Neste trabalho,nos apresentamos a K-teoria algébrica a qual é um ramo da álgebra que associa para cada anel comutativo comunidade R, uma sequencia de grupos abelianos ditos de n-ésimos K-grupos do anel R, denotada por Kn(R) . A meados da década de 1950,Alexander Grothendieck da a definição do K0(R) de um anel R. Em 1962, Hyman Bass e Stephen Schanuel apresenta a primeira definição adequada do K1(R) de um anel R. Em 1970, Daniel Quillen da uma definição geral dos K-grupos de um anel R a partir da +- construção do espaço classificante BGL(R). Nosso interesse é o estudo dos K-grupos sobre o anel de inteiros OF sobre um corpo numérico F. Usando alguns resultados de homologia dos grupos lineares, neste trabalho daremos a definição do mapa regulador de Borel. / In this paper,we present the algebraic K-theory,which is a branch of algebra that associates to any ring with unit R a sequence of abelian groups called n-th K-groups of R, denoted by Kn(R). The mid-1950s, Alexander Grothendieck gave a definition of the K0(R) of any ring R. In1962, Hyman Bass and Stephen Schanuel gave the first adequate definition of K1 of any ring R. In 1970, Daniel Quillen gave a general definition of K-groups of any ring R using the +- construction of the classifying space BGL(R). Our interest is the study of the K-groups on the ring of integers OF over a number field F. Using some results of homology of linear groups, this work will give the definition of Borel\'s regulator map.

Depuis 1960 et le résultat fondateur de Karatsuba, on sait que la complexité de la multiplication (d’entiers ou de polynômes) est sous-quadratique : étant donné un anneau R quelconque, le produit sur R[X] des polynômes a_0 + a_1 X et b_0 + b_1 X, pour tous a_0, a_1, b_0 et b_1 dans R, peut être calculé en seulement trois et non pas quatre multiplications sur R : (a_0 + a_1 X)(b_0 + b_1 X) = m_0 + (m_2 - m_0 - m_1)X + m_1 X^2, avec les trois produits m_0 = a_0b_0, m_1 = a_1b_1 et m_2 = (a_0 + a_1)(b_0 + b_1). De la même manière, l’algorithme de Strassen permet de multiplier deux matrices 2nx2n en seulement sept produits de matrices nxn. Les deux exemples précédents tombent dans la catégorie des applications bilinéaires : des fonctions de la forme Phi : K^m x K^n -> K^l, pour un corps donné K, linéaires en chacune des deux variables. Parmi les applications bilinéaires les plus classiques, on trouve ainsi la multiplication de polynômes, de matrices, ou encore d’éléments d’extensions algébriques de corps finis. Étant donnée une application bilinéaire Phi, calculer le nombre minimal de multiplications nécessaires au calcul de cette application est un problème NP-difficile. L'objectif de cette thèse est de proposer des algorithmes minimisant ce nombre de multiplications. Deux angles d'attaques ont été suivis. Un premier aspect de cette thèse est l'étude du problème du calcul de la complexité bilinéaire sous l'angle de la reformulation de ce problème en termes de recherche de sous-espaces vectoriels de matrices de rang donné. Ce travail a donné lieu à un algorithme tenant compte de propriétés intrinsèques aux produits considérés tels que les produits matriciels ou polynomiaux sur des corps finis. Cet algorithme a permis de trouver toutes les décompositions possibles, sur F_2, pour le produit de polynômes modulo X^5 et le produit de matrices 3x2 par 2x3. Un autre aspect de ma thèse est celui du développement d’algorithmes asymptotiquement rapides pour la multiplication entière. Une famille particulière d'algorithmes récents ont été proposés suite à un article de Fürer publié en 2007, qui proposait un premier algorithme, reposant sur la transformée de Fourier rapide (FFT) permettant de multiplier des entiers de n bits en O(n log n 2^{O(log^* n)}), où log^* est la fonction logarithme itéré. Dans cette thèse, un algorithme dont la complexité dépend d'une conjecture de théorie des nombres est proposé, reposant sur la FFT et l'utilisation de premiers généralisés de Fermat. Une analyse de complexité permet d'obtenir une estimation en O(n log n 4^{log^* n}) / Since 1960 and the result of Karatsuba, we know that the complexity of the multiplication (of integers or polynomials) is sub-quadratic: given a ring R, the product in R[X] of polynomials a_0 + a_1 X and b_0 + b_1 X, for any a_0, a_1, b_0 and b_1 in R, can be computed with three and not four multiplications over R: (a_0 + a_1X)(b_0 + b_1X) = m_0 + (m_2 - m_0 - m_1)X + m_1X^2, with the three multiplications m_0 = a_0b_0, m_1 = a_1b_1 et m_2 = (a_0 + a_1)(b_0 + b_1). In the same manner, Strassen's algorithm allows one to multiply two matrices 2nx2n with only seven products of matrices nxn. The two previous examples fall in the category of bilinear maps: these are functions of the form Phi : K^m x K^n -> K^l, given a field K, linear in each variable. Among the most classical bilinear maps, we have the multiplication of polynomials, matrices, or even elements of algebraic extension of finite fields. Given a bilinear map Phi, computing the minimal number of multiplications necessary to the evaluation of this map is a NP-hard problem. The purpose of this thesis is to propose algorithms minimizing this number of multiplications. Two angles of attack have been studied. The first aspect of this thesis is to study the problem of the computation of the bilinear complexity under the angle of the reformulation of this problem in terms of research of matrix subspaces of a given rank. This work led to an algorithm taking into account intrinsic properties of the considered products such as matrix or polynomial products over finite fields. This algorithm allows one to find all the possible decompositions, over F_2, for the product of polynomials modulo X^5 and the product of matrices 3x2 by 2x3. Another aspect of this thesis was the development of fast asymptotic methods for the integer multiplication. There is a particular family of algorithms that has been proposed after an article by Fürer published in 2007. This article proposed a first algorithm, relying on fast Fourier transform (FFT), allowing one to multiply n-bit integers in O(n log n 2^{O(log^* n)}), where log^* is the iterated logarithm function. In this thesis, an algorithm, relying on a number theoretical conjecture, has been proposed, involving the use of FFT and generalized Fermat primes. With a careful complexity analysis of this algorithm, we obtain a complexity in O(nlog n 4^{log^* n})

