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Hardware generation of cryptographic-safe primes.
Kabelková, Barbora ; Smékal, David (referee) ; Cíbik, Peter (advisor)
The bachelor's thesis deals with the topic of prime numbers and their generation. It briefly introduces prime numbers and points out the importance of secure primes in cryptography. It gives examples of asymmetric ciphers and closely analyses RSA algorithm. The thesis then presents some pseudo-random and true-random methods of generating sequences of numbers and compares their properties. It evaluates the most used primality tests, both probabilistic and real, based on their applicability in practice. It suggests several combinations of primality tests with generating methods and chooses one to implement on FPGA. The thesis describes the implementation of a generator that generates a sequence of numbers using the von Neumann middle-square method and subsequently uses the Miller-Rabin test to find primes between those numbers. Key processes of the proposed generator are explained and illustrated. The proposed implementation is simulated and synthesized in the Xilinx Viavado environment. The individual parts of the generator are tested using several behavioral simulations. Finally, the thesis comments on the conducted simulations and evaluates the properties of the proposed implementation.
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Parallelization of Integer Factorization from the View of RSA Breaking
Breitenbacher, Dominik ; Henzl, Martin (referee) ; Homoliak, Ivan (advisor)
This paper follows up the factorization of integers. Factorization is the most popular and used method for RSA cryptoanalysis. The SIQS was chosen as a factorization method that will be used in this paper. Although SIQS is the fastest method (up to 100 digits), it can't be effectively computed at polynomial time, so it's needed to look up for options, how to speed up the method as much as possible. One of the possible ways is paralelization. In this case OpenMP was used. Other possible way is optimalization. The goal of this paper is also to show, how easily is possible to use paralelizion and thanks to detailed analyzation the source codes one can reach relatively large speed up. Used method of iterative optimalization showed itself as a very effective tool. Using this method the implementation of SIQS achieved almost 100 multiplied speed up and at some parts of the code even more.
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Hardware generation of cryptographic-safe primes.
Kabelková, Barbora ; Smékal, David (referee) ; Cíbik, Peter (advisor)
The bachelor's thesis deals with the topic of prime numbers and their generation. It briefly introduces prime numbers and points out the importance of secure primes in cryptography. It gives examples of asymmetric ciphers and closely analyses RSA algorithm. The thesis then presents some pseudo-random and true-random methods of generating sequences of numbers and compares their properties. It evaluates the most used primality tests, both probabilistic and real, based on their applicability in practice. It suggests several combinations of primality tests with generating methods and chooses one to implement on FPGA. The thesis describes the implementation of a generator that generates a sequence of numbers using the von Neumann middle-square method and subsequently uses the Miller-Rabin test to find primes between those numbers. Key processes of the proposed generator are explained and illustrated. The proposed implementation is simulated and synthesized in the Xilinx Viavado environment. The individual parts of the generator are tested using several behavioral simulations. Finally, the thesis comments on the conducted simulations and evaluates the properties of the proposed implementation.
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Primality testing using elliptic curves
Pashchenko, Olha ; Barto, Libor (advisor) ; Šťovíček, Jan (referee)
In the present work we study primality tests. A primality test is an algorithm for determining whether an input number is prime. In the first part of this work we recapitulate the basic definitions and facts about number theory and study Pocklington's algorithm, that based on the group (Z/nZ)∗ . Then we study Generalized Pocklington's primality test and Pépin's primality test for Fermat numbers. In the second part of this work we represent the basic definitions and facts about elliptic curves. Then we study Goldwasser-Killian primality test, that based on elliptic curves. One part of this work is experementation with Goldwasser-Killian primality test. 1
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Parallelization of Integer Factorization from the View of RSA Breaking
Breitenbacher, Dominik ; Henzl, Martin (referee) ; Homoliak, Ivan (advisor)
This paper follows up the factorization of integers. Factorization is the most popular and used method for RSA cryptoanalysis. The SIQS was chosen as a factorization method that will be used in this paper. Although SIQS is the fastest method (up to 100 digits), it can't be effectively computed at polynomial time, so it's needed to look up for options, how to speed up the method as much as possible. One of the possible ways is paralelization. In this case OpenMP was used. Other possible way is optimalization. The goal of this paper is also to show, how easily is possible to use paralelizion and thanks to detailed analyzation the source codes one can reach relatively large speed up. Used method of iterative optimalization showed itself as a very effective tool. Using this method the implementation of SIQS achieved almost 100 multiplied speed up and at some parts of the code even more.
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