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Frobenius tests of primality
Hora, Jan ; Drápal, Aleš (advisor) ; Holub, Štěpán (referee)
Title: Frobenius primality tests Author: Jan Hora Department: Department of algebra Supervisor: prof. RNDr. Aleš Drápal, CSc. Dsc. Supervisor's e-mail address: drapal@karlin.mff.cuni.cz Abstract: In the present work we study Extended Quadratic Frobenius primality test. We study its functionality, error probability and taken time. We will define ring R(n, c), in which the test works. We will describe its structure depending up primality of tested number n, and algorithm of Frobenius test. We will show, that test succeeds anytime the tested number n is prime. We will study the upper bound for error probability of the test. We will show that test fails iff certain elements of set R(n, c) are chosen, Set of that elements will be denoted G(n, c). We will find conditions that G(n, c) must fulfil, and with them we will discover, that Frobenius test eroor probability is at most 1 24 . We will analyse individual parts of algorithm and discover, that one cyclus of Frobenius test can be done with approximately 2 log n multiplications in Zn. Finaly the teoretical estimates will be compared with practical results. Keywords: primality test, Frobenius automorfism, cyklic group, roots of one 1
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Extended binary Golay code
Uchytilová, Vendula ; Drápal, Aleš (advisor) ; Hora, Jan (referee)
This work deals with three different constructions of the extended binary Golay code G24. The first construction is based on a projective plane of order four. In terms of it Steiner system (5, 8, 24) is built. Linear span of its blocks forms a linear binary [24, 12, 8] code C. Every binary [24, 12, 8] code is isomorphic to C which is known as extended binary Golay code G24. The second construction uses so-called Miracle Octad Generator (MOG). All MOG-words of weight eight form Steiner system (5, 8, 24). The third construction uses impartial combinatorial game Mogul. In terms of its P-positions one can create a linear binary [24, 12, 8] code. The fact that is is also a lexikographic code is useful for parametres estimate. 1
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MDS codes conjecture
Kesely, Michal ; Drápal, Aleš (advisor) ; Lisoněk, Petr (referee)
In this thesis, we study some properties of MDS codes and we mainly focus on the MDS codes conjecture. In the first chapter we define MDS codes, show some examples and basic properties of MDS codes, for example a link between MDS codes and Latin squares or rectangles. Afterwards we state the MDS codes conjecture and prove it in several cases. In the third chapter we can observe the relationship between MDS codes and narcs in projective geometries. Finally we present those known cases, for which the MDS conjecture holds.
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Algebraická teorie S-boxů
Ďuránová, Elena ; Tůma, Jiří (advisor) ; Drápal, Aleš (referee)
The thesis focuses on an algebraic description of S-boxes by the special type of quadratic equations, defined as biaffine equations. Biaffine equations satisfying S-boxes of higher order may not even exist. However, the special type of S-boxes en- ables to find such equations also for S-boxes of higher order. The S-box in the block cipher Rijndael, composed of the inverse function and the affine transformation, is an example of such special type of S-boxes. The thesis proves that a number of biaffine equations satisfying an S-box of this type does not depend on the affine function. The thesis also proves that for every S-box of order n formed by the in- verse function there exist at least 3n − 1 biaffine equations satisfying this S-box. 1
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Applications of error-correcting codes in steganography
Cinkais, Roman ; Lisoněk, Petr (referee) ; Drápal, Aleš (advisor)
Modern steganography is a relatively new discipline with many applications in information security. Contrary to the cryptography which is trying to make a message unreadable to third party using cryptographic algorithms, the aim of steganography is to hide a communication between parties. Applications of error-correcting codes and covering functions markedly increases abilities and security of steganographic algorithms. This thesis is attended to steganography using error-correcting codes which has the best results nowadays regarding embedding efficiency. New constructions will help us to work with non-linear codes and providing new steganographic algorithms. We will see that these algorithms have a better ability to hide communication, resp. a message in a digital medium. Further improvements can be made using applications of general q-ary codes. Many new questions are coming out with that which need to be answered. Several comparisons are showing that the area of steganography is in a beginning and we will be participants of such a progress as cryptography experienced in the last two decades.
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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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