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Extension of fields and solvability of algebraic equations
Štíchová, Kateřina ; Stanovský, David (referee) ; Růžička, Pavel (advisor)
Xaxev praee: Hoxsifeni teles a reseni algebraickyeh rovnie Autor: Katerina fit iehova Katedra (ustav): Kafedra algebry Matematicko-fyxikalni f a k u l t y \ Pnixe Vedouoi bakalarske praee: Mgr. Pavel Kuxioka, Ph.I), Kaledra algebry e-mail vedoueiho: Pavel. Huxieka " i n f f . e u n i . e / Abstrakt: V pfedlo/ene praei se x'rnujciiK1 i'rsciii al^rbrairkvrh rovnic di'iilirhu ax pa- trho sT.u])iir a, CJaluisuvr ti'oiii. \i rasli jr po])s;i.n xpusnli rrscni ai^chraickyrh rovnic. s])ocialnr ( 'a.rda.novy vxnrcu. Dale jc nvcdcna rrorir trios, ktcrii jc stavcliniin prvkrin ])ro iiiishxlujici ka pi loin vlasl ni (ialoisovy t.curir, jcv, jc jadrcni t.rt.u prii.cc. I'o- slcdni ka.])itola sc 7a.l)yv;i aplika.ci (JaJoisovy t.coric. konkrctnc nt'n'sit.i'luosti rovnicc pateho stupno v ra.dikalccli. V Tlvodu. ]>ro Aplikaer a. ])rik!ady a jsinc vyeliaxeli lila.vnc xc /.droje ,7 . Vc 2. ka- ])ilolc jsme sc inspirovali f3] a. |0j. v Tcorii rclcs jsnic pou/ili -'2 . Dalo -1. ka.pitola byla. na])s;ina jioinoci 1 Title: Field extensions and solution of algebraic equations Author: Katerina Stiehova Department: I)c])art incut of Algebra Sii]>crvisor: Mj;r, l'a\'el Huxieka. Pli.D Supervisor's e-mail addre>s Pavel.l\u/icka u infi.cuni.cx Abstract: In the ]ji'cseiit \vork \vc s( udy solut ion of general algebraic c(|iiat Jcjns of...
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Qudratic field based cryptography
Straka, Milan ; Žemlička, Jan (referee) ; Stanovský, David (advisor)
Imaginary quadratic fields were first suggested as a setting for public-key cryptography by Buchmann and Williams already in 1988 and more cryptographic schemes followed. Although the resulting protocols are currently not as efficient as those based on elliptic curves, they are comparable to schemes based on RSA and, moreover, their security is believed to be independent of other widely-used protocols including RSA, DSA and elliptic curve cryptography. This work gathers present results in the field of quadratic cryptography. It recapitulates the algebraic theory needed to work with the class group of imaginary quadratic fields. Then it investigates algorithms of class group operations, both asymptotically and practically effective. It also analyses feasible cryptographic schemes and attacks upon them. A library implementing described cryptographic schemes is a part of this work.
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Testing perfect powers
Straková, Hana ; Jedlička, Přemysl (referee) ; Stanovský, David (advisor)
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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