In this digital age, cryptography is largely built in computer hardware or software as discrete structures. One of the most useful of these structures is finite fields. In this thesis, we explore a variety of applications of the theory and applications of arithmetic and computation in finite fields in both the areas of cryptography and cryptanalysis. First, multiplication algorithms in finite extensions of prime fields are explored. A new algebraic description of implementing the subquadratic Karatsuba algorithm and its variants for extension field multiplication are presented. The use of cy- clotomic fields and Gauss periods in constructing suitable extensions of virtually all sizes for efficient arithmetic are described. These multiplication techniques are then applied on some previously proposed public key cryptosystem based on exten- sion fields. These include the trace-based cryptosystems such as XTR, and torus- based cryptosystems such as CEILIDH. Improvements to the cost of arithmetic were achieved in some constructions due to the capability of thorough optimisation using the algebraic description. Then, for symmetric key systems, the focus is on algebraic analysis and attacks of stream ciphers. Different techniques of computing solutions to an arbitrary system of boolean equations were considered, and a method of analysing and simplifying the system using truth tables and graph theory have been investigated. Algebraic analyses were performed on stream ciphers based on linear feedback shift registers where clock control mechanisms are employed, a category of ciphers that have not been previously analysed before using this method. The results are successful algebraic attacks on various clock-controlled generators and cascade generators, and a full algebraic analyses for the eSTREAM cipher candidate Pomaranch. Some weaknesses in the filter functions used in Pomaranch have also been found. Finally, some non-traditional algebraic analysis of stream ciphers are presented. An algebraic analysis on the word-based RC4 family of stream ciphers is performed by constructing algebraic expressions for each of the operations involved, and it is concluded that each of these operations are significant in contributing to the overall security of the system. As far as we know, this is the first algebraic analysis on a stream cipher that is not based on linear feedback shift registers. The possibility of using binary extension fields and quotient rings for algebraic analysis of stream ciphers based on linear feedback shift registers are then investigated. Feasible algebraic attacks for generators with nonlinear filters are obtained and algebraic analyses for more complicated generators with multiple registers are presented. This new form of algebraic analysis may prove useful and thereby complement the traditional algebraic attacks. This thesis concludes with some future directions that can be taken and some open questions. Arithmetic and computation in finite fields will certainly be an important area for ongoing research as we are confronted with new developments in theory and exponentially growing computer power.

In the thesis we investigate two problems on Partial Differential Equations (PDEs) in differential geometry and fluid mechanics. First, we prove the weak L^p continuity of the Gauss-Codazzi-Ricci (GCR) equations, which serve as a compatibility condition for the isometric immersions of Riemannian and semi-Riemannian manifolds. Our arguments, based on the generalised compensated compactness theorems established via functional and micro-local analytic methods, are intrinsic and global. Second, we prove the vanishing viscosity limit of an incompressible fluid in three-dimensional smooth, curved domains, with the kinematic and Navier boundary conditions. It is shown that the strong solution of the Navier-Stokes equation in H^r+1 (r > 5/2) converges to the strong solution of the Euler equation with the kinematic boundary condition in H^r, as the viscosity tends to zero. For the proof, we derive energy estimates using the special geometric structure of the Navier boundary conditions; in particular, the second fundamental form of the fluid boundary and the vorticity thereon play a crucial role. In these projects we emphasise the linkages between the techniques in differential geometry and mathematical hydrodynamics.