In this dissertation, we consider six Prufer-like conditions on acommutative ring R. These conditions form a hierarchy. Being a Prufer ring is not a local property: a Prufer ring may not remain a Prufer ring when localized at a prime or maximal ideal. We introduce a seventh condition based on this fact and extend the hierarchy. All the conditions of the hierarchy become equivalent in the case of a domain, namely a Prufer domain. We also seek the relationship of the hierarchy with strong Prufer rings. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2015 / FAU Electronic Theses and Dissertations Collection

Neste trabalho,nos apresentamos a K-teoria algébrica a qual é um ramo da álgebra que associa para cada anel comutativo comunidade R, uma sequencia de grupos abelianos ditos de n-ésimos K-grupos do anel R, denotada por Kn(R) . A meados da década de 1950,Alexander Grothendieck da a definição do K0(R) de um anel R. Em 1962, Hyman Bass e Stephen Schanuel apresenta a primeira definição adequada do K1(R) de um anel R. Em 1970, Daniel Quillen da uma definição geral dos K-grupos de um anel R a partir da +- construção do espaço classificante BGL(R). Nosso interesse é o estudo dos K-grupos sobre o anel de inteiros OF sobre um corpo numérico F. Usando alguns resultados de homologia dos grupos lineares, neste trabalho daremos a definição do mapa regulador de Borel. / In this paper,we present the algebraic K-theory,which is a branch of algebra that associates to any ring with unit R a sequence of abelian groups called n-th K-groups of R, denoted by Kn(R). The mid-1950s, Alexander Grothendieck gave a definition of the K0(R) of any ring R. In1962, Hyman Bass and Stephen Schanuel gave the first adequate definition of K1 of any ring R. In 1970, Daniel Quillen gave a general definition of K-groups of any ring R using the +- construction of the classifying space BGL(R). Our interest is the study of the K-groups on the ring of integers OF over a number field F. Using some results of homology of linear groups, this work will give the definition of Borel\'s regulator map.

