Soient $k$ un corps de caractéristique nulle et $G$ un $k$-groupe algébrique linéaire. Il est bien connu que si $G$ est abélien, les torseurs sous $G_(X)$ sur un $k$-schéma $\pi:X\rightarrow \textup(Spec)\;k$ fournissent une obstruction à l'existence de points $k$-rationnels sur $X$, puisque la suite spectrale de Leray donne dans les bons cas (\textit(e.g.) $X$ propre) une suite exacte de groupes sur laquelle on peut directement lire l'obstruction à ce qu'un $\bar(G)_(X)$-torseur $\bar(P)\rightarrow\bar(X)$ de corps des modules $k$ soit défini sur $k$, \textit(i.e.) qu'il provienne par extension des scalaires à la cl(ô)ture algébrique $\bar(k)$ de $k$ d'un $G_(X)$-torseur $P\rightarrow X$. Le point crucial est que cette obstruction est mesurée par une gerbe, qui est neutre lorsque $X$ possède un point $k$-rationnel. On essaye ici d'étendre ce résultat au cas non-commutatif, et on en déduit (sous certaines conditions) des obstructions cohomologiques non-abéliennes à l'existence de points $k$-rationnels sur $X$, et des résultats sur la descente des torseurs.

Presburger arithmetic is the first-order theory of the integers with addition and ordering, but without multiplication. This theory is decidable and the sets it defines admit several different representations, including formulas, generators, and finite automata, the latter being the focus of this thesis. Finite-automata representations of Presburger sets work by encoding numbers as words and sets by automata-defined languages. With this representation, set operations are easily computable as automata operations, and minimized deterministic automata are a canonical representation of Presburger sets. However, automata-based representations are somewhat opaque and do not allow all operations to be performed efficiently. An ideal situation would be to be able to move easily between formula-based and automata-based representations but, while building an automaton from a formula is a well understood process, moving the other way is a much more difcult problem that has only attracted attention fairly recently. The main results of this thesis are new algorithms for extracting information about Presburger-definable sets represented by finite automata. More precisely, we present algorithms that take as input a finite-automaton representing a Presburger definable set S and compute in polynomial time the affine hull over Q or over Z of the set S, i.e., the smallest set defined by a conjunction of linear equations (and congruence relations in Z) which includes S. Also, we present an algorithm that takes as input a deterministic finite-automaton representing the integer elements of a polyhedron P and computes a quantifier-free formula corresponding to this set. The algorithms rely on a very detailed analysis of the scheme used for encoding integer vectors and this analysis sheds light on some structural properties of finite-automata representing Presburger definable sets. The algorithms presented have been implemented and the results are encouraging : automata with more than 100000 states are handled in seconds.

Cette thèse aborde différents aspects de la cryptologie, subsumant des champs aussi variés que la conception de protocoles, l’amélioration d’outils algorithmiques et les attaques. Les deux principales focales de cette étude sont : un protocole de co-signature prouvé irréfragable et un système de chiffrement authentifié à sécurité prouvée. Notre protocole de co-signature permet l’équité légale. L’équité légale est une nouvelle variante de la notion d’équité, ne reposant pas sur des tiers. Notre instanciation d’équité légale est construite à l’aide des signatures de Schnorr. Nous présenterons également un protocole d’authentification distribué de type Fiat-Shamir. La deuxième partie de cette thèse est consacrée aux améliorations algorithmiques. Nous introduisons une méthode permettant de doubler la vitesse de l’algorithme de Barrett en utilisant des modules composites spécifiques et un nouvel algorithme de multiplication à retour sur trace, particulièrement adapté aux microprocesseurs bon marché. Nous nous intéresserons ensuite à la sécurité des composants en étudiant la régulation du débit des correcteurs de von Neumann et les attaques en fautes sur des implémentations de cryptographie à courbes elliptiques. Enfin, un des actes novatoires incidents notre travail sera d’adapter aux codes correcteurs d’erreurs deux techniques empruntées à la cryptographie : un premier résultat améliore l’efficacité calculatoire des codes BCH grâce à une version de l’algorithme de Barrett étendue aux polynômes. Le second est un nouveau code correcteur d’erreurs basé sur la théorie des nombres. / This thesis addresses various topics in cryptology, namely protocol design, algorithmic improvements and attacks. In addition, we venture out of cryptography and propose two new applications of cryptographic techniques to error correcting codes. Our main results comprise a provably secure co-signature protocol and a provably secure authenticated encryption scheme. Our co-signature protocol achieves legal fairness, a novel fairness variant that does not rely on third parties. Legal fairness is implemented using Schnorr signatures. We also present a distributed Fiat-Shamir authentication protocol. The second part of the thesis is devoted to computational improvements, we discuss a method for doubling the speed of Barrett’s algorithm by using specific composite moduli, devise new BCH speed-up strategies using polynomial extensions of Barrett’s algorithm, describe a new backtracking-based multiplication algorithm suited for lightweight microprocessors and present a new number theoretic error-correcting code. Fault injection attacks are further overviewed and a new fault attack on ECC implementations is proposed.

This thesis studies the properties of finite automata recognizing sets of real vectors encoded in positional notation using an integer base. We consider both general infinite-word automata, and the restricted class of weak deterministic automata, used, in particular, as symbolic data structures for representing the sets of vectors definable in the first order additive theory of real and integer numbers. <br><br> In previous work, it has been established that all sets definable in the additive theory of reals and integers can be handled by weak deterministic automata regardless of the chosen numeration base. In this thesis, we address the reciprocal property, proving that the sets of vectors that are simultaneously recognizable in all bases, by either weak deterministic or Muller automata, are those definable in the additive theory of reals and integers. <br><br> Precisely, for weak deterministic automata, we establish that the sets of real vectors simultaneously recognizable in two multiplicatively independent bases are necessarily definable in the additive theory of reals and integers. For general automata, we show that the multiplicative independence is not sufficient, and we prove that, in this context, the sets of real vectors that are recognizable in two bases that do not share the same set of prime factors are exactly those definable in the additive theory of reals and integers. <br><br> Those results lead to a precise characterization of the sets of real vectors that are recognizable in multiple bases, and provide a theoretical justification to the use of weak automata as symbolic representations of sets. <br><br> As additional contribution, we also obtain valuable insight into the internal structure of automata recognizing sets of vectors definable in the additive theory of reals and integers.