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The goal of this thesis is to present a description of a highly transitive group M12 in two different ways and a comparison of these methods. First of them is a reproduction of a construction of M12 in the context of shuffle groups. We use this opportunity to pro- vide a description of shuffle groups including known results as well as open problems. The second method of choice is a new construction based on extended ternary Golay codes over a projective plane of order 3. This also gives us a proof of a part of a well known theorem linking monomial automorphisms of Golay codes and Mathieu groups. Part of the text is therefore dedicated to an introduction to affine and projective planes and error-correcting codes. Both approaches take advantage of a projection of monomial matrices over a field of order 3 onto the symmetric group S12 given by forgetting the signs. 1

The thesis systematizes the enumeration of essentially different Sudoku squares using Burnside's lemma. This enumeration significantly depends on the description of conju- gacy classes of symmetry group of Sudoku square, which was in previous works provided only with the strong help of mathematical software. The first chapter of this thesis sums up facts about a group action and about conjugacy classes and proposes the descrip- tion of conjugacy classes of wreath product, in the second chapter Sudoku square of symmetric group is formally defined and with the help of presented theory are counted representatives and sizes of its conjugacy classes. 1

Title: Generating polynomials for number field sieve Author: Anežka Pejlová Department: Department of Algebra Supervisor: prof. RNDr. Aleš Drápal, CSc., DSc., Department of Algebra Abstract: The topic of this thesis is mainly focused on Kleinjung algorithm for generating polynomials within the General Number Field Sieve, which is the most efficient factorization algorithm nowadays. Commonly used consecu- tions are explained with respect to the fact whether they can be rigorously proven or they are based only on heuristic assumptions. Another contribution of this thesis is the attached implementation of Kleinjung algorithm develo- ped as a part of the Number Field Sieve project led by the Department of Algebra. The appropriateness of some heuristics used in the theory beyond the Kleinjung algorithm is supported by empirical data obtained from this implementation. Keywords: Number field sieve, Kleinjung algorithm

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)

After Sidelnikov proved in 1992 that the cryptosystem of Niederreiter is vulnera- ble, he designed his own cryptosystem in 1993. This new cryptosystem was based on McEliece schema, it was to be resistant to quantum computers and faster than McEliece cryptosystem. However, in 2007, Minder and Shokrollah proposed an attack proving that the cryptosystem of Sidelnikov was vulnerable as well. This thesis uses several well-known and several new theorems to describe algebraic characteristics of the Reed-Muller code, especially from the affinity point of view. It proves that the attack proposed by Minder and Shokrollah really breaks the cryptosystem of Sidelnikov. Implementation of this attack in C/C++ language is presented in the conclusion of the thesis along with a table of duration of this attack on a personal computer.

This thesis focuses upon how to calculate local components of Weil differentials of an elliptic function field. Because Weil differentials constitute a one-dimension vector space then one Weil differential is fixed. An algorithm calculating a local component is developed for the fixed one. The first algorithm computes local components of places of degree one. It is based upon elementary properties of local components. The definition of the Weil differential does not say enough about why it is defined in this way and about why it is useful. Thus there is the relationship between the Weil differential and some objects from complex analysis like the Laurent series and the residue. It provides a better understanding of properties of the Weil differential. The result of this thesis are other two algorithms calculating local components of Weil differentials. The algorithms employ the residue. 1

The aim of the thesis is to desribe and implement the elliptic curve factorization method using curves in Edwards form. The thesis can be notionally divided into two parts. The first part deals with the theory of Edwards curves especially with properties of elliptic function fields. The second part deals with the factorization algorithm using Edwards form both formally and practically in the way the algorithm is really implemented. The contribution of this thesis is the enclosed implementation of the elliptic curve factorisation algorithm which can be run on a graphic card and which is faster than the state-of-the-art implementation GMP-ECM. Powered by TCPDF (www.tcpdf.org)

