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Linear Diophantine equations and congruences
Kaňáková, Natálie ; Beran, Filip (advisor) ; Jančařík, Antonín (referee)
This bachelor's thesis summarizes and systematizes knowledge about congruences and linear Diophantine equations. This work is divided into two parts. The first part is dedicated to congruences. At first, it shows where we can find congruences in real life, congruence as a relation, its properties, and applications in calculating the last ciphers of large numbers, proofs of divisibility rules, or calculating the date of Easter. Afterward, we look into congruences containing unknowns - linear congruence equations. It looks into methods of solving linear congruences and illustrates them in exercises. The last topic of the first part is oriented on systems of linear congruences and the Chinese remainder theorem, both for non-coprime and coprime moduli, the algebraic version, applications in various types of problems, and modular representation of numbers. The second part of this thesis is dedicated to linear Diophantine equations - equations with integer solutions. It shows various methods of solving linear Diophantine equations with two, three, or more unknowns - the extended Euclidean algorithm, reduction method, substitution method, and others. This part also describes the relationship between linear congruences and linear Diophantine equations and the use of this relationship in solving both linear...
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Multilinear Maps Over the Integers
Havránek, František ; Žemlička, Jan (advisor) ; Šaroch, Jan (referee)
The thesis aims to describe the [CLT15] scheme, which is based on the Diffie- Hellman scheme and uses multilinear maps over integers. This scheme enables an exchange of a key among several participants. The level κ scheme (using a κ-linear map) enables the exchange of a key among κ + 1 participants. The thesis introduces the basic terms, describes the needed theory, the base of which is the Chinese Remainder Theorem, and also the preparation and usage of the scheme. The correctness of the scheme is proved as well and the related requirements on the basic parameters are discussed.
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