Many of today's cryptographic systems are based on the discrete logarithm problem, e.g. the Diffie-Hellman protocol. The number field sieve algorithm (NFS) is the algorithm solving the problem of factorization of integers, but latest works show, it can be also applied to the discrete logarithm problem. In this work, we study the number field sieve algorithm for discrete logarithm and we also compare the NFS for discrete logarithm with the NFS for factoriza- tion. Even though these NFS algorithms are based on the same principle, many differences are found. 1

O gás de Bose carregado foi estudado em duas e três dimensões, sendo que no caso bidimensional foram considerados dois tipos distintos de interação: l/r e ln(r). Aplicamos a esses sistemas o método do campo auto-consistente que leva em consideração a interação de curto alcance entre os bosons através de uma correção de campo local. Por meio de cálculos numéricos auto-consistentes determinamos o fator de estrutura S(&#8594k) em um amplo intervalo de densidades. A partir de S(&#8594k) obtivemos a função de correlação dos pares, a energia do estado fundamental que é essencialmente a energia de correlação, a pressão do gás e o espectro de excitações elementares. Calculamos ainda a densidade de blindagem induzida por uma impureza carregada fixada no gás. No limite de altas densidades nossos cálculos reproduzem os resultados da teoria de perturbação de Bogoliubov. Na região de densidades intermediárias em que os sistemas são fortemente correlacionados nossos resultados apresentam uma boa concordância com cálculos baseados na aproximação de HNC e no método de Monte Carlo. Nossos resultados são em várias situações confrontados com os de RPA demonstrando que o método que utilizamos é muito mais adequado para tratar o sistema. Os sistemas bidimensionais mostraram-se mais correlacionados que o tridimensional, sendo que o gás com interação l/r é mais correlacionado que o logarítmico a altas densidades, mas na região de densidades baixas essa situação se inverte. Finalmente calculamos as funções termodinâmicas dos sistemas bi e tridimensionais a temperaturas finitas próximas do zero absoluto baseando-nos nos espectros de excitação do gás a temperatura zero. / The two and three-dimensional charged Bose gas have been studied. In the bidimensional case two different types of interaction were considered: l/r and ln(r).We have applied to these systems the self-consistent-field method, which takes into account the short range correlations between the bosons through a local-field correction. By using self-consistent numerical calculations we determinate the structure factor S(&#8594k) in a wide range of densities. From S(&#8594k) we obtained the pair-correlation function, the ground-state energy, the pressure of the gas and the spectrum of elementary excitations. In addition we calculated the screening density induced by a fixed charged impurity. In the high-density limit our calculations reproduce the results given by Bogoliubov\'s perturbation theory. In the intermediate-density region, corresponding to the strongly coupled systems, our results are in very good agreement with calculations based on HNC approximation as well as Monte Carlo method. Our results are compared in several situations with RPA results showing that the self-consistent method is much more accurate. The two-dimensional systems showed to be more correlated than the three-dimensional one; the gas with interaction l/r is also more correlated than the logarithmic one at high densities, but it begins to be less correlated than this one in the low-density region. Finally we calculated the thermodynamic functions of the two and three-dimensional systems at finite temperatures near absolute zero, based upon the gas excitation spectra at zero temperature.

