
Application of invertible elements in a zeroknowledge proof
Kučerová, Karolína ; Žemlička, Jan (advisor) ; Příhoda, Pavel (referee)
This work is focused on the description of one verifiable encryption scheme, specifically a zeroknowledge proof of knowledge protocol. Verifiable encryption allows us to prove properties of data without revealing its content. The main goal of the presented method is verification of knowledge of a secret key. This method can be used for group signa tures, multiple steps secret sharing, key escrow protocols, and many others cryptographic protocols. It is based on the hardness of the RingLWE problem and problems of finding solutions to linear relations over some ring. It uses the principle of rejection sampling. The method is build on two closely described cryptographic protocols, RingLWE and FiatShamir with aborts. It uses the construction of polynomial rings R = Z[x]/(xn + 1) a Rq = Zq[x]/(xn + 1). 1


Multivariate polynomial commitment schemes
Bžatková, Kateřina ; Hubáček, Pavel (advisor) ; Žemlička, Jan (referee)
This thesis focuses on polynomial commitment schemes  cryptographic protocols that allow committing to a polynomial and, subsequently, proving the correctness of evaluations of the committed polynomial at requested points. As our main results, we present new schemes that enable committing to multivariate polynomials and efficiently proving the correctness of evaluations at multiple points. As the main technical tools for our constructions, we use theorems from abstract algebra related to ideals of polynomial rings and some grouptheoretic properties. Compared to the stateoftheart that inspired our work, our main contribution is the improved communication complexity achieved by our protocol.


Representing Images by Weighted Finite Automata
Hurtišová, Viktória ; Holub, Štěpán (advisor) ; Žemlička, Jan (referee)
The goal of this thesis is to introduce weighted finite automata (WFA) as a means of representing multiresolution raster images. We explain the basic concepts of weighted finite automata. Then we describe an encoding algorithm that converts an image into a WFA and a decoding algorithm that can generate back the original image. We then provide our implementation of the encoding and decoding algorithms. 1


Structural and categorical description of classes of abelian groups
Dvořák, Josef ; Žemlička, Jan (advisor) ; Modoi, George Ciprian (referee) ; Příhoda, Pavel (referee)
The work presents results concerning the selfsmallness and relative smallness properties of products in the category of abelian groups and in Ab5categories. Criteria of relative smallness of abelian groups and a characterization of selfsmall products of finitely generated abelian groups are also given. A decomposition theory of UDcategories and in consequence a unified theory of decomposition in categories of Sacts is developed with applications on (auto)compactness proper ties of Sacts. A structural description of compact objects in two categories of Sacts is provided. The existence of projective covers in categories S − Act and S − Act0 and the issue of perfectness of a monoid with zero are discussed. 1


Rprojectivity
Fuková, Kateřina ; Trlifaj, Jan (advisor) ; Žemlička, Jan (referee)
A module over a ring R is Rprojective if it is projective relative to R. This module theoretic notion is dual to the notion of an Rinjective module that plays a key role in the classic Baer's Criterion for Injectivity. This Thesis is concerned with the validity of dual version of Baer's Criterion. It also introduces a concept of projectivity in a general categorytheoretic setting. DBC is known to hold for all perfect rings. However, DBC either fails or it is undecid able in ZFC for nonperfect rings. In this Thesis we deal with the subclass of nonperfect rings, which are small, regular, semiartinian and have primitive factors artinian. Trlifaj showed that there is an extension of ZFC in which DBC holds for such rings. Especially, it is enough to consider extension of ZFC in which the weak version of Jensen's Diamond Principle holds. This combinatorial principle is known as the Weak Diamond Principle. Apart from an overview of the properties of rings mentioned above and introduction of the necessary settheoretic notions, the Thesis also contains a proof of this new result by Trlifaj published in the paper "Weak diamond, weak projectivity, and transfinite extensions of simple artinian rings" in the J. Algebra in 2022. 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

 
 
 
 