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Dans cette thèse, nous exploitons les outils de la théorie des probabilités et de la géométrie stochastique pour analyser des algorithmes opérant sur les tessellations. Ce travail est divisé entre deux thèmes principaux, le premier traite de la navigation dans une tessellation de Delaunay et dans son dual, le diagramme de Voronoï avec des implications pour les algorithmes de localisation spatiales et de routage dans les réseaux en ligne. Nous proposons deux nouveaux algorithmes de navigation dans la triangulation de Delaunay, que nous appelons Pivot Walk et Cone Walk. Pour Cone Walk, nous fournissons une analyse en moyenne détaillée avec des bornes explicites sur les propriétés de la pire marche possible effectuée par l'algorithme sur une triangulation de Delaunay aléatoire d'une région convexe bornée. C'est un progrès significatif car dans l'algorithme Cone Walk, les probabilités d'utiliser un triangle ou un autre au cours de la marche présentent des dépendances complexes, dépendances inexistantes dans d'autres marches. La deuxième partie de ce travail concerne l'étude des propriétés extrémales de tessellations aléatoires. En particulier, nous dérivons les premiers et derniers statistiques d'ordre pour les boules inscrites dans les cellules d'un arrangement de droites Poissonnien; ce résultat a des implications par exemple pour le hachage respectant la localité. Comme corollaire, nous montrons que les cellules minimisant l'aire sont des triangles. / In this thesis, we leverage the tools of probability theory and stochastic geometry to investigate the behavior of algorithms on geometric tessellations of space. This work is split between two main themes, the first of which is focused on the problem of navigating the Delaunay tessellation and its geometric dual, the Voronoi diagram. We explore the applications of this problem to point location using walking algorithms and the study of online routing in networks. We then propose and investigate two new algorithms which navigate the Delaunay triangulation, which we call Pivot Walk and Cone Walk. For Cone Walk, we provide a detailed average-case analysis, giving explicit bounds on the properties of the worst possible path taken by the algorithm on a random Delaunay triangulation in a bounded convex region. This analysis is a significant departure from similar results that have been obtained, due to the difficulty of dealing with the complex dependence structure of localized navigation algorithms on the Delaunay triangulation. The second part of this work is concerned with the study of extremal properties of random tessellations. In particular, we derive the first and last order-statistics for the inballs of the cells in a Poisson line tessellation. This result has implications for algorithms involving line tessellations, such as locality sensitive hashing. As a corollary, we show that the cells minimizing the area are triangles.

A Mobile Ad Hoc Network (MANETs) is a collection of wireless hosts that can be rapidly deployed as a multi hop packet radio network without the aid of any established infrastructure or centralized administration. Such networks can be used to enable next generation of battlefield applications envisioned by the military, including situation awareness systems for maneuvering war fighters, and remotely deployed unmanned microsensor networks. Ad Hoc networks can also provide solutions for civilian applications such as disaster recovery and message exchanges among safety and security personnel involved in rescue missions. Existing solutions for wired network Intrusion Detection Systems (IDSs) do not suit wireless Ad Hoc networks. To utilize either misuse detection or anomaly detection to monitor any possible compromises, the IDS must be able to distinguish normal from anomaly activities. To enable intrusion detection in wireless Ad Hoc networks, the research problems are: • How to efficiently collect normal and anomaly patterns of Ad Hoc networks? The lifetime of the hosts is short and Ad Hoc networks do not have traffic concentration points (router, switch). • How to detect anomalies? The loss could be caused by host movement instead of attacks. Unexpectedly long delay could be caused by unreliable channel instead of malicious discard. In this thesis, we have proposed a novel architecture that uses specification based intrusion detection techniques to detect active attacks against the routing protocols of mobile Ad Hoc networks. Our work analyzes some of the vulnerabilities and discuss the attacks against the AODV protocol. Our approach involves the use of an FSM (Finite State Machine) for specifying the AODV routing behavior and the distributed network monitors for detecting the sequence number attack. Our method can detect most of the bad nodes with low false positive rate and the packet delivery ratio can also be increased with high detection rate. For packet dropping attack, we present a distributed technique to detect this attack in wireless Ad Hoc networks. A bad node can forward packets but in fact it fails to do so. In our technique, every node in the network will check the neighboring nodes to detect if any of them fail to forward the packets. Our technique can detect most of the bad nodes with low false positive rate and the packet delivery ratio can also be increased. The proposed solution can be applied to identify multiple malicious nodes cooperating with each other in MANETs and discover secure routes from source to destination by avoiding malicious nodes acting in cooperation. Our technique will detect the sequence number and Packet Dropping attacks in real time within its radio range with no extra overhead. For resource consumption attack, the proposed scheme incurs no extra overhead, as it makes minimal modifications to the existing data structures and functions related to bad listing a node in the existing version of pure AODV. The proposed scheme is more efficient in terms of the resultant routes established, resource reservations, and computational complexity. If multiple malicious nodes collaborate, they in turn will be restricted and isolated by their neighbors, because they monitor and exercise control over forwarding RREQs by nodes. Hence, the scheme successfully prevents Distributed attacks. The proposed scheme shifts the responsibility of monitoring this parameter to the node's neighbor, ensuring compliance of this restriction. This technique solves all of the problems caused due to unnecessary RREQs from a compromised node. Instead of self-control, the control exercised by a node's neighbor results in preventing this attack. Experiments show that the tool provides effective intrusion detection functionality while using only a limited amount of resources. The loop freedom property has been reduced to an invariant on pairs of nodes. Each node decides & transmits its decision to a control center. Robustness to Threats, Robustness to nodes destruction: Loss of Performance (in terms of ratio) is least for Distributed Option and highest for Centralized Option and Robustness to observations deletion. All the proposed schemes were analyzed and tested under different topologies and conditions with varying number of nodes .The proposed algorithms for improving the robustness of the wireless Ad Hoc networks using AODV protocol against Packet Dropping Attack, Sequence Number attack and resource consumption attack have been simulated for an illustrative network of about 30 nodes. Our experiments have shown that the pattern extracted through simulation can be used to detect attacks effectively. The patterns could also be applied to detect similar attacks on other protocols.