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Algorithm for word morphisms fixed points
Matocha, Vojtěch ; Holub, Štěpán (advisor) ; Žemlička, Jan (referee)
In the present work we study the first polynomial algorithm, which tests if the given word is a fixed point of a nontrivial morphism. This work contains an improved worst-case complexity estimate O(m · n) where n denotes the word length and m denotes the size of the alphabet. In the second part of this work we study the union-find problem, which is the crucial part of the described algorithm, and the Ackermann function, which is closely linked to the union-find complexity. We summarize several common methods and their time complexity proofs. We also present a solution for a special case of the union-find problem which appears in the studied algorithm. The rest of the work focuses on a Java implementation, whose time tests correspond to improved upper bound, and a visualization useful for particular entries.
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Advanced techniques for calculations of discrete logarithm
Matocha, Vojtěch ; Příhoda, Pavel (advisor) ; Jedlička, Přemysl (referee)
Let G be a finite cyclic group. Solving the equation g^x = y for a given generator g and y is called the discrete logarithm problem. This problem is at the core of many modern cryptographic transformations. In this paper we provide a survey of algorithms to attack this problem, including the function field sieve, the fastest known algorithm applicable to the multiplicative group of a finite field. We also discuss the index calculus algorithm and some techniques improving its performance: the Coppersmith's algorithm and the polynomial sieving. The most important contribution of this paper is a C-language implementation of the function field sieve and its application to real inputs.
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Advanced techniques for calculations of discrete logarithm
Matocha, Vojtěch ; Příhoda, Pavel (advisor) ; Jedlička, Přemysl (referee)
Let G be a finite cyclic group. Solving the equation g^x = y for a given generator g and y is called the discrete logarithm problem. This problem is at the core of many modern cryptographic transformations. In this paper we provide a survey of algorithms to attack this problem, including the function field sieve, the fastest known algorithm applicable to the multiplicative group of a finite field. We also discuss the index calculus algorithm and some techniques improving its performance: the Coppersmith's algorithm and the polynomial sieving. The most important contribution of this paper is a C-language implementation of the function field sieve and its application to real inputs.
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Algorithm for word morphisms fixed points
Matocha, Vojtěch ; Holub, Štěpán (advisor) ; Žemlička, Jan (referee)
In the present work we study the first polynomial algorithm, which tests if the given word is a fixed point of a nontrivial morphism. This work contains an improved worst-case complexity estimate O(m · n) where n denotes the word length and m denotes the size of the alphabet. In the second part of this work we study the union-find problem, which is the crucial part of the described algorithm, and the Ackermann function, which is closely linked to the union-find complexity. We summarize several common methods and their time complexity proofs. We also present a solution for a special case of the union-find problem which appears in the studied algorithm. The rest of the work focuses on a Java implementation, whose time tests correspond to improved upper bound, and a visualization useful for particular entries.
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