
Construction of MDS matrices
Belza, Lukáš ; Žemlička, Jan (advisor) ; Šťovíček, Jan (referee)
This thesis focuses on Maximum Distance Separable (MDS) matrices over finite fields, with emphasis on circulant MDS matrices. At the beginning, concepts related to MDS codes and their characterization are introduced. This is directly followed by an introduction into circulant matrices and their relation to factor algebras of polynomials. In the second part, we shift our focus specifically on circulant MDS matrices. We start from the construction of such matrices in dimensions 3×3 and 4×4 and then proceed to a general construction of MDS matrices from Vandermond matrices.Finally, we find some restrictions on the existence of orthogonal circulant MDS matrices, namely that there are no such 2d × 2d matrices over any finite field of characteristic two. 1


Rings with restricted minimum condition
Krasula, Dominik ; Žemlička, Jan (advisor) ; Šaroch, Jan (referee)
Ring is artintian if and only if all of its factors are artinian. We say that ring R satisfies the restricted minimum condition, if for every essenctial ideal, corresponding factor ring is artinian. We will call such ring RM ring for short. Similarly as the class of artinian rings, the class of RM rings is closed under fac tors and finite direct products. In this thesis we prove that restricted minimum condition is satisfied in coordinate rings, ring (R × R)[x] and noetherian CDR domains. We investigate the relation between unique factorization domains and RM domains. In last chpater, we will focus are attention to polynomial rings, proving that if ring R[x] is RM then R is semisimple. Laurents polynomials over domain R are RM rings if and only if R is a field. 1


Polynomial time primality testing
Bednaříková, Alžběta ; Žemlička, Jan (advisor) ; Čech, Martin (referee)
The topic of my thesis is the testing of prime numbers in polynomial time. The text focuses on the specific algorithm published in 2002 by Manindra Agrawal, Neeraj Kayal and Nitin Saxena and it is known as the AKS primality test. In the introduction of this work, important properties and concepts essential for the text understanding are revised. Then the basic idea of the test is explained. The description of the algorithm itself continues. The aim of the work is to prove the Theorem on the correctness of the AKS test from a gradually builtup theory and to calculate the time complexity of the algorithm. Finally, it is proved that the calculated time complexity is polynomial.


Small roots of multivariate polynomials with integral coefficients
Todorovová, Dora ; Příhoda, Pavel (advisor) ; Žemlička, Jan (referee)
This thesis focuses on the Coppersmith method for finding roots of mo dular polynomials. The method is based on the base reduction of a lattice. Firstly, we define a lattice and show a simplified form of the LLL algorithm. Then we describe the Coppersmith method and related theorems. Further more, we introduce a solved example from the article written by D. Boneh and G. Durfee. The general form of the method from the article written by E. Jochemsz and A. May snd we add some proofs. In the last chapter we solve examples using the method in general form. 1


On a matrix approach for constructing quadratic almost perfect nonlinear functions
Rezková, Zuzana ; Göloglu, Faruk (advisor) ; Žemlička, Jan (referee)
Search for new APN functions is an important topic in symmetric cryptography. The matrix approach for constructing quadratic APN functions was described by Y. Yu, M. Wang and Y. Li in 2014. The approach takes advantage of the one to one correspondence between quadratic homogenous APN functions and quadratic APN matrices. The aim of this thesis is to explain the matrices used in the original paper and show that similar matrices can be constructed directly from the algebraic normal form of the APN function. In Chapter 2 we introduce the original method adding extra theorems and expanding the proofs for better understanding. In Chapter 3 we define the matrices obtained simply from the algebraic normal form. In Chapter 4 we give examples of the matrices for chosen APN functions and show how they are related. 1


Compact modules over nonsingular rings
Kálnai, Peter ; Žemlička, Jan (advisor) ; Breaz, Simion (referee) ; Příhoda, Pavel (referee)
This doctoral thesis provides several new results in which we leverage the inner structure of nonsingular rings, in particular of selfinjective von Neumann regular rings. First, we describe categorical and settheoretical conditions under which all products of compact objects remain compact, where the notion of compactness is relativized with respect to a fixed subclass of objects. A special instance when such closure property holds are the classic module categories over rings of our interest. Moreover, we show that a potential counterexample for Köthe's Conjecture might be in the form of a countable local subring of a suitable simple selfinjective von Neumann regular ring. 1


Structure of pureinjective abelian groups
Jankovec, Filip ; Šaroch, Jan (advisor) ; Žemlička, Jan (referee)
In this thesis, we study the structure of the pureinjective abelian groups. We de scribe some equivalent characterizations of the pureinjective modules. Furthermore, we thoroughly discuss the special case of the pureinjective modules over principal ideal do main. We show that every pureinjective abelian group can be written unambiguously only using cyclic groups, Prüfer groups and the group of rational numbers. Moreover, an abelian group can be written in this form if and only if it is a pureinjective abelian group. 1


Multiplication in a finite field of characteristic 2 and XORmetrics
Carulkov, Nikita Edward ; Žemlička, Jan (advisor) ; Göloglu, Faruk (referee)
XORcounts measure the efficiency of multiplication in finite fields of characteristic 2. In the first chapter we define two XORcounts (the direct XORcount and the sequential XORcount) and present detailed proofs of some propositions from the paper from Lukas Kolsch about the XORcounts of inverse matrices and permutation similar matrices. It seems that the case when the direct XORcount is lower than the sequential XORcount is rare. We will explore those cases in the second chapter. Some of them were already described in the paper from Lukas Kolsch and we prove that they occur only for matrices with order higher or equal to six. 1


Linear version of Holub's algorithm
Tvrdý, David ; Holub, Štěpán (advisor) ; Žemlička, Jan (referee)
This work studies a linear agorithm which decides if a given word is a fixed point of any nontrivial morphism. This work also contains a description of auxiliary data structures which are crucial for linear time complexity of the algorithm. A Java implementation of the algorithm is provided along with a stepbystep visualization for particular input words. 1


Generalized integral property
Hrúzová, Jana ; Žemlička, Jan (advisor) ; Příhoda, Pavel (referee)
This thesis is based on an article C. Boura and A. Canteaut, Another View of the Division Property, which is focused on division property of sets from Fn 2 . In this thesis we introduce important definitions and propositions about boolean function, polynomials and ReedMuller codes at the beginning. Then we define parity set of a set from Fn 2 , which helps us to simplify the division property. We also show how sets, which satisfy division property of certain order, look like. From that we could follow how the division property propagate through the substitutionpermutation network. 1
