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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 Edon-80.
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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.
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Qudratic field based cryptography
Straka, Milan ; Stanovský, David (advisor) ; Žemlička, Jan (referee)
Imaginary quadratic fields were first suggested as a setting for public-key cryptography by Buchmann and Williams already in 1988 and more cryptographic schemes followed. Although the resulting protocols are currently not as efficient as those based on elliptic curves, they are comparable to schemes based on RSA and, moreover, their security is believed to be independent of other widely-used protocols including RSA, DSA and elliptic curve cryptography. This work gathers present results in the field of quadratic cryptography. It recapitulates the algebraic theory needed to work with the class group of imaginary quadratic fields. Then it investigates algorithms of class group operations, both asymptotically and practically effective. It also analyses feasible cryptographic schemes and attacks upon them. A library implementing described cryptographic schemes is a part of this work.
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Binary equality words
Hadravová, Jana ; Holub, Štěpán (advisor) ; Stanovský, David (referee)
Binary equality language is a set consisting of all solutions of equation g(w) = h(w), where g, h are arbitrary binary morphisms. Recently, it has been prooved that equality set for each pair of morphisms g, h is generated by at most two words. Structure of binary equality language has been already known in the case that at least one of morphisms g, h is periodic or if their equality set is generated exactly by two words. The main objective of the paper was to find a structure of solutions for morphisms whose equality set is generated by one word. The problem in general case remains unsolved but special result for solutions consisting of just one block for marked morphisms was discovered. Using methods established in this paper (covering by the same pattern to find n-multiple p-overflows and working with the cyclic pair (e, f, z)) it is believed that some more results can be achieved in the near future.
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Park's conjecture
Lauschmannová, Anna ; Stanovský, David (advisor) ; Ježek, Jaroslav (referee)
A finite algebra of finite type (i.e. in a finite language) is finitely based iff the variety it generates can be axiomatized by finitely many equations. Park's conjecture states that if a finite algebra of finite type generates a variety in which all subdirectly irreducible members are finite and of bounded size, then the algebra is finitely based. In this thesis, I reproduce some of the finite basis results of this millennium, and give a taster of older ones. The main results fall into two categories: applications of Jonsson's theorem from 1979 (Baker's theorem in the congruence distributive setting, and its extension by Willard to congruence meet-semidistributive varieties), whilst other proofs are syntactical in nature (Lyndon's theorem on two element algebras, Je·zek's on poor signatures, Perkins's on commutative semigroups and the theorem on regularisation). The text is self-contained, assuming only basic knowledge of logic and universal algebra, and stating the results we build upon without proof.
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Quasigroup based cryptography
Christov, Adam ; Stanovský, David (advisor) ; Drápal, Aleš (referee)
Public-key cryptographic schemes based on the complexity of solving multivariate quadratic equations over a finite field represent an alternative to widely used schemes relying on the complexity of factorization or on the discrete logarithm. Such a scheme was proposed by D. Gligoroski et al. [8]. Keys in this scheme are constructed using a special kind of quasigroups, the socalled quadratic quasigroups. In this paper we try and describe the quadratic quasigroups and classify them according to their properties. Finally, we present a theory which can be used to generate such quasigroups.
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Gröbnerovy báze v kryptografii
Hubáček, Pavel ; Stanovský, David (advisor) ; Šťovíček, Jan (referee)
The thesis focuses on the use of GrÄobner bases in cryptography and especially on applications in cryptanalysis of block ciphers. Some elementary concepts from the theory of GrÄobner bases are introduced together with Buchberger's algorithm, a method for constructing such bases. The principle of solving of poly nomial systems using suitable GrÄobner bases is explained. This is followed by pre sentation of modern algorithms that improve the Buchberger's algorithm. In the last part of the thesis present results achieved by GrÄobner bases are summarised and the notion of algebraic cryptanalysis is introduced. In algebraic cryptanalysis we transform breaking of given cryptosystem into a problem of solving polynomial equations over some nite eld. Examples of polynomial descriptions of block ciphers are provided together with some experimental result on arising polynomial systems.
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