Made available in DSpace on 2014-06-11T19:31:42Z (GMT). No. of bitstreams: 0 Previous issue date: 2012-04-26Bitstream added on 2014-06-13T18:42:44Z : No. of bitstreams: 1 noelfilho_a_dr_rcla.pdf: 5554552 bytes, checksum: d7ac8c9e804b77fa358a56a9655da58b (MD5) / Este trabalho é resultado de uma pesquisa que vislumbra encontrar relações entre a Cartografia e a Geometria Diferencial. Toma como ponto de partida os problemas adjacentes à Projeção de Mercator e explicita sua influencia na história do Cálculo e da Geometria Diferencial nas análises das obras de Pedro Nunes, Edward Wright e Gauss. A falta do trabalho original impediu a análise do verdadeiro método usado por Mercator na construção de sua projeção. Nos tratados, Sobre Certas Dúvidas da Navegação e em Defensam da Carta de Marear, são encontrados vestígios da contribuição da obra de Pedro Nunes na construção da Projeção de Mercator e em Certaine Errors in Navegation, Edward Wright apresenta uma justificativa matemática para o problema. O estudo da obra General Investigations of Curved Surfaces revela que o tratamento cartográfico dado aos resultados obtidos por Gauss no levantamento geodésico da cidade de Hannover serviu como base para muitos dos seus trabalhos. Os conhecimentos de Cartografia e de Astronomia adquiridos na experiência de campo, podem ter levado Gauss à formalização da teoria geral das superfícies curvas e com esta foi possível traduzir a lei da projeção de Mercator em linguagem moderna / This work is a result of research that envisions finding relations between Cartography and Differential Geometry. It takes as its starting point the problems surrounding the Mercator Projection and explains their influence in the history of calculus and differential geometry in the analysis of works of Pedro Nunes, Edward Wright and Gauss. The lack of labor prevented the original analysis of the true method used by Mercator in the construction of its projection. In the treaties, on Certain Questions of Navigation and the Letter of Defensam Marear traces of the contribution of the work of Pedro Nunes are found in the construction of the Mercator Projection and Certaine Errors in Navegation, Edward Wright presents a mathematical justification for the problem. The study of the book General Investigations of Curved Surfaces reveals that the treatment given to the mapping results obtained by the Gauss geodesic survey of Hannover city was the basis for many of his works. The knowledge of Cartography and Astronomy acquired in the field experience, may have taken Gauss to the formalization of the general theory of the surfaces curves and with this it was possible to translate the law of the projection of Mercator in modern language

Made available in DSpace on 2014-10-09T12:46:58Z (GMT). No. of bitstreams: 0 / Made available in DSpace on 2014-10-09T14:10:10Z (GMT). No. of bitstreams: 1 08315.pdf: 2209007 bytes, checksum: da772613fbbd4e5110a0d2f25639c69c (MD5) / Dissertacao (Mestrado) / IPEN/D / Instituto de Pesquisas Energeticas e Nucleares - IPEN/CNEN-SP

Neste trabalho foi realizado um grande levantamento de técnicas de esparsidade relacionadas a sistemas estáticos de energia elétrica. Tais técnicas visam, do ponto de vista computacional, ao aumento da eficiência na solução de rede elétrica objetivando, além da resolução em si, a redução dos requisitos de memória, armazenamento e tempo de processamento. Para tanto, uma extensa revisão bibliográfica foi compilada, apresentando um posicionamento histórico e uma ampla visão do desenvolvimento teórico. Os testes comparativos realizados para sistemas de 14, 30, 57 e 118 barras, sobre a implantação de três das técnicas mais empregadas, apontou a Bi-fatoração como tendo o melhor desempenho médio. Para sistemas pequenos, a Eliminação Esparsa e Sintética de Gauss apresentou melhores resultados. Este trabalho fornecerá subsídios conceituais e metodológicos a técnicos e pesquisadores da área. / In this work a great survey of sparsity techniques related to static systems of electric power was accomplished. Such techniques seek, for of the computational point of view, the increase of the efficiency in the solution of the electric net aiming, besides the resolution of itself, the reduction of memory requirements, the storage and time processing. For that, an extensive bibliographic review was compiled providing a historic positioning and a broad view of theoretic development. The comparative tests accomplished for systems of 14,30, 57 and 118 buses, on the implementation of three of the most employed techniques, it pointed out an bi-factorisation as best medium performance. For small systems, the sparse symmetric Gaussian elimination showed the best results. This work will supply conceptual and methodological subsidies to technicians and researchers of the area.

