Let n and l be positive integers and f (x) be an irreducible polynomial over Fq such that ldeg( f (x)) &lt / 2n - 1, where q is 2 or 3. We obtain an effective upper bound for the multiplication complexity of n-term polynomials modulo f (x)^l. This upper bound allows a better selection of the moduli when Chinese Remainder Theorem is used for polynomial multiplication over Fq. We give improved formulae to multiply polynomials of small degree over Fq. In particular we improve the best known multiplication complexities over Fq in the literature in some cases. Moreover, we present a method for multiplication in finite fields improving finite field multiplication complexity muq(n) for certain values of q and n. We use local expansions, the lengths of which are further parameters that can be used to optimize the bounds on the bilinear complexity, instead of evaluation into residue class field. We show that we obtain improved bounds for multiplication in Fq^n for certain values of q and n where 2 &lt / = n &lt / =18 and q = 2, 3, 4.

Time Memory Trade O (TMTO) is a cryptanalytic method that aims to develop an attack which has a lower memory complexity than lookup table and a lower online time complexity than exhaustive search. TMTO methods are widely studied in the literature and used for inverting various cryptosystems. We focus on the design and the analysis of TMTO on symmetric ciphers in this thesis. Firstly, the summary of the random mapping statistics from the view point of TMTO is presented. We also recalculate some expected values with a simpler approach than the existing proofs. Then, we propose some variant constructions and also present three new distinguishers based on random mappings. Next, we provide a detailed analysis of the success rate of two main improvements of the attack / Distinguished Point Method and Rainbow Method. Finally, we discuss the adjustment of the parameters to achieve a high success rate. To support our theoretical framework, we also present empirical results of our analysis to actual ciphers.

In this thesis we study low correlation zone (LCZ) sequence sets and a class of quadratic forms. In the first part we obtain two new classes of optimal LCZ sequence sets. In our first construction using a suitable orthogonal transformation we extend some results of [21]. We give new classes of LCZ sequence sets defined over Z4 in our second construction. We show that our LCZ sequence sets are optimal with respect to the Tang, Fan and Matsufiji bound [37]. In the second part we consider some special linearized polynomials and corresponding quadratic forms. We compute the number of solutions of certain equations related to these quadratic forms and we apply these result to obtain curves with many rational points.

In this thesis, the asymptotic behavior and oscillation of solutions of dynamic equations on time scales are studied. In the first part of the thesis, asymptotic equivalence and asymptotic equilibrium of dynamic systems are investigated. Sufficient conditions are established for the asymptotic equivalence of linear systems and linear and quasilinear systems, respectively, and for the asymptotic equilibrium of quasilinear systems by unifying and extending some known results for differential systems and difference systems to dynamic systems on arbitrary time scales. In particular, for the asymptotic equivalence of differential systems, the well-known theorems of Levinson and Yakubovich are improved and the well-known theorem of Wintner for the asymptotic equilibrium of linear differential systems is generalized to arbitrary time scales. Some of our results for asymptotic equilibrium are new even for difference systems. In the second part, the oscillation of solutions of a particular class of second order nonlinear delay dynamic equations and, more generally, two-dimensional nonlinear dynamic systems, including delay-dynamic systems, are discussed. Necessary and sufficient conditions are derived for the oscillation of solutions of nonlinear delay dynamic equations by extending some continuous results. Specifically, the classical theorems of Atkinson and Belohorec are generalized. Sufficient conditions are established for the oscillation of solutions of nonlinear dynamic systems by unifying and extending the corresponding continuous and discrete results. Particularly, the oscillation criteria of Atkinson, Belohorec, Waltman, and Hooker and Patula are generalized.

Differential cryptanalysis and linear cryptanalysis are the first significant methods used to attack on block ciphers. These concepts compose the keystones for most of the attacks in recent years. Also, while designing a cipher, these attacks should be taken into consideration and the cipher should be created as secure against them. Although differential cryptanalysis and linear cryptanalysis are still important, they started to be inecient due to the improvements in the technology. So, these attacks are extended. For instance, higher order differential cryptanalysis, truncated differential cryptanalysis, generalized linear cryptanalysis, partitioning linear cryptanalysis, linear cryptanalysis using multiple linear approximations are introduced as the extended versions of these attacks. There exists significant applications of these extended attacks. Algebraic attack is a method of cryptanalysis that consists of obtaining a representation of the cipher as a system of equations and then, solving this system. Up to today, just a few attacks that are practically possible to mount are presented. However, due to the fact that algebraic cryptanalysis requires only a handful of known plaintexts to perform, it is a promising and significant attack. This thesis is a survey covering all the methods of attacks described above. Illustrations and summaries of some important papers including these cryptanalysis techniques are given.

