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Lattice based cryptography
Divišová, Jana ; Barto, Libor (referee) ; Stanovský, David (advisor)
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)
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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Park's conjecture
Lauschmannová, Anna ; Ježek, Jaroslav (referee) ; Stanovský, David (advisor)
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 ; Drápal, Aleš (referee) ; Stanovský, David (advisor)
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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