Brief Announcement: Effects of Topology Knowledge and Relay Depth on Asynchronous Consensus

Consider an asynchronous incomplete directed network. We study the feasibility and efficiency of approximate crash-tolerant consensus under different restrictions on topology knowledge and relay depth, i.e., the maximum number of hops any message can be relayed

Effects of Topology Knowledge and Relay Depth on Asynchronous Appoximate Consensus

Consider a point-to-point message-passing network. We are interested in the asynchronous crash-tolerant consensus problem in incomplete networks. We study the feasibility and efficiency of approximate consensus under different restrictions on topology knowledge and the relay depth, i.e., the maximum number of hops any message can be relayed. These two constraints are common in large-scale networks, and are used to avoid memory overload and network congestion respectively. Specifically, for positive integer values k and k\u27, we consider that each node knows all its neighbors of at most k-hop distance (k-hop topology knowledge), and the relay depth is k\u27. We consider both directed and undirected graphs. More concretely, we answer the following question in asynchronous systems: "What is a tight condition on the underlying communication graphs for achieving approximate consensus if each node has only a k-hop topology knowledge and relay depth k\u27?" To prove that the necessary conditions presented in the paper are also sufficient, we have developed algorithms that achieve consensus in graphs satisfying those conditions: - The first class of algorithms requires k-hop topology knowledge and relay depth k. Unlike prior algorithms, these algorithms do not flood the network, and each node does not need the full topology knowledge. We show how the convergence time and the message complexity of those algorithms is affected by k, providing the respective upper bounds. - The second set of algorithms requires only one-hop neighborhood knowledge, i.e., immediate incoming and outgoing neighbors, but needs to flood the network (i.e., relay depth is n, where n is the number of nodes). One result that may be of independent interest is a topology discovery mechanism to learn and "estimate" the topology in asynchronous directed networks with crash faults

Reliable Broadcast with Respect to Topology Knowledge

We study the Reliable Broadcast problem in incomplete networks against a Byzantine adversary. We examine the problem under the locally bounded adversary model of Koo (2004) and the general adversary model of Hirt and Maurer (1997) and explore the tradeoff between the level of topology knowledge and the solvability of the problem. We refine the local pair-cut technique of Pelc and Peleg (2005) in order to obtain impossibility results for every level of topology knowledge and any type of corruption distribution. On the positive side we devise protocols that match the obtained bounds and thus, exactly characterize the classes of graphs in which Reliable Broadcast is possible. Among others, we show that Koo\u27s Certified Propagation Algorithm (CPA) is unique against locally bounded adversaries in ad hoc networks, that is, it can tolerate as many local corruptions as any other non-faulty algorithm; this settles an open question posed by Pelc and Peleg. We also provide an adaptation of CPA against general adversaries and show its uniqueness in this case too. To the best of our knowledge this is the first optimal algorithm for Reliable Broadcast in generic topology ad hoc networks against general adversaries

On the Resilience and Uniqueness of CPA for Secure Broadcast

We consider the Secure Broadcast problem in incomplete networks. We study the resilience of the Certified Propagation Algorithm (CPA), which is particularly suitable for ad hoc networks. We address the issue of determining the maximum number of corrupted players $t^{\mathrm{CPA}}_{\max}$ that CPA can tolerate under the $t$-locally bounded adversary model, in which the adversary may corrupt at most $t$ players in each player\u27s neighborhood. For any graph $G$ and dealer-node $D$ we provide upper and lower bounds on $t^{\mathrm{CPA}}_{\max}$ that can be efficiently computed in terms of a graph theoretic parameter that we introduce in this work. Along the way we obtain an efficient 2-approximation algorithm for $t^{\mathrm{CPA}}_{\max}$. We further introduce two more graph parameters, one of which matches $t^{\mathrm{CPA}}_{\max}$exactly. Our approach allows to provide an affirmative answer to the open problem of CPA Uniqueness posed by Pelc and Peleg in 2005