Made available in DSpace on 2016-04-27T16:57:39Z (GMT). No. of bitstreams: 1 Flavio Cabral de Souza.pdf: 3290710 bytes, checksum: 0fca20522f91a2be80f46598226cf404 (MD5) Previous issue date: 2015-08-04 / The aim of this paper was to assess how students at sixth grade of elementary school in Brazil, who have never had any formal contact with integers numbers and operations, use their prior knowledge to solve situations involving this mathematical object and how they develop their knowledge autonomously. To achieve this goal, a review of the literature was made in order to check how researchers have treated the integers numbers and which strategies for the approach of integers numbers and their operations were used in the past. This review was crucial to develop activities that allowed the students to engage with problem solving and to bring out their prior knowledge. The series of activities developed for this project included visual resources with the aid of technology, problem situation and objective questions. During the development of the activities, it was possible to identify, through the records and conversations in the classroom, the singularities related to the understanding of integers numbers and their operations by the students. Therefore, this research has identified the scope of the students prior knowledge in the understanding of integers numbers and their operations, and the obstacles they faced when addressing this content. The results of this project can help us to rethink how integers numbers and their operations can be addressed in the teaching practice / Este trabalho tem como objetivo verificar como os alunos do 6º ano do ensino fundamental, que não tiveram contato formal com os números inteiros e suas operações, mobilizam seus conhecimentos prévios para resolver situações que envolvam esse objeto matemático e se os mesmos poderiam se desenvolver de forma autônoma para a sua compreensão. Para atingir esse objetivo foi feita uma revisão da literatura a fim de verificar como os pesquisadores têm tratado os números inteiros, procurando identificar quais estratégias para a abordagem dos números inteiros e suas operações foram utilizadas. Na sequência de atividades construída para esse trabalho são apresentadas situações nas quais foram explorados recursos visuais, com o auxílio de tecnologia, situações-problema e questões objetivas. A sequência de atividades procurou enfatizar o objeto matemático a partir de situações concretas, possibilitando que o aluno abstraia e generalize o conhecimento construído. Durante o desenvolvimento das atividades foram identificadas, nos registros e nos diálogos dos alunos, situações que permitiram inferir as singularidades referentes à compreensão que os alunos podem obter sobre os números inteiros e suas operações. Assim, essa pesquisa permitiu identificar o alcance dos conhecimentos prévios dos alunos para compreender os números inteiros e suas operações, e também os obstáculos que os mesmos enfrentaram para o desenvolvimento desse conteúdo. Os resultados obtidos revelam informações que nos permitem repensar sobre como os números inteiros e suas operações podem ser abordados na prática docente

A new class of moduli called the low-weight polynomial form integers (LWPFIs) is introduced. LWPFIs are expressed in a low-weight, monic polynomial form, <em>p</em> = <em>f</em>(<em>t</em>). While the generalized Mersenne numbers (GMNs) proposed by Solinas allow only powers of two for <em>t</em>, LWPFIs allow any positive integers. In our first proposal of LWPFIs, we limit the coefficients of <em>f</em>(<em>t</em>) to be 0 and ±1, but later we extend LWPFIs to allow any integer of less than <em>t</em> for the coefficients of <em>f</em>(<em>t</em>). Modular multiplication using LWPFIs is performed in two phases: 1) polynomial multiplication in Z[<em>t</em>]/<em>f</em>(<em>t</em>) and 2) coefficient reduction. We present an efficient coefficient reduction algorithm based on a division algorithm derived from the Barrett reduction algorithm. We also show a coefficient reduction algorithm based on the Montgomery reduction algorithm. We give analysis and experimental results on modular multiplication using LWPFIs. <br /><br /> New three, four and five-way squaring formulae based on the Toom-Cook multiplication algorithm are presented. All previously known squaring algorithms are symmetric in the sense that the point-wise multiplication step involves only squarings. However, our squaring algorithms are asymmetric and use at least one multiplication in the point-wise multiplication step. Since squaring can be performed faster than multiplication, our asymmetric squaring algorithms are not expected to be faster than other symmetric squaring algorithms for large operand sizes. However, our algorithms have much less overhead and do not require any nontrivial divisions. Hence, for moderately small and medium size operands, our algorithms can potentially be faster than other squaring algorithms. Experimental results confirm that one of our three-way squaring algorithms outperforms the squaring function in GNU multiprecision library (GMP) v4. 2. 1 for certain range of input size. Moreover, for degree-two squaring in Z[<em>x</em>], our algorithms are much faster than any other squaring algorithms for small operands. <br /><br /> We present a side channel attack on XTR cryptosystems. We analyze the statistical behavior of simultaneous XTR double exponentiation algorithm and determine what information to gather to reconstruct the two input exponents. Our analysis and experimental results show that it takes <em>U</em>^{1. 25} tries, where <em>U</em> = max(<em>a</em>,<em>b</em>) on average to find the correct exponent pair (<em>a</em>,<em>b</em>). Using this result, we conclude that an adversary is expected to make <em>U</em>^{0. 625} tries on average until he/she finds the correct secret key used in XTR single exponentiation algorithm, which is based on the simultaneous XTR double exponentiation algorithm.

