Exact Byzantine Consensus on Undirected Graphs under Local Broadcast Model

Abstract

We consider Byzantine consensus in a synchronous system where nodes are connected by a network modeled as a directed graph, i.e., communication links between neighboring nodes are not necessarily bi-directional. The directed graph model is motivated by wireless networks wherein asymmetric communication links can occur. In the classical point-to-point communication model, a message sent on a communication link is private between the two nodes on the link. This allows a Byzantine faulty node to equivocate, i.e., send inconsistent information to its neighbors. This paper considers the local broadcast model of communication, wherein transmission by a node is received identically by all of its outgoing neighbors. This allows such neighbors to detect a faulty node's attempt to equivocate, effectively depriving the faulty nodes of the ability to send conflicting information to different neighbors.Prior work has obtained sufficient and necessary conditions on undirected graphs to be able to achieve Byzantine consensus under the local broadcast model. In this paper, we obtain tight conditions on directed graphs to be able to achieve Byzantine consensus with binary inputs under the local broadcast model. The results obtained in the paper provide insights into the trade-off between directionality of communication and the ability to achieve consensus. * 1 consensus in the presence of up to f Byzantine faulty nodes. The nodes are connected by a communication network represented by a graph. In the classical point-to-point communication model, a message sent by a node to one of its neighboring nodes is received only by that node. This allows a Byzantine faulty node to send inconsistent messages to its neighbors without the inconsistency being observed by the neighbors. For instance, a faulty node u may report to neighbor v that its input is 0, whereas report to neighbor w that its input is 1. Node v will not hear the message sent by node u to node w. This ability of a faulty node to send conflicting information on different communication links is called equivocation [4]. The problem of Byzantine consensus in point-to-point networks is well-studied [2,7,16,19,21]. In undirected graphs, it is known that n > 3f and connectivity ≥ 2f + 1 are necessary and sufficient conditions to achieve Byzantine consensus [7].This paper considers the local broadcast model of communication [3,14]. Under local broadcast, a message sent by a node is received identically by all neighbors of that node. This allows the neighbors of a faulty node to detect its attempts to equivocate, effectively depriving the faulty node of the ability to send conflicting information to its neighbors. In the example above, in the local broadcast model, if node u attempts to send different input values to different neighbors, the neighbors will receive all the messages, and can detect the inconsistency. Recent work has shown that this ability to detect equivocation reduces network requirements for Byzantine consensus in undirected graphs. In particular, for the local bro...

Resilient Consensus with Multi-hop Communication

“…Moreover, we present examples to illustrate how multi-hop communication helps to improve graph robustness without changing the network topology. As a side result, we prove that for the case of unbounded path length in message relaying, our graph condition is equivalent to the necessary and sufficient graph condition for binary consensus under malicious attacks studied in [30].…”

“…In this subsection, we discuss the relation of the graph conditions used in this paper and in the recent work [30]. The authors there studied the Byzantine binary consensus under the local broadcast model, which is essentially equivalent to the f -total malicious mode studied in the current paper.…”

“…In [29], the authors tackled the same problem under asynchronous updates based on rounds, which is different from the asynchrony setting used in this paper (see the discussions in Section VI). The work [30] studied Byzantine binary consensus under the local broadcast model (malicious model) using a flooding algorithm, where nodes relay their values over the entire network.…”

“…It is hence of significant importance to extend our algorithm to the case of asynchronous updates with time delays. We must note that such a case is not considered in [10] and [30].…”

Boosting the Efficiency of Byzantine-Tolerant Reliable Communication

“…On top of the communication network, two alternative primitives are available: unicast (UL) and local broadcast (LBL) links [1,2,11]: the former interconnect single pairs of processes p i , p j ; the latter attach a process p i to many others, such that if a message is sent by p i then it is received by all of its neighbors, thus preventing a faulty process to equivocate (i.e., to transmit conflicting messages to different neighbors).…”