Generating functions are important tools that are used in many areas of mathematics and especially statistics. Besides analyzing the general structure of sequences and their asymptotic behavior / these functions, which can be roughly thought as the transformation of sequences into functions, are also used effciently to solve combinatorial problems. In this thesis, the effects of the transformations of generating functions on their corresponding sequences and the effects of the change in sequences on the generating functions are examined. With these knowledge, the generating functions for the resulting sequence of some combinatorial problems such as number of partitions, number of involutions, Fibonacci numbers and Catalan numbers are found. Moreover, some mathematical identities are proved by using generating functions. The sequences are the bases of especially symmetric key cryptosystems in cryptography. It is seen that by using generating functions, linear complexities and periods of sequences generated by constant coeffcient linear homogeneous recursions, which are used in linear feedback shift register (LFSR) based stream ciphers, can be calculated. Hence studying generating functions leads to have a better understanding in them. Therefore, besides combinatorial problems, such recursions are also examined and the results are used to observe the linear complexity and the period of LFSR&rsquo / s combined in different ways to generate &ldquo / better&rdquo / system of stream cipher.

High-dimensional data take place from digital image processing, gene expression micro arrays, neuronal population activities to financial time series. Dimensionality Reduction - extracting low dimensional structure from high dimension - is a key problem in many areas like information processing, machine learning, data mining, information retrieval and pattern recognition, where we find some data reduction techniques. In this thesis we will give a survey about modern data reduction techniques, representing the state-of-the-art of theory, methods and application, by introducing the language of mathematics there. This needs a special care concerning the questions of, e.g., how to understand discrete structures as manifolds, to identify their structure, preparing the dimension reduction, and to face complexity in the algorithmically methods. A special emphasis will be paid to Principal Component Analysis, Locally Linear Embedding and Isomap Algorithms. These algorithms are studied by a research group from Vilnius, Lithuania and Zeev Volkovich, from Software Engineering Department, ORT Braude College of Engineering, Karmiel, and others. The main purpose of this study is to compare the results of the three of the algorithms. While the comparison is beeing made we will focus the results and duration.

One of the most basic properties expected from block ciphers and hash functions is passing statistical randomness testing, as they are supposed to behave like random mappings. Previously, testing of AES candidate block ciphers was done by using the statistical tests defined in the NIST Test Suite. As some of the tests in this suite require long sequences, data sets are formed by concatenating the outputs of the algorithms obtained from various input types. However, the nature of block cipher and hash function algorithms necessitates devising tests and test parameters focused particularly on short sequences, therefore we propose a package of statistical randomness tests which produce reliable results for short sequences and test the outputs of the algorithms directly rather than concatenations. Moreover, we propose an alternative method to evaluate the test results and state the required computations of related probabilities for the new evaluation method. We also propose another package of statistical tests which are designed basing on certain cryptographic properties of block ciphers and hash functions to evaluate their randomness, namely the cryptographic randomness testing. The packages are applied to the AES finalists, and produced more precise results than those obtained in similar applications. Moreover, the packages are also applied to SHA-3 second round candidate algorithms.

In this study, a bi-objective capacitated facility location problem is presented which includes partial coverage concept and relocation of facility nodes. In partial coverage, a predefined distance between a demand node and a facility node is assumed to be fully covered. After the predefined distance, the service level commences to decay linearly. The problem is designed to consider the existence of already functioning facility nodes. It is allowed to close these existing facilities and open new facilities in potential sites. However, existing facility nodes are strongly favored against new facility nodes. The objectives are the maximization of the weighted total coverage and the minimization of number of facility nodes. A novel interactive multi-objective evolutionary algorithm is proposed to solve this problem, I-TREA. I-TREA is originated from NSGA-II and designed for interactive methods benefiting from quality infeasible solutions. The performance of I-TREA is benchmarked with a modified version of NSGA-II on randomly generated problems with various sizes and utility functions.

ABSTRACT PORTFOLIO SELECTION AND RETURN PERFORMANCE: An Application of the Black-Litterman Method in the Istanbul Stock Exchange Bozdemir, Mehmet Burak M.Sc, Department of Financial Mathematics Supervisor : Assist. Prof. Dr. Seza Dani