Title: Hadamard matrices and their applications in cryptography Author: Jan Luber Department: Department of Algebra Supervisor: prof. RNDr. Aleš Drápal, CSc., DSc., Department of Algebra Abstract: This thesis takes interest in Hadamard matrices, their constructions and application in cryptography. Firstly, we introduce basic properties of Hadamard matrices and selected summary of classical constructions is presented. Then we show a table of constructions that can be used to construct Hadamard matrix of given order. In the next part, we get concerned with Hadamard matrices with circulant cores with detailed description of construction Hadamard matrices with two circulant cores from GL-pair. In the end, we present cryptosystem using Hadamard matrices, we show its essential weaknesses and simple attacks. We propose several improvements in the form of adding other security elements. Keywords: Hadamard matrix, Hadamard conjecture, symmetric cryptography

The main purpose of this bachelor thesis is the description of the attack on the Niederreiter cryptosystem based on GRS codes. This attack was published by Sidelnikov and Shestakov in 1992. In the beginning the problem of group action, which is used in the attack, is introduced. A short preface into the coding theory follows, GRS codes are described and McEliece and Niederreiter cryptosystems are introduced, both as representatives of post-quantum cryptography. The following part of the thesis is dedicated to the attack itself. It is showed how one uses the group action, the process of the attack is also described in detail and its computing complexity is mentioned. Everything is illustrated by examples. Powered by TCPDF (www.tcpdf.org)

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

Title: Goppa codes and their applications Author: Bc. Jaroslav Kotil Department: Department of algebra Supervisor: prof. RNDr. Aleš Drápal, CSc., DSc. Abstract: In this diploma paper we introduce Goppa codes, describe their para- metres and inclusion in Alternant codes, which are residual Generalized Reed- Solomon codes, and Algebraic-geometry codes. Aftewards we demonstrate deco- ding of Goppa codes and introduce Wild Goppa codes. We also describe post- quantum cryptography member: McEliece cryptosystem for which no effective attacks with quantum computers are known. We outline a usage of this crypto- system with Goppa codes and describe the security of the cryptosystem together with possible attacks of which the most effective ones are based on information- set decoding. Keywords: Goppa codes, Generalized Reed-Solomon codes, Algebraic-geometry codes, Post-quantum cryptography, McEliece cryptosystem 1

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

The aim of this paper is to descript QR codes. First of all we focus on a description of a structure and then we deal with decoding, more specifically with the part which uses the theory of error-correcting codes. From the viewpoint of this theory QR codes can be perceived as an instance of Reed-Solomon codes. For that reason the paper deals with elementary properities of RS codes and with one of the technique of decoding, concretely Euclidean decoding algorithm. An example shows the knowledge and futhermore the paper includes detailed description of mathematical principle of used algorithm.

Title: Supporting algorithms of number field sieve Author: Adéla Skoková Department: Department of Algebra Supervisor: prof. RNDr. Aleš Drápal, CSc., DSc. Abstract: In this work we study the first part of the algorithm of number field sieve, generating of polynomials. At first we describe all the algorithm of the sieve for understanding of the role of polynomials and their impact on the entire algorithm. Then we present their characteristics and evaluation. The last part is about the most effective know algorithms of generating polynomials, invented by Thorsen Klinjung. The second Kleinjung algoritm was also programmed. Keywords: GNFS, Number sieve, Number field, Kleinjung algorithm Powered by TCPDF (www.tcpdf.org)

Projective plane of order 10 does not exist. Proof of this assertion was finished in 1989 and is based on the nonexistence of a binary code C generated by the incidence vectors of the plane's lines. As part of the proof of the nonexistence of code C, the coefficients of its weight enumerator were studied. It was shown that coefficients A12, A15, A16 and A19 have to be equal to zero, which contradicted other findings about the relationship among the coefficients. Presented diploma thesis elaborately analyses the phases of the proof and, in several places, enhances them with new observations and simplifications. Part of the proof is generalized for projective planes of order 8m + 2. 1

This thesis deals with problems of comparing the safety and running time of digital signatures DSA and Schnorr. Digital signature is almost full, legally recognized alternative to physical sign, intended for use in a digital environment. Digital signature uses asymmetric codes and hash functions which are easily described, as well as other basic concepts such as discrete logarithm and cyclic groups. The thesis deals with the analysis of possible attacks on DSA and compares DSA and Schnorr algorithm. Digital signature history and its implementation is part of the thesis.