A new class of moduli called the low-weight polynomial form integers (LWPFIs) is introduced. LWPFIs are expressed in a low-weight, monic polynomial form, <em>p</em> = <em>f</em>(<em>t</em>). While the generalized Mersenne numbers (GMNs) proposed by Solinas allow only powers of two for <em>t</em>, LWPFIs allow any positive integers. In our first proposal of LWPFIs, we limit the coefficients of <em>f</em>(<em>t</em>) to be 0 and ±1, but later we extend LWPFIs to allow any integer of less than <em>t</em> for the coefficients of <em>f</em>(<em>t</em>). Modular multiplication using LWPFIs is performed in two phases: 1) polynomial multiplication in Z[<em>t</em>]/<em>f</em>(<em>t</em>) and 2) coefficient reduction. We present an efficient coefficient reduction algorithm based on a division algorithm derived from the Barrett reduction algorithm. We also show a coefficient reduction algorithm based on the Montgomery reduction algorithm. We give analysis and experimental results on modular multiplication using LWPFIs. <br /><br /> New three, four and five-way squaring formulae based on the Toom-Cook multiplication algorithm are presented. All previously known squaring algorithms are symmetric in the sense that the point-wise multiplication step involves only squarings. However, our squaring algorithms are asymmetric and use at least one multiplication in the point-wise multiplication step. Since squaring can be performed faster than multiplication, our asymmetric squaring algorithms are not expected to be faster than other symmetric squaring algorithms for large operand sizes. However, our algorithms have much less overhead and do not require any nontrivial divisions. Hence, for moderately small and medium size operands, our algorithms can potentially be faster than other squaring algorithms. Experimental results confirm that one of our three-way squaring algorithms outperforms the squaring function in GNU multiprecision library (GMP) v4. 2. 1 for certain range of input size. Moreover, for degree-two squaring in Z[<em>x</em>], our algorithms are much faster than any other squaring algorithms for small operands. <br /><br /> We present a side channel attack on XTR cryptosystems. We analyze the statistical behavior of simultaneous XTR double exponentiation algorithm and determine what information to gather to reconstruct the two input exponents. Our analysis and experimental results show that it takes <em>U</em>^{1. 25} tries, where <em>U</em> = max(<em>a</em>,<em>b</em>) on average to find the correct exponent pair (<em>a</em>,<em>b</em>). Using this result, we conclude that an adversary is expected to make <em>U</em>^{0. 625} tries on average until he/she finds the correct secret key used in XTR single exponentiation algorithm, which is based on the simultaneous XTR double exponentiation algorithm.

En 1993, Conway et Schneeberger fournirent un critère simple permettant de déterminer si une forme quadratique donnée représente tous les entiers positifs ; le théorème des 15. Dans ce mémoire, nous nous intéressons à un problème analogue, soit la recherche d’un critère similaire permettant de détecter si une forme quadratique en trois variables représente tous les entiers impairs. On débute donc par une introduction générale à la théorie des formes quadratiques, notamment en deux variables, puis on expose différents points de vue sous lesquels on peut les considérer. On décrit ensuite le théorème des 15 et ses généralisations, en soulignant les techniques utilisées dans la preuve de Bhargava. Enfin, on démontre deux théorèmes qui fournissent des critères permettant de déterminer si une forme quadratique ternaire représente tous les entiers impairs. / In 1993, Conway and Schneeberger gave a simple criterion allowing one to determine whether a given quadratic form represents all positive integers ; the 15-theorem. In this thesis, we investigate an analogous problem, that is the search for a similar criterion allowing one to detect if a quadratic form in three variables represents all odd integers. We start with a general introduction to the theory of quadratic forms, namely in two variables, then, we expose different points of view under which quadratic forms can be considered. We then describe the 15-theorem and its generalizations, with a particular emphasis on the techniques used in Bhargava’s proof of the theorem. Finally, we give a proof of two theorems which provide a criteria to determine whether a ternary quadratic form represents all odd integers. / Les calculs numériques ont été effectués à l'aide du logiciel SAGE.

