|
|
Gröbner basis, Zhuang-Zi algorithm and attacks of multivariable cryptosystems
Doktorová, Alice ; Tomáš, Jiří (referee) ; Kureš, Miroslav (advisor)
This diploma thesis is devoted to the multivariate cryptosystems. It includes an overview of commutative algebra with emphasis on Gröbner bases. Of all algorithms, especially the ones using Gröbner bases are studied, i.e. Buchberger's algorithm, which is already implemented in Wolfram Mathematica, and F4 algorithm, for which a program package has been created in the Wolfram Mathematica environment. Also Zhuang-Zi algorithm is described. To simplify its steps a program to compute the Lagrange interpolation polynomial has been created in Python.
|
|
|
Elliptic curves in cryptography
Geyer, Lukáš ; Burda, Karel (referee) ; Lambertová, Petra (advisor)
The objective of this bachelor thesis is to decribe the role of the elliptic curves in modern cryptosystems, explain the mathematical fundamentals upon which the elliptic curves are based along with their advantages and disadvantages, followed by application in the digital signature. The project is concluded by a software solution demonstrating the use of elliptic curves in digital signature scheme ECDSA
|
|
| |
|
|
Algorithms of the interpolation by multivariate polynomials
Doktorová, Alice ; Čermák, Libor (referee) ; Kureš, Miroslav (advisor)
This bachelor's work concerns to algorithms of the multivariate interpolation. The problem of the interpolation over the plane is studied in the first part of this work. In the next section, the multivariate Lagrange interpolation is described and the polynomial degree is discussed. A Mathematica program package was developed, by this, the multivariate interpolation over an arbitrary field can be solved.
|
|
|
Two types of septic trinomials and their use in hyperelliptic cryptography
Felcmanová, Adéla ; Tomáš, Jiří (referee) ; Kureš, Miroslav (advisor)
This thesis deals with two types of septic trinomials and genus three hyperelliptic curves constructed from them. It includes an introduction to the theory of hyperelliptic curves and divisors, as well as terms and algorithms necessary for their implementation in hyperelliptic cryptosystems. The principle of the hyperelliptic curve cryptography is presented along with two examples of cryptosystems. It also contains several exercises, some of which were programmed in Python language.
|
|
| |
|
|
Two types of septic trinomials and their use in hyperelliptic cryptography
Felcmanová, Adéla ; Tomáš, Jiří (referee) ; Kureš, Miroslav (advisor)
This thesis deals with two types of septic trinomials and genus three hyperelliptic curves constructed from them. It includes an introduction to the theory of hyperelliptic curves and divisors, as well as terms and algorithms necessary for their implementation in hyperelliptic cryptosystems. The principle of the hyperelliptic curve cryptography is presented along with two examples of cryptosystems. It also contains several exercises, some of which were programmed in Python language.
|
|
|
Fast multiplication in the field GF(2n)
Bajtoš, Marek ; Žemlička, Jan (advisor) ; Šaroch, Jan (referee)
Title: Fast multiplication in the field GF(2n ) Author: Marek Bajtoš Department: Department of Algebra Supervisor: doc. Mgr. et Mgr. Žemlička Jan, Ph.D., Department of Algebra Abstract: In this bachelor thesis we research how to optimize multiplication with a fixed element of finite field which can be useful for implementation of crypto- graphic algorithms in lightweight cryptography. We will represent effectivity of multiplication by number of XOR operation needed for implementation of matrix which represent some fixed element of finite field. We prove that some matrix re- presents multiplication with some element of finite field if and only if the minimal polynomial of matrix is irreducible. We also prove theorems describing conditi- ons which matrix must satisfy so matrix can be implemented with only 1 or 2 XOR operations. At the end of the thesis we show construction of circulant MDS matrices which uses elements of finite field with low XOR count so they can be easily implemented. Keywords: lightweight cryptography, finite field, XOR, MDS matrix
|
|
|
Gröbner basis, Zhuang-Zi algorithm and attacks of multivariable cryptosystems
Doktorová, Alice ; Tomáš, Jiří (referee) ; Kureš, Miroslav (advisor)
This diploma thesis is devoted to the multivariate cryptosystems. It includes an overview of commutative algebra with emphasis on Gröbner bases. Of all algorithms, especially the ones using Gröbner bases are studied, i.e. Buchberger's algorithm, which is already implemented in Wolfram Mathematica, and F4 algorithm, for which a program package has been created in the Wolfram Mathematica environment. Also Zhuang-Zi algorithm is described. To simplify its steps a program to compute the Lagrange interpolation polynomial has been created in Python.
|
|
|
Elliptic curves in cryptography
Geyer, Lukáš ; Burda, Karel (referee) ; Lambertová, Petra (advisor)
The objective of this bachelor thesis is to decribe the role of the elliptic curves in modern cryptosystems, explain the mathematical fundamentals upon which the elliptic curves are based along with their advantages and disadvantages, followed by application in the digital signature. The project is concluded by a software solution demonstrating the use of elliptic curves in digital signature scheme ECDSA
|
guest ::