This thesis deals with problems of comparing the safety and running time of digital signatures DSA and Schnorr. Digital signature is almost full, legally recognized alternative to physical sign, intended for use in a digital environment. Digital signature uses asymmetric codes and hash functions which are easily described, as well as other basic concepts such as discrete logarithm and cyclic groups. The thesis deals with the analysis of possible attacks on DSA and compares DSA and Schnorr algorithm. Digital signature history and its implementation is part of the thesis.

Title: Even triangulations and commutative groups Author: Jan Luber Department: Department of Algebra Supervisor: prof. RNDr. Aleš Drápal, CSc., DSc. Abstract: This thesis takes interest in latin bitrades and triangulations construc- ted from them. Firstly, we introduce needed definitions, properties of the latin bitrades, detailed construction of the triangulation and mainly possibility of em- bedding latin bitrades into abelian groups. These groups are determined by the relations definied on vertices of the triangulation. Then we get concerned with a particular kind of 3-homogeneous latin bitrades which correspond to toroidal tri- angulation whose each vertex has degree six. For these groups we express relation matrix and complement to their torsion ranks. In case of simple triangulations we present explicit description of the groups and with modular arithmetic we get partial description even for more complex triangulations. Keywords: latin bitrade, eulerian triangulation, abelian group

This thesis deals with problems of comparing the safety and running time of digital signatures DSA and Schnorr. Digital signature is almost full, legally recognized alternative to physical sign, intended for use in a digital environment. Digital signature uses asymmetric codes and hash functions which are easily described, as well as other basic concepts such as discrete logarithm and cyclic groups. The thesis deals with the analysis of possible attacks on DSA and compares DSA and Schnorr algorithm. Digital signature history and its implementation is part of the thesis.

This thesis deals with problems of comparing the safety and running time of digital signatures DSA and Schnorr. Digital signature is almost full, legally recognized alternative to physical sign, intended for use in a digital environment. Digital signature uses asymmetric codes and hash functions which are easily described, as well as other basic concepts such as discrete logarithm and cyclic groups. The thesis deals with the analysis of possible attacks on DSA and compares DSA and Schnorr algorithm. Digital signature history and its implementation is part of the thesis.

This work deals with three different constructions of the extended binary Golay code G24. The first construction is based on a projective plane of order four. In terms of it Steiner system (5, 8, 24) is built. Linear span of its blocks forms a linear binary [24, 12, 8] code C. Every binary [24, 12, 8] code is isomorphic to C which is known as extended binary Golay code G24. The second construction uses so-called Miracle Octad Generator (MOG). All MOG-words of weight eight form Steiner system (5, 8, 24). The third construction uses impartial combinatorial game Mogul. In terms of its P-positions one can create a linear binary [24, 12, 8] code. The fact that is is also a lexikographic code is useful for parametres estimate. 1

In the presented work we define a class of error-correcting codes based on incidence vectors of projective geometries, including the necessary basis of coding theory and projective geometries. A detailed calculation is performed to show the dimension of these codes. In conclusion we concern ourselves with majority decoding. This work is a summary of the results of some known authors engaged in this field. We continue on some of these results and we present evidence of some of the statements, which have been proven differently by other authors.

Title: Frobenius primality tests Author: Jan Hora Department: Department of algebra Supervisor: prof. RNDr. Aleš Drápal, CSc. Dsc. Supervisor's e-mail address: drapal@karlin.mff.cuni.cz Abstract: In the present work we study Extended Quadratic Frobenius primality test. We study its functionality, error probability and taken time. We will define ring R(n, c), in which the test works. We will describe its structure depending up primality of tested number n, and algorithm of Frobenius test. We will show, that test succeeds anytime the tested number n is prime. We will study the upper bound for error probability of the test. We will show that test fails iff certain elements of set R(n, c) are chosen, Set of that elements will be denoted G(n, c). We will find conditions that G(n, c) must fulfil, and with them we will discover, that Frobenius test eroor probability is at most 1 24 . We will analyse individual parts of algorithm and discover, that one cyclus of Frobenius test can be done with approximately 2 log n multiplications in Zn. Finaly the teoretical estimates will be compared with practical results. Keywords: primality test, Frobenius automorfism, cyklic group, roots of one 1

