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This paper considers the Byzantine consensus problem for nodes with binary inputs. The nodes are interconnected by a network represented as an undirected graph, and the system is assumed to be synchronous. Under the classical point-to-point communication model, it is well-known that the following two conditions are both necessary and sufficient to achieve Byzantine consensus among n nodes in the presence of up to ƒ Byzantine faulty nodes: n & 3 #8805; 3 ≥ ƒ+ 1 and vertex connectivity at least 2 ƒ + 1. In the classical point-to-point communication model, it is possible for a faulty node to equivocate, i.e., transmit conflicting information to different neighbors. Such equivocation is possible because messages sent by a node to one of its neighbors are not overheard by other neighbors. This paper considers the local broadcast model. In contrast to the point-to-point communication model, in the local broadcast model, messages sent by a node are received identically by all of its neighbors. Thus, under the local broadcast model, attempts by a node to send conflicting information can be detected by its neighbors. Under this model, we show that the following two conditions are both necessary and sufficient for Byzantine consensus: vertex connectivity at least ⌋ 3 fƒ / 2 ⌊ + 1 and minimum node degree at least 2 ƒ. Observe that the local broadcast model results in a lower requirement for connectivity and the number of nodes n, as compared to the point-to-point communication model. We extend the above results to a hybrid model that allows some of the Byzantine faulty nodes to equivocate. The hybrid model bridges the gap between the point-to-point and local broadcast models, and helps to precisely characterize the trade-off between equivocation and network requirements.
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This content will become publicly available on July 29, 2025
Fault-tolerant Consensus in Anonymous Dynamic Network
This paper studies the feasibility of reaching consensus in an anonymous dynamic network. In our model, n anonymous nodes proceed in synchronous rounds. We adopt a hybrid fault model in which up to f nodes may suffer crash or Byzantine faults, and the dynamic message adversary chooses a communication graph for each round. We introduce a stability property of the dynamic network – (T,D)-dynaDegree for T ≥ 1 and n−1 ≥ D ≥ 1 – which requires that for every T consecutive rounds, any fault-free node must have incoming directed links from at least D distinct neighbors. These links might occur in different rounds during a T -round interval. (1, n−1)-dynaDegree means that the graph is a complete graph in every round. (1, 1)-dynaDegree means that each node has at least one incoming neighbor in every round, but the set of incoming neighbor(s) at each node may change arbitrarily between rounds. We show that exact consensus is impossible even with (1, n − 2)-dynaDegree. For an arbitrary T , we show that for crash-tolerant approximate consensus, (T , ⌊n/2⌋)-dynaDegree and n > 2f are together necessary and sufficient, whereas for Byzantine approximate consensus, (T , ⌊(n + 3f )/2⌋)- dynaDegree and n > 5f are together necessary and sufficient.
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- Award ID(s):
- 2334021
- PAR ID:
- 10508478
- Publisher / Repository:
- IEEE
- Date Published:
- Journal Name:
- IEEE International Conference on Distributed Computing Systems
- Format(s):
- Medium: X
- Location:
- Jersey City, New Jersey, USA
- Sponsoring Org:
- National Science Foundation
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