 
 

Quasigroups, oneway functions and hash mappings
Machek, Ivo ; Drápal, Aleš (advisor) ; Stanovský, David (referee)
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 EdonRI,II and we deduce the complexity of the corresponding algorithm for inverting the hash function EdonR.

 
 

Some questions of definability
Lechner, Jiří ; Stanovský, David (advisor) ; Kepka, Tomáš (referee)
We focus on firstorder definability in the quasiordered class of finite digraphs ordered by embeddability. At first we will prove definability of each digraph up to size three. We will need to add to the quasiorder structure some digraphs as constants, so we try to find the needed set of constants as small as possible with small digraph as well. Gradually we make instruments that we can use to express the inner structure of each digraphs in the language of embeddability. At the end we investigate definability in the closely related lattice of universal classes of digraphs. We show that the set of finitely generated and also the set of finitely axiomatizable universal classes are definable subsets of the lattice.

 
 

Quasigroup based cryptography
Frisová, Andrea ; Stanovský, David (advisor) ; Drápal, Aleš (referee)
In this work, we study some properties of an in nite matrix, which consists of quasigroup elements. This matrix is generated from a certain sequence X using left iterated translations. We suppose that the sequence X is periodic and we examine how the periods of the rows of our matrix behave for various types of quasigroups. We show that for central quasigroups the periods increase at most linearly. Further, we try to apply our result to the stream cipher Edon80.


Lattice based cryptography
Divišová, Jana ; Stanovský, David (advisor) ; Barto, Libor (referee)
The aim of this work is several faces of lattices in cryptography. After the section in which we describe lattices in general and lattice problems, we turn to the lattice based cryptosystems. We describe their mathematical background and also formulations of encryption and decryption algorithms. In the next part we describe the usage of lattice in cryptanalysis. It is mainly attacks against knapsack system a solving hidden number problem. The signi cant part of this work is to compare two cryptosytems RSA a NTRU for the similar level of security. We compare the speed of encryption, decryption and key generation.
