
Rank Two Commutative Semifields
Tittl, Ondřej ; Göloglu, Faruk (advisor) ; Růžička, Pavel (referee)
In this thesis we will explain what are semifields and what interesting properties these algebraic objects possesses. In the first chapter we will go over some basics and preliminaries to understand what semifields are. In the second chapter we will prove some useful lemmata for either commutative and noncommutative case of semifields and provide some examples. At last we will try to do some research by ourselves, where we will try to find some examples of semifields. 1


Ultrafilters and their monads
Hladil, Josef ; Slávik, Alexander (advisor) ; Růžička, Pavel (referee)
Generalising the notion of an ultrafilter to structured sets, we construct the ultrafilter monad in the categories of partially ordered sets and finitely colourable graphs. This is done similarly to codensity monads, knowing that the codensity monad of the inclusion of finite sets into sets is the ultrafilter monad. We derive an equivalent definition of an ultrafilter on an object applicable for general graphs, also giving rise to a monad. We show that ultrafilters on a poset can be completely characterised in terms of suprema or infima of directed subsets when the poset has only finite antichains. We attempt to classify algebras over the poset ultrafilter monad; our results completely classify the algebras with all antichains finite as posets with a particular compact Hausdorff topology. 1


Functional encryption for quadratic functions
Sýkora, Josef ; Žemlička, Jan (advisor) ; Růžička, Pavel (referee)
In this thesis we will be studying functional encryption for quadratic functions. We want to encrypt a message, in form of a vector, z to a plaintext ct and create a secret key skf for a quadratic function f, which will allow us to decrypt ct to f(z), while the ct and skf will not leak any information about z. We will introduce one concrete design. The aim of this thesis will be the preparation of necessary preliminaries, which will allow us to describe the design, and to verify correctness of the algorithms. We will describe Arithmetic Branching Programs. Such objects will help us represent function f. Furthermore, we will introduce Garbling and Partial Garbling schemes. Those will allow us to "randomize"a part of the algorithm. We will also specify an encryption algorithm for linear transformation and use it to describe the main encryption algorithm for quadratic functions. 1


Rank Two Commutative Semifields
Tittl, Ondřej ; Göloglu, Faruk (advisor) ; Růžička, Pavel (referee)
In this thesis we will explain what are semifields and what interesting properties these algebraic objects possesses. In the first chapter we will go over some basics and preliminaries to understand what semifields are. In the second chapter we will prove some useful lemmata for either commutative and noncommutative case of semifields and provide some examples. At last we will try to do some research by ourselves, where we will try to find some examples of semifields. 1


Security of an Electronic Voting System
Fritzová, Petra ; Růžička, Pavel (advisor) ; Hojsík, Michal (referee)
Electronic elections, also known as ivoting might help in removing the crisis in our democracy, which is reflected in noncooperation in the opportunity of expressing their opinions during direct elections. A reasonable set up of information and communication technologies in technical and financial terms could help that elections would be attended by more voters. The implementation of electronic elections could achieve that the way of governance in the democratic republic will be truly represented by the view of the vast majority of people who are authorized to elect. The introduction of the ivoting system could be efficient from the financial point of view. This electoral process could reduce the risk of human error as well as the risk of manipulation of votes since most of the processes would be automated. Thesis proposes a definition of the basic requirements for an ideal ivoting system which compares the requirements for ensuring the safety of two previously proposed electronic electoral systems. Thanks to a deeper analysis of these two systems the thesis also describes the imperfection in safety and it raises the possibility of basic attacks on components and systems properties due to imperfections in security.


The arity of NU polymorphisms
Draganov, Ondřej ; Barto, Libor (advisor) ; Růžička, Pavel (referee)
This paper deals with an arity of NU polymorphisms of relational structures. The goal is to simplify and clearly describe an already existing example of a relational structure, which has an NU polymorphism, but no NU polymorphisms of low arity in respect to arity of relations and to a number of elements in the relational structure. We explicitly describe mary relational structures with n elements, n ≥ 2, m ≥ 3, which have no NU polymorphisms of arity (m − 1)2n−2 , but have an NU polymorphism of arity (m − 1)2n−2 + 1, which is constructed in the paper, and binary relational structures with n elements, n ≥ 3, which have no NU polymorphisms of arity 22n−3 , but have an NU polymorphism of arity 22n−3 + 1.

 

Combinatorial group theory and cryptography
Ferov, Michal ; Příhoda, Pavel (advisor) ; Růžička, Pavel (referee)
In the presented work we focus on applications of decision problems from combinatorial group theory. Namely we analyse the ShpilrainZapata pro tocol. We give formal proof that small cancellation groups are good platform for the protocol because the word problem is solvable in linear time and they are generic. We also analyse the complexity of the brute force attack on the protocol and show that in a theoretical way the protocol is immune to attack by adversary with arbitrary computing power.

 

Banschewského funkce na komplementárních modulárních svazech
Mokriš, Samuel ; Růžička, Pavel (advisor) ; Žemlička, Jan (referee)
Title: Banaschewski function on countable complemented modular lattices Author: Samuel Mokriš Department: Department of Algebra Supervisor of the bachelor thesis: Mgr. Pavel Růžička, Ph.D., Department of Algebra Abstract: A Banaschewski function on a bounded lattice L is an antitone selfmap on L that picks a complement for each element of L. On any at most countable complemented modular lattice L, there exists a Banaschewski function with a Boolean range M. Moreover, such M is a maximal Boolean sublattice of L and is uniquely determined up to isomorphism. In the thesis we give a negative answer to the related question whether all maximal Boolean sublattices in an arbitrary countable complemented modular lattice are isomorphic and whether every max imal Boolean sublattice in an arbitrary countable complemented modular lattice L is the range of some Banaschewski function on L. We also generalize the coun terexample to greater cardinalities; for a given infinite cardinal κ we construct a complemented modular lattice L of cardinality κ and maximal Boolean sublat tices B and E of L such that B is not the range of any Banaschewski function on L, that there exists a Banaschewski function on L of range E, and that B is not isomorphic to E. Keywords: complemented modular lattice, Banaschewski function, von...