Made available in DSpace on 2014-10-09T12:46:58Z (GMT). No. of bitstreams: 0 / Made available in DSpace on 2014-10-09T14:10:10Z (GMT). No. of bitstreams: 1 08315.pdf: 2209007 bytes, checksum: da772613fbbd4e5110a0d2f25639c69c (MD5) / Dissertacao (Mestrado) / IPEN/D / Instituto de Pesquisas Energeticas e Nucleares - IPEN/CNEN-SP

Submitted by Luciana Ferreira (lucgeral@gmail.com) on 2015-05-07T13:34:15Z No. of bitstreams: 2 Dissertação - Carmencita Ferreira Silva Assis - 2014.pdf: 1032992 bytes, checksum: dcfbc22b53a2352c6e65a7615ffb72b5 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2015-05-07T13:40:12Z (GMT) No. of bitstreams: 2 Dissertação - Carmencita Ferreira Silva Assis - 2014.pdf: 1032992 bytes, checksum: dcfbc22b53a2352c6e65a7615ffb72b5 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Made available in DSpace on 2015-05-07T13:40:12Z (GMT). No. of bitstreams: 2 Dissertação - Carmencita Ferreira Silva Assis - 2014.pdf: 1032992 bytes, checksum: dcfbc22b53a2352c6e65a7615ffb72b5 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Previous issue date: 2014-03-20 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This work aims to present te hniques for solving systems of linear equations, in its traditional formulation, where it sought to explore the referen es ommonly used in ourses in linear algebra and numeri al omputation, fo using on the dire t methods of Gauss elimination and LU fa torization. Troubleshooters established in the literature are ondu ted, in order to illustrate the operation and appli ation of su h methods to real problems, thus highlighting the possibility of inserting them in high s hool. The ontents were treated and exposed so that exemplify the diversity of areas in luding linear systems, su h as engineering, e onomi s and biology, showing the gains that an be a hieved by students if they have onta t with the methods as soon as possible. At the end we suggest the use of omputational resour es in math lasses, sin e the redu tion of time spent in algebrai manipulation will allow the tea her to deepen the on epts and to address larger systems, to enhan e the resolution perspe tive, and motivate the student in the learning pro ess. / Este trabalho tem por objetivo apresentar té ni as de resolução de sistemas de equações lineares, em sua formulação tradi ional, onde se bus ou explorar as referên ias usualmente utilizadas em ursos de álgebra linear e ál ulo numéri o, enfo ando os métodos diretos de Eliminação de Gauss e Fatoração LU. Resoluções de problemas onsolidados na literatura são realizadas, om a nalidade de ilustrar o fun ionamento e apli ação de tais métodos em problemas reais, desta ando assim a possibilidade de inserção dos mesmos no Ensino Médio. Os onteúdos foram tratados e expostos de modo que exempli quem a diversidade de áreas que abrangem os sistemas lineares, tais omo engenharia, e onomia e biologia, mostrando os ganhos que podem ser al ançados pelos alunos, se tiverem ontato om os métodos o quanto antes. Ao nal sugere- se a utilização de re ursos omputa ionais nas aulas de matemáti a, uma vez que a redução do tempo empregado na manipulação algébri a permitirá que o professor possa aprofundar os on eitos e abordar sistemas de maior porte, que ampliem a perspe tiva de resolução, além de motivar o aluno no pro esso de aprendizagem.

Submitted by Franciele Moreira (francielemoreyra@gmail.com) on 2017-12-27T13:50:44Z No. of bitstreams: 2 Tese - Wesley da Silva Ruys - 2017.pdf: 2660976 bytes, checksum: 3c6402ac0974e65da560b50e1f65b52e (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2017-12-28T09:42:07Z (GMT) No. of bitstreams: 2 Tese - Wesley da Silva Ruys - 2017.pdf: 2660976 bytes, checksum: 3c6402ac0974e65da560b50e1f65b52e (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2017-12-28T09:42:07Z (GMT). No. of bitstreams: 2 Tese - Wesley da Silva Ruys - 2017.pdf: 2660976 bytes, checksum: 3c6402ac0974e65da560b50e1f65b52e (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2017-12-07 / Fundação de Amparo à Pesquisa do Estado de Goiás - FAPEG / In this work we present a classification of the Laguerre minimal surfaces with flat curvature lines. We introduce three classes of hypersurfaces that generalize the Laguerre minimal surfaces with the prescribed Gaussian normal application. The first class is associated to biharmonic applications and is related by a Legendre transformation to hypersurfaces that in the isotropic model has harmonic isotropic mean curvature. As an application, we classify the hypersurfaces of rotation and we present examples of these hypersurfaces parameterized by flat curvature lines. We obtain a characterization of the other two classes of hypersurfaces, we study the rotation ones and we present examples. / Neste trabalho apresentamos uma classificação das superfícies mínimas de Laguerre com linhas de curvatura planas. Introduzimos três classes de hipersuperfícies que generalizam as superfícies mínimas de Laguerre com aplicação normal de Gauss prescrita. A primeira classe está associada a aplicações biharmônicas e está relacionada por uma transformação de Legendre a hipersuperfícies que no modelo isotrópico tem curvatura média isotrópica harmônica. Como aplicação, classificamos as hipersuperfícies de rotação e apresentamos exemplos destas hipersuperfícies parametrizadas por linhas de curvatura planas. Obtemos uma caracterização das outras duas classes de hipersuperfícies, estudamos as de rotação e apresentamos exemplos.

