In this thesis, we extend the construction of the formal (affine) Demazure algebra due to Hoffnung, Malagón-López, Savage and Zainoulline in two directions. First, we introduce and study the notion of formal Demazure lattices of a Kac-Moody root system and show that the definitions and properties of the formal (affine) Demazure operators and algebras hold for such lattices. Second, we show that for the hyperbolic formal group law the formal Demazure algebra is isomorphic (after extending the coefficients) to the Hecke algebra.

Elliptic curve cryptography has gained much popularity in the past decade and has been challenging the dominant RSA/DSA systems today. This is mainly due to elliptic curves offer cryptographic systems with higher speed, less memory and smaller key sizes than older ones. Among the various arithmetic operations required in implementing public key cryptographic algorithms based on elliptic curves, the elliptic curve scalar multiplication has probably received the maximum attention from the research community in the past a few years. Many methods for efficient and secure implementation of scalar multiplication have been proposed by many researchers. In this thesis, many scalar multiplication methods are studied in terms of their mathematical, computational and implementational points.

In this thesis, we compute minimal presentations in terms of generators and relations for the oriented cohomology rings of several semisimple linear algebraic groups of ranks 1 and 2 over algebraically closed fields of characteristic 0. The main tools we use in this thesis are the combinatorics of Coxeter groups and formal group laws, and recent results of Calm\`es, Gille, Petrov, Zainoulline, and Zhong, which relate the oriented cohomology rings of flag varieties and semisimple linear algebraic groups to the dual of the formal affine Demazure algebra.

La prise en compte du phénomène des groupes de sociétés par le droit est, depuis longtemps, un sujet qui divise la doctrine et qui ne trouve pas de réponse uniforme au niveau des diverses législations. L'idée d'instituer un droit des groupes, initié dans l'Allemagne des années 1960, continue à faire débat, aujourd'hui même, dans plusieurs pays. En Tunisie, précisément, la question a été tranchée sans grands débats et de manière circonstancielle par l'adoption d'une réglementation d'ensemble. La présente thèse a pour objet d'analyser la manière dont le Droit tunisien a appréhendé et organisé le phénomène des groupes de sociétés avec pour objectif principal d'assurer la protection des intérêts en présence. A cet effet, l'analyse a porté sur l'émergence de la notion en droit tunisien et le régime spécifique des groupes. Menée à la lueur du droit français, elle a permis de donner à l'étude un éclairage particulièrement utile. / The legal framework of the groups companies has long been a subject which divides the doctrine keeping in mind that is not resolved in a uniform manner by legislators. The idea of establishing a special group law was initiated in Gemrany in the 1960s, and continues .to be a subject to debate even today in several countries. ln Tunisia, specifically, the issue was resolved without major debates and in a circumstantial manner through the introduction of an appropriate regulatory framework. In this context, his thesis aims to analyze the ways in which the Tunisian lav,I was able to organize the phenomenon of groups of companies and protect the stakeholder's interests. This research was backed up by the analysis of the emergence of the concept of groups in Tunisian law, as well as its legal framework in the light of French law to allow deepen knowledge an understanding of it.

En 1947, après l’obtention de l’indépendance, l’Inde est devenue une république séculière et démocratique proposant ainsi une nouvelle organisation de la société. Sans faire l’unanimité, des lois familiales différenciées selon l’identité religieuse ainsi que des droits socioéconomiques associés à l’appartenance à un groupe ont été reconnus par l’État. Dans le climat politique des années 90, le secularism et les droits de groupe ont été le sujet de débats. À partir d’une considération du contexte sociohistorique, cette recherche porte sur les manières de se représenter la laïcité indienne et sur ses rapports potentiels avec des lois différenciées selon l’appartenance religieuse. À travers la notion d’égalité, cette recherche explore les droits et les devoirs associés à la juste approche de la diversité religieuse en Inde indépendante. Une attention particulière est accordée à la période contemporaine et aux droits des Indiens musulmans. Dans cette recherche, les représentations juridiques qui se dégagent de l’analyse des débats publics et intellectuels sont mises en parallèle avec les points de vue de dix-sept répondants de la classe moyenne de Kolkata (été 2011). À travers cette démarche, cette analyse du discours informe sur les conceptions du secularism débattues en Inde indépendante et dans la période contemporaine. Parallèlement à un accent mis sur l’amour de la diversité, les droits individuels, les devoirs et l’auto régulation, les droits différenciés pour les musulmans sont, pour la majorité des répondants, rejetés. Deux approches de l’État sont soulevées dans les définitions du secularism, une version plus dirigiste et l’autre laissant plus de souveraineté aux groupes. / In 1947, India became a secular democratic republic proposing a new organization of society. Family laws, established according to religious affiliation and affirmative action policies, were recognized by the state without unanimity. In the political climate of the nineties, such secularism and group rights were questioned. From a socio-historical perspective, this research focuses on the practices & representations of Indian secularism and its potential relationship with laws differentiated by religious affiliation. Based on the notion of equality, the present research explores the rights and obligations associated with the just approach to religious diversity in independent India. Particular attention is given to the contemporary period and to Indian Muslims’ group rights. In this research, legal representations that emerge from the analysis of public and intellectual debates are paralleled with the viewpoints of seventeen of Kolkata’s middle-class informants (summer 2011). Using this approach, the discourse analysis informs the reader on the conception of secularism discussed in contemporary, independent India. Recognising the value of diversity, duties and self regulation, the majority of respondents reject differentiated rights for Muslims. Two approaches of the state emerge; one in which the State is more directive and one which provides more autonomy to the group.

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Let C be the class of finite nilpotent, solvable, symmetric, simple or semi-simple groups and n be a positive integer. We discuss the following question on group laws: What is the length of the shortest non-trivial law holding for all finite groups from the class C of order less than or equal to n?:Introduction 0 Essentials from group theory 1 The two main tools 1.1 The commutator lemma 1.2 The extension lemma 2 Nilpotent and solvable groups 2.1 Definitions and basic properties 2.2 Short non-trivial words in the derived series of F_2 2.3 Short non-trivial words in the lower central series of F_2 2.4 Laws for finite nilpotent groups 2.5 Laws for finite solvable groups 3 Semi-simple groups 3.1 Definitions and basic facts 3.2 Laws for the symmetric group S_n 3.3 Laws for simple groups 3.4 Laws for finite linear groups 3.5 Returning to semi-simple groups 4 The final conclusion Index Bibliography / Sei C die Klasse der endlichen nilpotenten, auflösbaren, symmetrischen oder halbeinfachen Gruppen und n eine positive ganze Zahl. We diskutieren die folgende Frage über Gruppengesetze: Was ist die Länge des kürzesten nicht-trivialen Gesetzes, das für alle endlichen Gruppen der Klasse C gilt, welche die Ordnung höchstens n haben?:Introduction 0 Essentials from group theory 1 The two main tools 1.1 The commutator lemma 1.2 The extension lemma 2 Nilpotent and solvable groups 2.1 Definitions and basic properties 2.2 Short non-trivial words in the derived series of F_2 2.3 Short non-trivial words in the lower central series of F_2 2.4 Laws for finite nilpotent groups 2.5 Laws for finite solvable groups 3 Semi-simple groups 3.1 Definitions and basic facts 3.2 Laws for the symmetric group S_n 3.3 Laws for simple groups 3.4 Laws for finite linear groups 3.5 Returning to semi-simple groups 4 The final conclusion Index Bibliography