Made available in DSpace on 2014-12-17T14:36:24Z (GMT). No. of bitstreams: 1 MerciaOP_TESE.pdf: 1133404 bytes, checksum: 81953bfb8e7bacc94a6b29fea8929367 (MD5) Previous issue date: 2010-12-22 / Coordena??o de Aperfei?oamento de Pessoal de N?vel Superior / In Mathematics literature some records highlight the difficulties encountered in the teaching-learning process of integers. In the past, and for a long time, many mathematicians have experienced and overcome such difficulties, which become epistemological obstacles imposed on the students and teachers nowadays. The present work comprises the results of a research conducted in the city of Natal, Brazil, in the first half of 2010, at a state school and at a federal university. It involved a total of 45 students: 20 middle high, 9 high school and 16 university students. The central aim of this study was to identify, on the one hand, which approach used for the justification of the multiplication between integers is better understood by the students and, on the other hand, the elements present in the justifications which contribute to surmount the epistemological obstacles in the processes of teaching and learning of integers. To that end, we tried to detect to which extent the epistemological obstacles faced by the students in the learning of integers get closer to the difficulties experienced by mathematicians throughout human history. Given the nature of our object of study, we have based the theoretical foundation of our research on works related to the daily life of Mathematics teaching, as well as on theorists who analyze the process of knowledge building. We conceived two research tools with the purpose of apprehending the following information about our subjects: school life; the diagnosis on the knowledge of integers and their operations, particularly the multiplication of two negative integers; the understanding of four different justifications, as elaborated by mathematicians, for the rule of signs in multiplication. Regarding the types of approach used to explain the rule of signs arithmetic, geometric, algebraic and axiomatic , we have identified in the fieldwork that, when multiplying two negative numbers, the students could better understand the arithmetic approach. Our findings indicate that the approach of the rule of signs which is considered by the majority of students to be the easiest one can be used to help understand the notion of unification of the number line, an obstacle widely known nowadays in the process of teaching-learning / Na literatura especializada na ?rea de Matem?tica, existem registros que ressaltam as dificuldades enfrentadas no processo de ensino/aprendizagem de n?meros inteiros. Tais dificuldades, vivenciadas e superadas pelos matem?ticos do passado por um longo per?odo, tornam-se obst?culos epistemol?gicos que se imp?em a alunos e professores na atualidade. Este trabalho cont?m os resultados de uma pesquisa desenvolvida na cidade de Natal (RN) no decorrer no primeiro semestre de 2010, em uma escola p?blica estadual de educa??o b?sica e em uma universidade p?blica federal e envolveu 45 alunos assim discriminados: 20 do ensino fundamental, 9 do ensino m?dio e 16 do ensino superior. Teve-se como objetivo central identificar, de um lado, a abordagem da justificativa da multiplica??o entre n?meros inteiros que ? mais bem compreendida pelos alunos e de outro, os elementos presentes nas justificativas que contribuem para a supera??o dos obst?culos epistemol?gicos nos processos de ensino e aprendizagem de n?meros inteiros. Para tanto, procurou-se determinar em que medida os obst?culos epistemol?gicos enfrentados pelos alunos na aprendizagem de n?meros inteiros aproximam-se das dificuldades vivenciadas pelos matem?ticos ao longo da hist?ria da humanidade. Em decorr?ncia da natureza do objeto de pesquisa buscaram-se, no referencial te?rico, os estudos relativos ao cotidiano do ensino de Matem?tica e os te?ricos que se dedicam ao processo de constru??o do conhecimento. Foram elaborados dois instrumentos de pesquisa com a finalidade de apreender as seguintes informa??es sobre os sujeitos pesquisados: vida estudantil; diagn?stico dos conhecimentos de n?meros inteiros e suas opera??es, em especial da multiplica??o de dois n?meros inteiros negativos; compreens?o de quatro justificativas diferentes elaboradas pelos matem?ticos para a regra dos sinais na multiplica??o. No trabalho de campo identificou-se, dentre as abordagens aritm?tica, geom?trica, alg?brica e axiom?tica dadas ao produto de dois n?meros negativos, que os alunos compreendiam melhor a que usava argumentos aritm?ticos. Os resultados obtidos indicam que a justificativa para a regra de sinais que ? considerada de mais f?cil compreens?o pela maioria dos alunos dos ensinos fundamental, m?dio e superior pode ser usada para facilitar a compreens?o da unifica??o da reta num?rica, um obst?culo amplamente identificado no processo de ensino/aprendizagem na atualidade

Cette thèse est centrée autour de la monogénéité de corps de nombres en situation relative puis à la conjecture de Collatz.\newline Premièrement on détermine l'ensemble de classes des générateurs de l'anneau des entiers des certaines extensions relatives de corps de nombres, en utilisant l'algorithme de Gaál & Phost et le logiciel PARI/GP. La deuxième partie propose différents formulations d'une généralisation de la conjecture de Collatz, aux entiers p-adiques. On étudie ensuite le comportement de suites analogues dans le cadre d'anneaux d'entiers de corps de nombres. / This thesis are centered around the monogeneity of number fields in a relative situation and the Collatz conjecture. Firstly, we determine the set of generator classes of the ring of integers of some relative extensions of number fields, using the Gaál& Phost algorithm and the PARI/GP software. The second part proposes different formulations of a generalization of the Collatz conjecture to p-adic integers. We then study the behavior of similar sequences in the framework of rings of integers of number fields.