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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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Problém realizace von Neumannovsky regulárních okruhů
Mokriš, Samuel ; Růžička, Pavel (advisor) ; Žemlička, Jan (referee)
Title: The realization problem for von Neumann regular rings Author: Samuel Mokriš Department: Department of Algebra Supervisor of the master thesis: Mgr. Pavel Růžička, Ph.D., Department of Algebra Abstract: With every unital ring R, one can associate the abelian monoid V (R) of isomor- phism classes of finitely generated projective right R-modules. Said monoid is a conical monoid with order-unit. Moreover, for von Neumann regular rings, it satisfies the Riesz refinement property. In the thesis, we deal with the question, under what conditions an abelian conical re- finement monoid with order-unit can be realized as V (R) for some unital von Neumann regular ring or algebra, with emphasis on countable monoids. Two generalizations of the construction of V (R) to the context of nonunital rings are presented and their interrelation is analyzed. To that end, necessary properties of rings with local units and modules over such rings are devel- oped. Further, the construction of Leavitt path algebras over quivers is presented, as well as the construction of a monoid associated with a quiver that is isomorphic to V (R) of the Leavitt path algebra over the same quiver. These methods are then used to realize directed unions of finitely generated free abelian monoids as V (R) of algebras over any given field. A method...
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Gröbner bases
Petržilková, Lenka ; Žemlička, Jan (advisor) ; Růžička, Pavel (referee)
In this thesis we remind you of the basic Buchberger algorithm for com- puting the Gröbner base over commutative polynomial rings. We also observe uniqueness of the Gröbner base for the ideal. Next we research less known, but more effective (for some instances) Faugère F4 algorithm. At the end of the first chapter we compare these two algorithms. In the second chapter we analyze a generalization of the Buchberger algorithm for noncommutative rings both for free algebra and factor algebra. On the contary to the commu- tative case, Gröbner bases can be infinite in this case, even for some finitely generated ideals. Among other things, we investigate quasi-zero elements,i.e. such elements, that we get zero by multiplying them with an arbitrary term, and their role in the division of a polynom by set of polynoms. 1
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Algebraický přístup k CSP
Bulín, Jakub ; Barto, Libor (advisor) ; Růžička, Pavel (referee)
For a finite relational structure A, the Constraint Satisfaction Problem with template A, or CSP(A), is the problem of deciding whether an input relational structure X admits a homomorphism to A. The CSP dichotomy conjecture of Feder and Vardi states that for any A, CSP(A) is either in P or NP-complete. In the first part we present the algebraic approach to CSP and summarize known results about CSP for digraphs, also known as the H-coloring problem. In the second part we study a class of oriented trees called special polyads. Using the algebraic approach we confirm the dichotomy conjecture for special polyads. We provide a finer description of the tractable cases and give a construction of a special polyad T such that CSP(T) is tractable, but T does not have width 1 and admits no near-unanimity polymorphisms.
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