The growing interest in computing structured matrix functions stems from the fact that preserving and exploiting the structure of matrices can help us gain physically meaningful solutions with less computational cost and memory requirement. The work presented here is divided into two parts. The first part deals with the computation of functions of structured matrices. The second part is concerned with the structured error analysis in the computation of matrix functions. We present algorithms applying the inverse scaling and squaring method and using the Schur-like form of the symplectic matrices as an alternative to the algorithms using the Schur decomposition to compute the logarithm of symplectic matrices. There are two main calculations in the inverse scaling and squaring method: taking a square root and evaluating the Padé approximants. Numerical experiments suggest that using the Schur-like form with the structure preserving iterations for the square root helps us to exploit the Hamiltonian structure of the logarithm of symplectic matrices. Some type of matrices are nearly structured. We discuss the conditions for using the nearest structured matrix to the nearly structured one by analysing the forward error bounds. Since the structure preserving algorithms for computing the functions of matrices provide advantages in terms of accuracy and data storage we suggest to compute the function of the nearest structured matrix. The analysis is applied to the nearly unitary, nearly Hermitian and nearly positive semi-definite matrices for the matrix logarithm, square root, exponential, cosine and sine functions. It is significant to investigate the effect of the structured perturbations in the sensitivity analysis of matrix functions. We study the structured condition number of matrix functions defined between smooth square matrix manifolds. We develop algorithms computing and estimating the structured condition number. We also present the lower and upper bounds on the structured condition number, which are cheaper to compute than the "exact" structured condition number. We observe that the lower bounds give a good estimation for the structured condition numbers. Comparing the structured and unstructured condition number reveals that they can differ by several orders of magnitude. Having discussed how to compute the structured condition number of matrix functions defined between smooth square matrix manifolds we apply the theory of structured condition numbers to the structured matrix factorizations. We measure the sensitivity of matrix factors to the structured perturbations for the structured polar decomposition, structured sign factorization and the generalized polar decomposition. Finally, we consider the unstructured perturbation analysis for the canonical generalized polar decomposition by using three different methods. Apart from theoretical aspect of the perturbation analysis, perturbation bounds obtained from these methods are compared numerically and our findings show an improvement on the sharpness of the perturbation bounds in the literature.

This thesis analyzes various attacks on underlying computational problem of asymmetric cryptosystems. First part introduces two of the most used problems asymmetric cryptography is based on, which are integer factorization and computation of discrete logarithm. Algorithms for solving these problems are described and for each of them there is a discussion about when the use of this particular algorithm is appropriate and when it isn't. In the next part computational problems are related to algorithms RSA and ECC and it is shown, how solving the underlying problem enables us to crack the cypher. As a part of this thesis an application was developed that measures the efficiency of described attacks and by providing easy-to-understand enumeration of algorithm's steps it can be used to demonstrate how the attack works. Based on the results of performed analysis, most secure asymmetric cryptosystem is selected along with some recommendations regarding key pair generation.

Este trabalho tem como objetivo principal contribuir para o aperfeiçoamento do professor de matemática seja ele em formação ou em atuação. Buscamos oferecer um material que possa servir de referência técnica, histórica e epistemológica para o estudo do Logaritmo Natural. Discutimos aqui o conceito de Conhecimento Especializado do Conteúdo, cunhado por pesquisadores da Universidade de Michigan e liderados por Deborah Ball. Em seu artigo Content Knowledge for Teaching: What Makes It Special? (2008), eles levantam a questão \"Qual matemática o professor deve conhecer para dar cabo do trabalho de ensinar?\", dado que o conhecimento matemático necessário para o docente difere do conhecimento matemático requerido em outras profissões. Fazemos aqui uma análise crítica da abordagem utilizada para o tema em alguns livros didáticos de Ensino Médio, descrevemos de modo detalhado a construção da Função Logarítmica como realmente ocorreu no século XVII, ou seja, por meio de áreas de regiões sob a curva xy = 1, e definimos a função exponencial como a inversa dela, enfoque esse com caráter fortemente geométrico e que deu origem à noção de integral definida. Mostramos também a estreita relação existente entre as Progressões Aritméticas, Geométricas, Trigonometria e o próprio tema principal. Obtemos ainda a formalização do número irracional e tanto pelo método tradicional usado em livros de Cálculo e Análise como a decorrente da teoria apresentada. Por fim, apresentamos algumas situações curiosas que envolvem direta ou indiretamente essa constante e que podem ser trabalhadas com alunos da Educação Básica. / The main objective of this work is to contribute to the improvement of the mathematics teacher, whether in training or acting. We seek to offer a material that can serve as a technical, historical and epistemological reference for the study of the Natural Logarithm. We discuss here the concept of Specialized Content Knowledge, coined by University of Michigan researchers and led by Deborah Ball. In your article Content Knowledge for Teaching: What Makes It Special? (2008), they raise the question \"What mathematics does the teacher need to know for teaching?\", since the mathematical knowledge required for the teacher differs from the mathematical knowledge required in other professions. Here we present a critical analysis of the approach used for the subject in some high school textbooks. We describe in detail the construction of the Logarithmic Function as actually occurred in the seventeenth century, that is, through areas of regions under the curve xy = 1, and we define the exponential function as the inverse of it, a focus with a strongly geometric character that gave rise to the notion of definite integral. We also show the close relationship between Arithmetic, Geometric, Trigonometry and the main theme itself. We also obtain the formalization of the irrational number e, both by the traditional method used in Calculus and Analysis books and by the theory presented. Finally, we present some curious situations that directly or indirectly involve this constant and that can be worked with Basic Education students.

