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Testing perfect powers
Straková, Hana ; Stanovský, David (advisor) ; Jedlička, Přemysl (referee)
A positive integer n is a perfect power if there exist integers x and k, both at least 2, such that n = x^k. Perfect power testing is important as preprocessing for number factorization and prime number testing, because many algorithms for that are not able to distinguish between prime number and power of prime number, so it is necessary to test it by perfect power tests. This thesis includes comparison of two algorithms for perfect power testing, one by Daniel J. Bernstein and the other by E. Bach & J. Sorenson. The goal is to implement described algorithms in C language with GMP library for multiple-precision arithmetics, to compare the theoretical results and running times of implemented algorithms.
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Factorization of polynomials over finite fields
Straka, Milan ; Žemlička, Jan (advisor) ; Stanovský, David (referee)
Nazcv prace: Faktorizace polynoinu nad konccnynii telesy Autor: Milan Straka Katcdra (ustav): Katcdra algebry Vedouci bakalarske prace: Mgr. Jan Zcmlicka, Ph.D. E-mail vedouciho: Jan.Zemlicka((hnff. cuni.cz Abstrakt: Cilem prace je prozkoumat problem rozkladu polynomn nad konecnym telc- scm na soucin ircducibilnich polynoinu. PopHanim nekolika algoritmu hledaji- cich tento rozklad se ukaze, ze tento problem je vzdy fcsitclny v polynornialnim case vzhleclem kc stupni polynomu a poctu prvku konecneho telcsa. U jeduoho z algoritnm je po])sana implenientace s vclnii clobrou asymptotic- kou casovou slozito.sti O(nLylD log c/}, kdc i\. jc stupen rozkladaneho polynuinn nad telesem « q prvky. Program pouzivajiei jcdnodnssi, ale prakticky rychlcjsi variantu tohoto algoritnm jc soucasti ])racc. Klicova slova: faktorizace, kouecna telesa, polynoniy, algoritmns Title: Factoring polynomials over finite fields Author: Milan Straka Department: Department of Algebra Supervisor: Mgr. Jan Zemlicka, Ph.D. Supervisor's e-mail address: Jan. Zcirilicka@mJJ.cum.cz Abstract: The goal of this work is to present the problem of the decomposition of a polyno- mial over a finite field into a product of irreducible polynomials. By describing algorithms solving this problem, we show that the decomposition can always be found in...
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Non-commutative Gröbner bases
Požárková, Zuzana ; Šťovíček, Jan (advisor) ; Stanovský, David (referee)
In the presented work we define non-commutative Gröbner bases including the necessary basis of non- commutative algebra theory and notion admissible ordering. We present non-commutative variant of the Buchberger algorithm and study how the algorithm can be improved. Analogous to the Gebauer-Möller criteria lead us to detect almost all unnecessary obstructions in the non-commutative case. The obstructions are graphically ilustrated. The Buchberger algorithm can be improved within redundant polynomials. This work is a summary and its specification of the results of some known authors engaged in this field. Presented definitions are ilustrated on examples. We perform proves of some of the statements which have been proven differently by other authors. Powered by TCPDF (www.tcpdf.org)
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Comparison of public key cryptography algorithms
Mareš, Jiří ; Stanovský, David (advisor) ; Žemlička, Jan (referee)
In the present work we study comparison of basic public key encryption algorithms - RSA, Rabin and ElGamel method. We derive theoretic complexity of encrypting / decrypting of one block and we derive an expected model of its behavior with the key of double size. We also take practical measurements of speed of each algorithm using keys sized 64 - 4096 bits and we statistically analyze the results. We also mention special cases of some algorithms and discuss the advantages and disadvantages of their practical usage. At the end of this thesis we make a comparison of the speed of algorithms and we also compare the measured data with theoretical hypothesis.
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Polynomial equations over finite fields and algebraic cryptanalysis
Seidl, Jan ; Stanovský, David (advisor) ; Drápal, Aleš (referee)
Title: Polynomial equations over finite fields and algebraic cryptanalysis Author: Jan Seidl Department: Department of Algebra Supervisor: doc. RNDr. David Stanovský, Ph.D., Department of Algebra Abstract: The present work deals with the procedure of algebraic crypta- nalysis, in which the problem of breaking cipher is at first converted to the problem of finding solutions to polynomial systems of equations and then the problem of finding a solution to this equation is converted to the SAT problem. The work specifically describes the methods that allow you to con- vert the problem of breaking cipher RC4 to the SAT problem. The individual methods were programmed in Mathematica programming language and then applied to RC4 with a word length of 2, 3. For finding of satisfiable evaluation of the resulting logical formula was used SAT-solver CryptoMiniSAT. In case of RC4 with word length 2 the solution was reached in the range from 0.09 to 0.34 second. In case of RC4 with word length 3 the solution was reached in the range from 1.10 to 1.23 second. Keywords: RC4, SAT, CryptoMiniSAT 1
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