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Searching optimal strategies for the number field sieve
Perůtka, Lukáš ; Drápal, Aleš (advisor) ; Růžička, Pavel (referee)
In this work we study the number field sieve algorithm. Our main focus is on its theoretical background. We present all important theorems which are needed for a full understanding of the algorithm. We also describe the most widely used implementation of the parts of the algorithm and we discuss in which situation they should be used. At the end we show results from measurements of sieving phase on the implementation which was written for our Department of Algebra.
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Linear codes and a projective plane of order 10
Liška, Ondřej ; Drápal, Aleš (advisor) ; Vojtěchovský, Petr (referee)
Projective plane of order 10 does not exist. Proof of this assertion was finished in 1989 and is based on the nonexistence of a binary code C generated by the incidence vectors of the plane's lines. As part of the proof of the nonexistence of code C, the coefficients of its weight enumerator were studied. It was shown that coefficients A12, A15, A16 and A19 have to be equal to zero, which contradicted other findings about the relationship among the coefficients. Presented diploma thesis elaborately analyses the phases of the proof and, in several places, enhances them with new observations and simplifications. Part of the proof is generalized for projective planes of order 8m + 2. 1
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Samoopravné kódy a rozpoznávání podle duhovky
Luhan, Vojtěch ; Šťovíček, Jan (advisor) ; Drápal, Aleš (referee)
Iris recognition constitutes one of the most powerful method for the iden- tification and authentication of people today. This thesis aims to describe the algorithms used in a sophisticated and mathematically correct way, while re- maining comprehensible. The description of these algorithms is not the only objective of this thesis; the reason they were chosen and potential improvements or substitutions are also discussed. The background of iris recognition, its use in cryptosystems, and the application of error-correcting codes are investigated as well.
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Point Counting on Elliptic and Hyperelliptic Curves
Vácha, Petr ; Šťovíček, Jan (advisor) ; Drápal, Aleš (referee)
In present work we study the algorithms for point counting on elliptic and hy- perelliptic curves. At the beginning we describe a few simple and ineffective al- gorithms. Then we introduce more complex and effective ways to determine the point count. These algorithms(especially the Schoof's algorithm) are important for the cryptography based on discrete logarithm in the group of points of an el- liptic or hyperelliptic curve. The point count is important to avoid the undesirable cases where the cryptosystem is easy to attack. 1
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Applications of error-correcting codes in steganography
Cinkais, Roman ; Drápal, Aleš (advisor) ; Lisoněk, Petr (referee)
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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Kódování a efektivita LDPC kódů
Kozlík, Andrew ; Drápal, Aleš (advisor) ; Holub, Štěpán (referee)
Low-density parity-check (LDPC) codes are linear error correcting codes which are capable of performing near channel capacity. Furthermore, they admit efficient decoding algorithms that provide near optimum performance. Their main disadvantage is that most LDPC codes have relatively complex encoders. In this thesis, we begin by giving a detailed discussion of the sum-product decoding algorithm, we then study the performance of LDPC codes on the binary erasure channel under sum-product decoding to obtain criteria for the design of codes that allow reliable transmission at rates arbitrarily close to channel capacity. Using these criteria we show how such codes are designed. We then present experimental results and compare them with theoretical predictions. Finally, we provide an overview of several approaches to solving the complex encoder problem.
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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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Audio steganography and IP telephony
Hrinčárová, Monika ; Drápal, Aleš (advisor) ; Kozlík, Andrew (referee)
Steganography is a technique which hides secret information. In this work, we will hide a secret information in the packets which are produced during a Skype call. Skype is one of the best known and the most widely used VoIP applications. We will propose, describe and implement a steganography method by which we will send the secret message during the Skype call. For embedding the message into packets and extracting them, we will use steganographic method called matrix encoding. To avoid packet loss, we will increase the robustness of this method by error-correcting and self-synchronising codes. As error-correcting codes, we will use the binary Hamming (7, 4) -codes and for the self-synchronising, we will use T-codes. 1
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