We live in an age where we willingly provide our social security number, credit card information, home address and countless other sensitive information over the Internet. Whether you are buying a phone case from Amazon, sending in an on-line job application, or logging into your on-line bank account, you trust that the sensitive data you enter is secure. As our technology and computing power become more sophisticated, so do the tools used by potential hackers to our information. In this paper, the underlying mathematics within ciphers will be looked at to understand the security of modern ciphers. An extremely important algorithm in today's practice is the Advanced Encryption Standard (AES), which is used by our very own National Security Agency (NSA) for data up to TOP SECRET. Another frequently used cipher is the RSA cryptosystem. Its security is based on the concept of prime factorization, and the fact that it is a hard problem to prime factorize huge numbers, numbers on the scale of 2^{2048} or larger. Cryptanalysis, the study of breaking ciphers, will also be studied in this paper. Understanding effective attacks leads to understanding the construction of these very secure ciphers.

This thesis identifies and extends techniques that can be linked to the principle of maximum entropy (maxent) and applied to parameter estimation in machine learning and statistics. Entropy functions based on deformed logarithms are used to construct Bregman divergences, and together these represent a generalization of relative entropy. The framework is analyzed using convex analysis to charac- terize generalized forms of exponential family distributions. Various connections to the existing machine learning literature are discussed and the techniques are applied to the problem of non-negative matrix factorization (NMF).

The purpose of the study was to investigate the effects of discovery and application based instruction (DABI) on students&rsquo / mathematics achievement and also to explore opinions of students toward DABI. The research was conducted by 118 ninth grade students from Etimesgut Anatolian High School, in Ankara, during the spring semester of 2001-2002 academic year. During the study, experimental groups received DABI and control groups received Traditionally Based Instruction (TBI). The treatment was completed in three weeks. Mathematics Achievement Test (MAT) and Logarithm Achievement Test (LAT) were administered as pre and posttest respectively. In addition, a questionnaire, Students&rsquo / Views and Attitudes About DABI (SVA) and interviews were administered to determine students&rsquo / views and attitudes toward DABI. Analysis of Covariance (ANCOVA), independent sample t-test and descriptive statistics were used for testing the hypothesis of the study. No significant difference was found between LAT mean scores of students taught with DABI and traditionally based instruction when MAT test scores were controlled. In addition, neither students&rsquo / field of study nor gender was a significant factor for LAT scores. Students&rsquo / gender was not a significant factor for SVA scores. However, there was significant effect of math grades and field selections of students on SVA scores.

