module category, projective module, self-small
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Set-theoretic Methods in the Theory of Modules
Šaroch, Jan ; Trlifaj, Jan (advisor) ; Příhoda, Pavel (referee) ; Struengmann, Lutz (referee)
The thesis collects my actual contributions to the theory of cotorsion pairs, with closer attention paid to the application of set-theoretic methods in this area. It consists of an introduction and three papers with coauthors. The first two, already published, deal with the so-called Telescope Conjecture for Module Categories. We prove here, for instance, that a hereditary cotorsion pair (A, B) with the class B closed under direct limits is generated by a set of countably presented modules. Moreover, if the class A is closed under direct limits too, then the pair (A, B) is cogenerated by a set of indecomposable pure-injective modules. In the third paper, we deal with the cotorsion pairs which provide us with non-trivial examples of abstract elementary classes (in the sense of Shelah). Then we study the class D of all 1-projective modules, proving e.g. that-regardless of the ring-it always forms a Kaplansky class.
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Attacks based on hardware bugs
Válková, Martina ; Holub, Štěpán (advisor) ; Příhoda, Pavel (referee)
Title: Attacks based on hardware bugs Author: Martina Válková Department: Department of Algebra Supervisor: Mgr. Štěpán Holub, Ph.D. Supervisor's e-mail address: Stepan.Holub@mff.cuni.cz Abstract: The study concerns hardware bugs producing computational errors and cryptanalytic attacks which utilize them. Particularly, the research is focused on attacks presented in the article by Biham E., Carmeli Y., Shamir A.: Bug Attacks [1] and their practical application in the case of schemes RSA and Pohlig-Hellman and various computational circumstances, which points out bigger vulnerability of schemes in the case of using the Right- to-Left modular exponentiation algorithm. The attacks have been tested against the software simulation of a faulty processor, which confirmed that they pose a real security threat in point of that situation. The mathematical part of this work concerns the problem of the finding any roots in Zp. Keywords: hardware bug, attack, RSA, roots modulo p
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