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Combinatorial group theory and cryptography
Ferov, Michal ; Příhoda, Pavel (advisor) ; Růžička, Pavel (referee)
In the presented work we focus on applications of decision problems from combinatorial group theory. Namely we analyse the Shpilrain-Zapata pro- tocol. We give formal proof that small cancellation groups are good platform for the protocol because the word problem is solvable in linear time and they are generic. We also analyse the complexity of the brute force attack on the protocol and show that in a theoretical way the protocol is immune to attack by adversary with arbitrary computing power.
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Compressing Pseudorandom sequences
Vald, Denis ; Holub, Štěpán (advisor) ; Růžička, Pavel (referee)
Generators of pseudorandom sequences are widely used objects, not in the least place because of their application in stream ciphers. One of the ways to improve resistance to different types of attack is to use compression on the generated sequence in order to remove redundant information, that might lead to an attack against the generator. In this work we try to explore from a wider perspective the theoretical foundations for compressing pseudorandom sequences created thus far. Using this general view we will examine some known attacks against the PRN generators and look for a way to resist such attacks.
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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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Cryptographic schemes based on the discrete logarithm problem
Kadlček, Tomáš ; Holub, Štěpán (advisor) ; Růžička, Pavel (referee)
In the paper we try to give a view of the discrete logarithm problem, especially of related problems that appear in literature since 2001. These problems are based on a computation of Weil and Tate pairing on eliptic curves. We give a view of these problems including some reductions. We mention some chosen schemes based on these problems that are iteresting because of their practical parametrs, primaci of security proofs or because these schemes introduced the new problem. We try to cover precisely the most important definitions in this sector of cryptography because these definition are omitted in the literature and it is often left up to reader to presume details by himself.
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