O gás de Bose carregado foi estudado em duas e três dimensões, sendo que no caso bidimensional foram considerados dois tipos distintos de interação: l/r e ln(r). Aplicamos a esses sistemas o método do campo auto-consistente que leva em consideração a interação de curto alcance entre os bosons através de uma correção de campo local. Por meio de cálculos numéricos auto-consistentes determinamos o fator de estrutura S(&#8594k) em um amplo intervalo de densidades. A partir de S(&#8594k) obtivemos a função de correlação dos pares, a energia do estado fundamental que é essencialmente a energia de correlação, a pressão do gás e o espectro de excitações elementares. Calculamos ainda a densidade de blindagem induzida por uma impureza carregada fixada no gás. No limite de altas densidades nossos cálculos reproduzem os resultados da teoria de perturbação de Bogoliubov. Na região de densidades intermediárias em que os sistemas são fortemente correlacionados nossos resultados apresentam uma boa concordância com cálculos baseados na aproximação de HNC e no método de Monte Carlo. Nossos resultados são em várias situações confrontados com os de RPA demonstrando que o método que utilizamos é muito mais adequado para tratar o sistema. Os sistemas bidimensionais mostraram-se mais correlacionados que o tridimensional, sendo que o gás com interação l/r é mais correlacionado que o logarítmico a altas densidades, mas na região de densidades baixas essa situação se inverte. Finalmente calculamos as funções termodinâmicas dos sistemas bi e tridimensionais a temperaturas finitas próximas do zero absoluto baseando-nos nos espectros de excitação do gás a temperatura zero. / The two and three-dimensional charged Bose gas have been studied. In the bidimensional case two different types of interaction were considered: l/r and ln(r).We have applied to these systems the self-consistent-field method, which takes into account the short range correlations between the bosons through a local-field correction. By using self-consistent numerical calculations we determinate the structure factor S(&#8594k) in a wide range of densities. From S(&#8594k) we obtained the pair-correlation function, the ground-state energy, the pressure of the gas and the spectrum of elementary excitations. In addition we calculated the screening density induced by a fixed charged impurity. In the high-density limit our calculations reproduce the results given by Bogoliubov\'s perturbation theory. In the intermediate-density region, corresponding to the strongly coupled systems, our results are in very good agreement with calculations based on HNC approximation as well as Monte Carlo method. Our results are compared in several situations with RPA results showing that the self-consistent method is much more accurate. The two-dimensional systems showed to be more correlated than the three-dimensional one; the gas with interaction l/r is also more correlated than the logarithmic one at high densities, but it begins to be less correlated than this one in the low-density region. Finally we calculated the thermodynamic functions of the two and three-dimensional systems at finite temperatures near absolute zero, based upon the gas excitation spectra at zero temperature.

CoordenaÃÃo de AperfeÃoamento de Pessoal de NÃvel Superior / Este trabalho apresenta os nÃmeros complexos com um enfoque que julgamos ser adequado para os professores do ensino mÃdio. O objetivo do trabalho à fornecer mais um texto sobre o tema e auxiliar os professores do ensino mÃdio em suas aulas. Iniciamos o trabalho com uma definiÃÃo de nÃmeros complexos que contempla o rigor matemÃtico necessÃrio e busca manter a simplicidade exigida para esse nÃvel de ensino. Utilizamos a representaÃÃo geomÃtrica de um nÃmero complexo sempre que possÃvel para motivar e simplificar as definiÃÃes e demonstraÃÃes. Abordamos as fÃrmulas trigonomÃtrica e de Moivre ressaltando a sua importÃncia. Apresentamos a deduÃÃo da fÃrmula da raiz n-Ãsima de um nÃmero complexo . No penÃltimo capÃtulo, abordamos alguns assuntos que nÃo sÃo contemplados nos livros didÃticos de matemÃtica do ensino mÃdio que sÃo a fÃrmula de Euler, a qual grande parte das aplicaÃÃes dos nÃmeros complexos existe devido a essa grande descoberta. O logaritmo complexo, cuja teoria explica como se calcular logaritmos de nÃmeros negativos ou complexos e tambÃm tratamos sobre potÃncias complexas, de tal maneira a explicar como se calcular potÃncias de nÃmero quando a base e o expoente sÃo nÃmeros complexos. Finalmente, encerramos este trabalho fazendo uma anÃlise de alguns livros didÃticos do ensino mÃdio. / This paper presents the complex numbers with an approach that we think to be appropriated for high school teachers. The aim is to provide one more text on the subject and assist high school teachers in their classes. We started working with a definition of complex numbers which includes the mathematical rigor necessary and seeks to maintain the simplicity required for this level of education. We used the geometric representation of a complex number, wherever possible, to motivate and simplify the definitions and demonstrations. We discussed the trigonometric and Moivre formulas emphasizing their importance. We presented the deduction of the formula for n-th root of a complex number. On the penultimate chapter, we discussed some issues that are not covered in mathematics textbooks from high school such as Eulerâs formula, which the majority of applications of complex numbers exists because of this great discovery. The complex logarithm,whose theory explains how to calculate logarithms of negative or complex numbers and we also worked on complex powers, in such a way to explain how to calculate power number when the base and the exponent are complex numbers. Finally, we concluded this paper by analyzing some high school textbooks.