|
|
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.
|
|
|
Rabin-Miller test and the choice of a basis
Franců, Martin ; Simon, Petr (advisor) ; Čunát, Vladimír (referee)
This thesis is dedicated to various choices of basis in Rabin-Miller test. Short overview of similar methods is shown and some properties of structure of the set of strong liars are proved in theoretical part. Selected innovative choices of basis are tested on the set of odd composite numbers in range of 100 and 200 000 000 and the results are compared to results of tests with usual choices of bases. Hypothesis about possible improvement of test through using basis of special form with regard to tested number is proposed. Program used for compu- tations of these results is included. The program allows user to compare results of tests with various ways of choosing basis. The second part of the thesis contains documentation of the program.
|
|
|
Lucas-Lehmer test
Vejpustek, Ondřej ; Holub, Štěpán (advisor) ; Žemlička, Jan (referee)
The aim of this thesis is to explain quadratic number field theory and prove correctness of the Lucas-Lehmer primality test. A quadratic number field is a field of the form Q( √ m). Chapter one describes elementary properties of such field's ring of integers focusing on characterisation of the ring's group of units. Chapter two studies ideal factorisation in this ring. It contains proofs of a theorem on unique factorisation of the ideals into prime ideals and a theorem describing all prime ideals. Chapter three employs quadratic number field theory to prove correctness of the Lucas-Lehmer prime test, which is a deterministic primality test for numbers of the form 2p − 1. 1
|
|
|
Rabin-Miller test and the choice of a basis
Franců, Martin ; Simon, Petr (advisor) ; Čunát, Vladimír (referee)
This thesis is dedicated to various choices of basis in Rabin-Miller test. Short overview of similar methods is shown and some properties of structure of the set of strong liars are proved in theoretical part. Selected innovative choices of basis are tested on the set of odd composite numbers in range of 100 and 200 000 000 and the results are compared to results of tests with usual choices of bases. Hypothesis about possible improvement of test through using basis of special form with regard to tested number is proposed. Program used for compu- tations of these results is included. The program allows user to compare results of tests with various ways of choosing basis. The second part of the thesis contains documentation of the program.
|
|
|
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
|
|
|
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.
|
guest ::
