Calculation of extended gcd by normalization

Travail indépendant. Ce document est en cours de lecture pour le journal "International Journal of Mathematics". Il a été déposé sur le site de "Science Research Association" (SCIREA).Public Science Framework http://www.aiscience.orgLa référence du document est [70400106].International audienceWe propose a new algorithm solving the extended gcd problem, which provides a solution minimizing one of the two coordinates. The algorithm relies on elementary arithmetic properties

Right-angled Artin groups and the cohomology basis graph

Let $\Gamma$ be a finite graph and let $A(\Gamma)$ be the corresponding right-angled Artin group. From an arbitrary basis $\mathcal B$ of $H^1(A(\Gamma),\mathbb F)$ over an arbitrary field, we construct a natural graph $\Gamma_{\mathcal B}$ from the cup product, called the \emph{cohomology basis graph}. We show that $\Gamma_{\mathcal B}$ always contains $\Gamma$ as a subgraph. This provides an effective way to reconstruct the defining graph $\Gamma$ from the cohomology of $A(\Gamma)$, to characterize the planarity of the defining graph from the algebra of $A(\Gamma)$, and to recover many other natural graph-theoretic invariants. We also investigate the behavior of the cohomology basis graph under passage to elementary subminors, and show that it is not well-behaved under edge contraction.Comment: 17 page

Profils de séparation et de Poincaré

The goal of this thesis report is to present my research concerning separation and Poincaré profiles. Separation profile first appeared in 2012 in a seminal article written by Benjamini, Schramm and Timár. This definition was based on preceding research, in the field of computer science, mainly work of Lipton and Trajan concerning planar graphs, and of Miller, Teng, Thurston and Vavasis concerning overlap graphs. The separation profile plays now a role in geometric group theory, where my personal interests lies, because of its property of monotonicity under coarse embeddings. It was generalized by Hume, Mackay and Tessera in 2019 to a spectrum of profiles, called the Poincaré profiles.Ce manuscrit de thèse récapitule mes travaux de recherche sur les profils de séparation et de Poincaré. Le profil de séparation est apparu en 2012 dans un l'article fondateur de Benjamini, Schramm et Timár. La définition donnée tirait ses origines dans des travaux antérieurs, dans le domaine du calcul formel : principalement des études de Lipton et Trajan concernant les graphes planaires, et de Miller, Teng, Thurston et Vavasis concernant des graphes d'intersection. Le profil de séparation est maintenant utilisé en théorie géométrique des groupes, mon domaine de recherche, à cause de sa propriété de monotonie par plongements grossiers. Il a été généralisé par Hume, Mackay et Tessera en 2019 en une gamme continue de profils, appelés profils de Poincaré

Post-quantum hash functions using SLn(Fp)\mathrm{SL}_n(\mathbb{F}_p)

We define new families of Tillich-Zémor hash functions, using higher dimensional special linear groups over finite fields as platforms. The Cayley graphs of these groups combine fast mixing properties and high girth, which together give rise to good preimage and collision resistance of the corresponding hash functions. We justify the claim that the resulting hash functions are post-quantum secure