published_or_final_version / Physics / Doctoral / Doctor of Philosophy

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Diss. (sammanfattning) Uppsala : Univ., 2003. / Härtill 7 uppsatser.

Thesis (Ph. D.)--Missouri University of Science and Technology, 2009. / Vita. The entire thesis text is included in file. Title from title screen of thesis/dissertation PDF file (viewed May 4, 2009) Includes bibliographical references (p. 103-113).

Thesis (Ph. D.)--University of Wisconsin, 1966. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

Made available in DSpace on 2014-06-11T19:26:55Z (GMT). No. of bitstreams: 0 Previous issue date: 2010-02-26Bitstream added on 2014-06-13T19:26:01Z : No. of bitstreams: 1 benedito_cwo_me_sjrp.pdf: 1004485 bytes, checksum: fd9cc4cec014a6fbfc619f640e7f98b5 (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) / Neste trabalho é feito um estudo sobre famílias de reticulados algébricos e reticulados ideais. Nosso principal objetivo é a construção de reticulados que são versões rotacioanadas de reticulados já conhecidos na literatura. Deste modo, apresentamos construções obtidas via polinômios, via perturbações do homomorfismo canônico e, também, construções ciclotômicas a partir fo reticulado Zn. / This work presents a study of algebraic and families of ideal lattices. Our main goal is the construction of lattices which are rotated versions of known lattices in the literature. In this way, we present constructions obtained via polynomials, via pertubations of the canonical homomorphism, and also cyclotomic construction from the lattice Zn.

This thesis aims to describe in detail the Coppersmith's algorithm for fin- ding small solutions to polynomial congruences which is based on lattice basis reduction. This algorithm is a cornerstone of several attacks on the most wi- despread asymmetric cryptosystem RSA, therefore, next aim of the thesis is a description of selected attacks. As an important and current example, we can mention so-called ROCA attack which factorizes RSA modulus whenever the pri- mes are specifically crafted. At the end of the thesis, we implement both the Coppersmith's algorithm and the ROCA and several measurements and experi- ments are done. From the resulting data, one can deduce how the running time of the algorithm is affected by different parameters or what are the ideal values for these parameters in various situations. 1

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Ce manuscrit présente les travaux que j'ai effectués au Laboratoire de Physique Théorique durant ma thèse. Ils portent sur l'interaction d'ondes de matière avec des réseaux optiques modulables en temps et en espace. L'utilisation de ces réseaux a permis de contrôler de manière cohérente les propriétés dynamiques d'un gaz d 'atomes ultra-froids. Cette étude théorique a été réalisée en collaboration avec le groupe Atomes Froids du Laboratoire LCAR. Les variations spatiales de l'enveloppe d u réseau créent, localement, des gaps spatiaux créant une cavité de Bragg pour onde de matière, dont nous avons étudié en détail les propriétés et qui a fait l'objet d'une réalisation expérimentale impliquant la propagation d'un condensat de Bose-Einstein de rubidium 85 dans un guide d'onde. Nous avons également étudié la propagation d'un nuage d 'atomes dans un réseau bichromatique qui permet de réaliser un simulateur quantique du modèle de Harper. Le spectre du hamiltonien de ce système a une dimension fractale pouvant être caractérisée nu­ mériquement. Nous avons montré, par ailleurs, qu'il est possible d'exploiter les interactions inter-atomiques répulsives d'un condensat de Bose-Einstein afin d'amplifier les corrélations position-vitesse lors de sa pro­ pagation dans un guide. Notre étude montre qu'une mesure des grandeurs dynamiques locales du nuage atomique permet de sonder expérimentalement les résonances d'un potentiel optique jusqu'à l'échelle du picoKelvin. Enfin, un nuage d'atomes en interaction attractive admet une solution d'équilibre : le soliton. Nous avons démontré, numériquement, que celui-ci peut être utilisé pour sonder des états liés d'un poten­ tiel de taille finie, en peuplant ces états lors d'une expérience de diffusion comme, par exemple, des états de surface. / This thesis presents the studies that I did at the Laboratoire de Physique Théorique. It concerns the interaction between matter waves and time and space depandant optical lattices. Using such lattices allows one to manipulate coherently the dynamical properties of ultra cold atoms. This theoretical study has been done in collaboration with the Cold Atoms group at the LCAR laboratory. The spatial variations of the lattice envelope locally create spatial gaps which create a Bragg cavity for matter waves. We have st udied in detail their properties and the cavity has been realized experimentally by using a Ru bid ium 85 Bose-Einstein condensate in a wave guide. We have also studied the propagation of an atomic cloud in a bichromatic optical lattice which allows us to make a quantum simulator of the Harper madel. The spectrum of the system Hamiltonian· posseses a fractal dimension which can be numerically characterized. We have also shawn that it is possible to use the repulsive interatomic interaction of a Bose-Einstein condensate in arder to amplify the momentum-position correlation during propagation in a guide. Our st udy shows that a mesure of local dynamical quantities of the atomic cloud enables one to experimentally probe resonances of an optical potential down to the picoKelvin scale. At last, an atomic cloud with attractive interactions admit a stable solution, the soliton. We have numerically demonstrated that this soliton can be used to probe bound states of a potential by populating those states through a scattering experiment, for example surface states.

This paper examines the theory of an adelization of Shintani's zeta function, especially as it relates to density theorems for discriminants of cubic extensions of number fields. / Mathematics