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Point Counting on Elliptic and Hyperelliptic Curves
Vácha, Petr ; Šťovíček, Jan (advisor) ; Drápal, Aleš (referee)
In present work we study the algorithms for point counting on elliptic and hy- perelliptic curves. At the beginning we describe a few simple and ineffective al- gorithms. Then we introduce more complex and effective ways to determine the point count. These algorithms(especially the Schoof's algorithm) are important for the cryptography based on discrete logarithm in the group of points of an el- liptic or hyperelliptic curve. The point count is important to avoid the undesirable cases where the cryptosystem is easy to attack. 1
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Weil pairing
Luňáčková, Radka ; Drápal, Aleš (advisor) ; Šťovíček, Jan (referee)
This work introduces fundamental and alternative definition of Weil pairing and proves their equivalence. The alternative definition is more advantageous for the purpose of computing. We assume basic knowledge of elliptic curves in the affine sense. We explain the K-rational maps and its generalization at the point at infinity, rational map. The proof of equivalence of the two mentioned definitions is based upon the Generalized Weil Reciprocity, which uses a concept of local symbol. The text follows two articles from year 1988 and 1990 written by L. Charlap, D. Robbins a R. Coley, and corrects a certain imprecision in their presentation of the alternative definition. Powered by TCPDF (www.tcpdf.org)
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Complex algebraic curves
Zvěřina, Adam ; Šťovíček, Jan (advisor) ; Kazda, Alexandr (referee)
The thesis describes the relationship between algebraic curves and Riemann surfaces. We define Weierstrass ℘-function and prove some of its properties. We further prove that every complex algebraic curve can be regarded as a Riemann surface. Finally, we demonstrate that an elliptic curve can be parametrised with Weierstrass ℘-function. 1
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Implementation of cryptographic protocols on smart cards
Moravanský, Michal ; Hajný, Jan (referee) ; Dzurenda, Petr (advisor)
The bachelor thesis is focused on cryptographic schemes using Attribute-Based Credentials, which try to minimize the negative impact on the protection of a user's privacy when using authentication systems. The aim of the bachelor's thesis was the implementation of two specified schemes on smart cards as a device with limited performance. Schemes differ only in the ability to revoke user. The practical part of this paper contains the analysis and selection of smart card platform and cryptographic libraries depending on performance. The work also describes the architecture of both schemes and individual protocols, including ongoing communication. The implementation of the Attribute-Based Credentials scheme was performed on a programmable smart card Multos (user side) and Raspberry Pi 2 (issuer and verifier side). The time complexity of the selected algorithms was also compared. In the end, the dependencies affecting the resulting efficiency and speed of the protocol were formulated.
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Counting the points on elliptic curves over finite fields
Eržiak, Igor ; Šťovíček, Jan (advisor) ; Stanovský, David (referee)
The goal of this thesis is to explain and implement Schoof's algorithm for counting points on elliptic curves over finite fields. We start by defining elliptic curve as a set of points satisfying certain equation and then proceeding to define an operation on this set. Theoretical background needed for the algorithm is presented in the second chapter. Finally, the Schoof's algorithm is introduced in the third chapter, supplemented by an implementation in SageMath open-source software.
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Cryptographic protocols for privacy protection
Hanzlíček, Martin ; Dzurenda, Petr (referee) ; Hajný, Jan (advisor)
This work focuses on cryptographic protocol with privacy protection. The work solves the question of the elliptic curves and use in cryptography in conjunction with authentication protocols. The outputs of the work are two applications. The first application serves as a user and will replace the ID card. The second application is authentication and serves as a user authentication terminal. Both applications are designed for the Android operating system. Applications are used to select user attributes, confirm registration, user verification and show the result of verification.
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Weil pairing
Luňáčková, Radka ; Drápal, Aleš (advisor) ; Šťovíček, Jan (referee)
This work introduces fundamental and alternative definition of Weil pairing and proves their equivalence. The alternative definition is more advantageous for the purpose of computing. We assume basic knowledge of elliptic curves in the affine sense. We explain the K-rational maps and its generalization at the point at infinity, rational map. The proof of equivalence of the two mentioned definitions is based upon the Generalized Weil Reciprocity, which uses a concept of local symbol. The text follows two articles from year 1988 and 1990 written by L. Charlap, D. Robbins a R. Coley, and corrects a certain imprecision in their presentation of the alternative definition. Powered by TCPDF (www.tcpdf.org)
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Rational points on elliptic curves
Raclavský, Marek ; Stanovský, David (advisor) ; Šťovíček, Jan (referee)
This thesis concerns with rational points on elliptic curves. By the Mordell theorem we know that the group of rational points on elliptic curve is finitely generated. First, we study torsion subgroup, which turns out to be well described by theorem of Nagell-Lutz. Next, we focus on torsion-free part, which is characterized by the notion of rank. The thesis consists of solved problems and we also provide a summary of theoretical foundations. We find points of finite order on particular elliptic curves and compute their ranks. Powered by TCPDF (www.tcpdf.org)
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Point Counting on Elliptic and Hyperelliptic Curves
Vácha, Petr ; Šťovíček, Jan (advisor) ; Drápal, Aleš (referee)
In present work we study the algorithms for point counting on elliptic and hy- perelliptic curves. At the beginning we describe a few simple and ineffective al- gorithms. Then we introduce more complex and effective ways to determine the point count. These algorithms(especially the Schoof's algorithm) are important for the cryptography based on discrete logarithm in the group of points of an el- liptic or hyperelliptic curve. The point count is important to avoid the undesirable cases where the cryptosystem is easy to attack. 1
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