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CRC-codes
Lorenc, Filip ; Žemlička, Jan (advisor) ; Šťovíček, Jan (referee)
This thesis deals with description of CRC codes, which is a type of polynomial error correction codes, and description of CAN and CAN FD protocols used in automobiles for data transmission between sensors. One of the security elements is usage of the CRC codes with the Hamming distance 6. Unfortunately, both protocols contain a design vulnerability which causes that some received messages with one wrong bit do not have to be detected by the protocol. The aim of the thesis was to describe this vulnerability and found out, if it was possible to eliminate it by using different CRC code. It managed to characterize all messages, which are not during this vulnerability detected by CRC code and based on that it was possible to prove, that the probability of error does not depend on a CRC code choice of a fixed length. 1
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Cryptosystems based on codes with rank metrics
Marko, Marek ; Žemlička, Jan (advisor) ; Šaroch, Jan (referee)
The first part of this paper explains the uses of the element's rank and the metric induced by it in linear error-correcting codes over finite fields. Describing the effective decoding algorithm of rank-metric codes without the use of exhaustive search is essential. This algorithm is applied in cryptographic systems based on codes with rank metric pre- sented in the next chapter. Apart from the scheme of cryptosystem, we focus on the de- tailed illustration of a possible structural attack. Comprehension of the attack will be significant in order to show some methods how to withstand it. 1
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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
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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
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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 built-up theory and to calculate the time complexity of the algorithm. Finally, it is proved that the calculated time complexity is polynomial.
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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
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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
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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 non-singular rings, in particular of self-injective von Neumann regular rings. First, we describe categorical and set-theoretical 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 self-injective von Neumann regular ring. 1
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Structure of pure-injective abelian groups
Jankovec, Filip ; Šaroch, Jan (advisor) ; Žemlička, Jan (referee)
In this thesis, we study the structure of the pure-injective abelian groups. We de- scribe some equivalent characterizations of the pure-injective modules. Furthermore, we thoroughly discuss the special case of the pure-injective modules over principal ideal do- main. We show that every pure-injective 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 pure-injective abelian group. 1
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Multiplication in a finite field of characteristic 2 and XOR-metrics
Carulkov, Nikita Edward ; Žemlička, Jan (advisor) ; Göloglu, Faruk (referee)
XOR-counts measure the efficiency of multiplication in finite fields of characteristic 2. In the first chapter we define two XOR-counts (the direct XOR-count and the sequential XOR-count) and present detailed proofs of some propositions from the paper from Lukas Kolsch about the XOR-counts of inverse matrices and permutation similar matrices. It seems that the case when the direct XOR-count is lower than the sequential XOR-count 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
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