Delivery delay and mobile faults

In this work we address the problem of reaching approximate consensus in a complete network of n nodes, where message deliveries can be delayed by at most d time-steps. We consider a mobile adversary, which corrupts at most f nodes in any step, modeled as a synchronous round. We explicitly study how d affects the feasibility of the problem. More precisely, we propose a framework to analyze mobile fault-tolerance in the presence of message delays. We prove that approximate consensus is feasible if and only if n \u3e 4df. We assume no knowledge of time (round index) by the nodes; instead, in our model, whenever a message is sent, it is timestamped by the communication channel. We propose the tight TimeStamps algorithm, which utilizes timestamps to optimally bound the number of faulty messages

Secure broadcast in generic and wireless networks

73 σ.Εθνικό Μετσόβιο Πολυτεχνείο--Μεταπτυχιακή Εργασία. Διεπιστημονικό-Διατμηματικό Πρόγραμμα Μεταπτυχιακών Σπουδών (Δ.Π.Μ.Σ.) “Εφαρμοσμένες Μαθηματικές Επιστήμες”Τα δίκτυα επικοινωνίας αποτελούνται από πολυάριθμες συσκευές που επικοινωνούν. Αυτές οι συσκευές συχνά επιθυμούν να συνεργαστούν για να πετύχουν κάποιον κοινό στόχο, ακόμα και αν κάποιες από αυτές τις συσκευές είναι κακόβουλες/διεφθαρμένες. Μία τέτοια θεώρηση δημιουργεί την ανάγκη για ασφαλείς κατανεμημένους υπολογισμούς. Είναι πλέον ευρέως αποδεκτό πως ένα αναπόσπαστο μέρος ενός κατανεμημένου συστήματος, είναι ένας μηχανισμός για την επίτευξη "συμφωνίας" μεταξύ όλων των μή διεφθαρμένωνμερών του συστήματος. Μία από τις σημαντικότερες παραλλαγές του προβλήματος είναι γνωστή στη βιβλιογραφία ως πρόβλημα της "Ασφαλούς Εκπομπής", όπου θεωρούμε την ύπαρξη ενός καθορισμένου συμμετέχοντος, στην αρχική τιμή του οποίου θα πρέπει να συμφωνήσει κάθε μη διεφθαρμένος συμμετέχων. Το πρόβλημα έχει μελετηθεί εκτενώς στο μοντέλο των πλήρων δικτύων. Στην περίπτωση των μη πλήρων δικτύων, η σχετική έρευνα εισήγαγε νέες παραμέτρους προς βελτιστοποίηση, σχετικές με την τοπολογία του δικτύου. Επιπλέον θεωρήθηκαν μοντέλα αντιπάλου με τοπολογικούς περιορισμούς και σχετικά προβλήματα παραμένουν άλυτα. Τέλος, το πρόβλημα της Ασφαλούς Εκπομπής σε ασύρματα δίκτυα εξετάζεται σε αυτήν την εργασία. Αφενός, η δομή των ασύρματων δικτύων επιτρέπει στους διεφθαρμένους συμμετέχοντες να προκαλέσουν παρεμβολές σήματος σε άλλους παραλήπτες το οποίο εισάγει νέες προκλήσεις σχετικά με τη διαχείρισή τους. Αφετέρου οι συμμετέχοντες δεσμεύονται να εκτελέσουν τοπικές εκπομπές, το οποίο διευκολύνει σε μεγάλο βαθμό την επίτευξη συμφωνίας.Communication networks consist of numerous interacting entities. These entities often wish to collaborate in order to achieve a certain task even if some of them are malicious/corrupted. Such considerations put forth the need for secure distributed computing. It is widely accepted that an integral part of a secure distributed system is a mechanism for reaching ``agreement'' between all non-corrupted parts of a system. One of the major variations of the agreement problem is the "Secure Broadcast" problem, where we assume the existence of a designated participant on whose input value every non-corrupted participant should agree. The problem has been extensively studied in the standard model where the communication network is assumed to be complete. In the case of incomplete communication networks, research has introduced new parameters for optimization, relative to the network's topology. Moreover topologically restricted corruption models have been considered, and related problems remain open so far. Finally secure Broadcast in wireless networks is considered in this thesis. The structure of wireless networks allows the corrupted entities to cause interference to other receivers, which brings up new challenges regarding their manipulation. On the other hand, participants are committed to perform local broadcasts, which greatly facilitates achieving agreement.Δημήτρης Κ. Σακαβάλα