In the present work we study a security of the AES cipher. We concern in a mathematical representation of a block cipher and how to use it to algebraic attacks. We show a summary of known algorithms which are useful for these attacks. We show how to convert problem of solving the system of polynomial equations to SAT problem and we describe how SAT solvers work.

In the rst part of this work we study the complexity of solving nonlinear quasigroup equations for di erent classes of quasigroups. In particular we study the application of principle of central quasigroups on the blocks of congruence. We show that these quasigroups can be shapeless and therefore we gain counterexample to the hypothesis which was stated by D. Gligoroski. In the second part of this work we apply previous results on the concrete quasigroups of the type Edon-R-I,II and we deduce the complexity of the corresponding algorithm for inverting the hash function Edon-R.

In the present work we study a class of generalised inner states of the cipher RC4, the so-called persistent states. The RC4 stream cipher is the most widely used software-based stream cipher and the existence of such a state would be a significant weakness of the cipher. We describe the Tabular model and using the model we prove the periodicity of these states. Then we study a new type of relationship between the tabular model and the equivalences on linearly ordered sets and we prove the regularity of the matrix determined by such an equivalence. Afterwards we apply the obtained result to the theory of persistent states and we prove that there exists no reachable persistent k-state for k equal to 2, 3, 4 in the specific case. Moreover, we present some new unreachable persistent states. Finally, we indicate the cryptanalytical significance of the persistent states.

In this work we study the number field sieve algorithm. Our main focus is on its theoretical background. We present all important theorems which are needed for a full understanding of the algorithm. We also describe the most widely used implementation of the parts of the algorithm and we discuss in which situation they should be used. At the end we show results from measurements of sieving phase on the implementation which was written for our Department of Algebra.

In the present work we study a security of the AES cipher. We concern in a mathematical representation of a block cipher and how to use it to algebraic attacks. We show a summary of known algorithms which are useful for these attacks. We show how to convert problem of solving the system of polynomial equations to SAT problem and we describe how SAT solvers work.

Xazev praee: Reed-Solomonovy kody a jejich aplikaco Autor: Pavel Iloral Katcdra (ustav): Katedra Algebry Vedouci bakalafsko prace: Doc. RNDr. Ales Drapal. CSc. c-nia.il vedoiiciho: drapar^karlin.mil', cuni.cz Abstrakt: Prace podava ucelenon definici klasiekyeh Reed-Solomonovych kodn, vcetne potfobuyeh zakladu tcxjric1 k()du. Je dokazana cykliciiost RS kodn delky q - I . Na cyklienosti jsou pak zalozeny tri inx'zontovane dekcklovac'i algoritmy fPetersonnv, Rorlrkain])-Mas.seyuv a Enklidi'iv dekodova.ci algoritmns), vcetne. dukazfi existcniee feseni. V ])oslodni ka])itole uvadiin nckolik a.]>likaci RS kodu, vcetne nejznainejyiho standardu CIRC' pouzivancho na Imdebnich CD. Klieova alova: .s;nnoo])ravny k(5d.1'X'C, Reed-Solomon. Pcterstm, Berlekanip-Massey. Euklid. CIRC1 Title: Rood-Solomon codes and applications Author: Pavel llora.1 Department: Department of Algebra Supervisor: Doc. HXDr. Ales Drapal, CSc. Supervisor's e-mail address: drapal (fkarlin.nirl.ouni.c/ Alislracl: This work presents compact definition of classic Heed-Solomon codes with necessary elements of coding theory. The ryclicity of RS codes of length q - 1 is prooved and there are comletely described three decoding algorithms (Peterson's, Berlekamp-Ma,ssoy and Euclid decoding algorithm) based on RS cyclirity. I also in- troduce a few RS...