La médecine moderne fait usage de divers appareils pour faciliter le diagnostic et le traitement des maladies. Parmi ceux-ci, l’imagerie par tomographie d’émission par positrons (TEP) se démarque par sa capacité d’imager des processus biologiques spécifiques comme le métabolisme du glucose. Cette caractéristique permet de mettre en évidence des signes distinctifs des maladies comme le cancer à l’aide de radiotraceurs capables de cibler certaines cellules. Dans le but de favoriser de meilleurs diagnostics et de mener des recherches de pointe dans le domaine médical, la qualité des images produites par les appareils TEP doit être améliorée. Les avancées des dernières années ont permis d’améliorer la résolution spatiale des images jusqu’à pratiquement atteindre la limite théorique imposée par le déplacement du positron lors du processus de désintégration. Depuis, les travaux s’orientent plutôt vers l’amélioration du contraste de l’image en intégrant la mesure du temps de vol (TdV) dans l’algorithme de reconstruction. Le défi dans cette mesure réside dans la conception d’un photodétecteur avec une résolution temporelle suffisante pour localiser le lieu d’émission du radiotraceur entre deux détecteurs coïncidents. La plupart des photodétecteurs actuels utilisent un seuil sur le nombre de photons de scintillation observé pour déterminer le temps d’arrivée des photons d’annihilation dans le scintillateur. Cependant, plusieurs travaux ont démontré qu’une meilleure résolution temporelle est atteignable en pondérant adéquatement l’information temporelle numérisée de plusieurs photons de scintillation à la place de n’en considérer qu’un seul. Dans le but d’améliorer la résolution temporelle des photodétecteurs, l’utilisation d’un estimateur statistique combinant l’information de plusieurs photons de scintillation se révèle une méthode prometteuse en considérant les résultats théoriques. Cependant, une implémentation matérielle pouvant être intégrée à un appareil TEP reste à être démontrée. Les travaux de recherche présentés dans ce mémoire portent sur l’intégration d’un tel estimateur statistique à un photodétecteur pour la TEP. Ces travaux ont mené au développement d’une chaine d’acquisition qui comporte 1) un circuit de lecture, 2) une trieuse, 3) un filtre de bruit thermique et 4) un estimateur statistique du temps d’interaction basé sur le Best Linear Unbiased Estimator (BLUE). L’intégration de cette chaine à même le circuit intégré du photodétecteur de 1 x 1 mm2 en CMOS 65 nm permet de réduire la bande passante de 250 Mbit/s à 0,5 Mbit/s et le temps mort de 68,6 μs à 1024 ns. Des simulations démontrent l’atteinte d’une résolution temporelle qui s’approche de la limite inférieure théorique (appelée borne de Cramér-Rao) quant à la résolution temporelle atteignable par ce photodétecteur.

This thesis examines how to find the best fit to a series of data points when curve fitting using power-exponential models. We describe the different numerical methods such as the Gauss-Newton and Levenberg-Marquardt methods to compare them for solving non-linear least squares of curve fitting using different power-exponential functions. In addition, we show the results of numerical experiments that illustrate the effectiveness of this approach.Furthermore, we show its application to the practical problems by using different sets of data such as death rates and rocket-triggered lightning return strokes based on the transmission line model.