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1. Exact Byzantine Consensus on Undirected Graphs under Local Broadcast Model

   https://doi.org/10.1145/3293611.3331619  

   Khan, Muhammad Samir ; Naqvi, Syed Shalan ; Vaidya, Nitin H. ( July 2019 , ACM Symposium on Principles in Distributed Computing)

   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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2. Byzantine Consensus with Local Multicast Channels

   https://doi.org/10.4230/LIPIcs.DISC.2021.26  

   Khan, Muhammad Samir ( October 2021 , Leibniz international proceedings in informatics)

   Gilbert, Seth (Ed.)

   Byzantine consensus is a classical problem in distributed computing. Each node in a synchronous system starts with a binary input. The goal is to reach agreement in the presence of Byzantine faulty nodes. We consider the setting where communication between nodes is modelled via an undirected communication graph. In the classical point-to-point communication model all messages sent on an edge are private between the two endpoints of the edge. This allows a faulty node to equivocate, i.e., lie differently to its different neighbors. Different models have been proposed in the literature that weaken equivocation. In the local broadcast model, every message transmitted by a node is received identically and correctly by all of its neighbors. In the hypergraph model, every message transmitted by a node on a hyperedge is received identically and correctly by all nodes on the hyperedge. Tight network conditions are known for each of the three cases. We introduce a more general model that encompasses all three of these models. In the local multicast model, each node u has one or more local multicast channels. Each channel consists of multiple neighbors of u in the communication graph. When node u sends a message on a channel, it is received identically by all of its neighbors on the channel. For this model, we identify tight network conditions for consensus. We observe how the local multicast model reduces to each of the three models above under specific conditions. In each of the three cases, we relate our network condition to the corresponding known tight conditions. The local multicast model also encompasses other practical network models of interest that have not been explored previously, as elaborated in the paper. 

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3. Testing Equality Under the Local Broadcast Model

   https://doi.org/https://doi.org/10.1007/978-3-030-79527-6_15  

   Khan M.S., Vaidya N.H. ( June 2021 , 28th International Colloquium on Structural Information and Communication Complexity)

   Jurdziński, T ; Schmid, S (Ed.)

   In the multiparty equality problem, each of the n nodes starts with a k-bit input. If there is a mismatch between the inputs, then at least one node must be able to detect it. The cost of a multiparty equality protocol is the total number of bits sent in the protocol. We consider the problem of minimizing this communication cost under the local broadcast model for the case where the underlying communication graph is undirected. In the local broadcast model of communication, a message sent by a node is received identically by all of its neighbors. This is in contrast to the classical point-to-point communication model, where a message sent by a node to one of its neighbors is received only by its intended recipient. Under point-to-point communication, there exists a simple protocol which is competitive within a factor 2 of the lower bound [1]. In this protocol, a rooted spanning tree is fixed and each node sends its entire input to its parent in the tree. On receiving a value from its child, a node compares it against its own input to check if the two values match. Ignoring lower order additive terms, a more complicated protocol comes within a factor 4/3 of the lower bound and is tight for certain classes of graphs [1]. Tight results, ignoring lower order terms, are also known for complete graphs [2, 9]. We study the multiparty equality problem under the local broadcast model. Recently, our work has shown that the connectivity requirements for Byzantine consensus are lower in the local broadcast model as compared to the classical model [7, 8]. In this work, 1. we identify a lower bound for the multiparty equality problem in this model. 2. we first identify simple protocols, wherein nodes are restricted to either transmit their entire input or not transmit anything at all, and find that these can cost Ω(logn) times the lower bound using existing example for the set cover problem [12]. 3. we then design a protocol to solve the problem within a constant factor of the lower bound. 

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4. Fault-tolerant Consensus in Anonymous Dynamic Network

   Zhang, Qinzi ; Tseng, Lewis ( July 2024 , IEEE International Conference on Distributed Computing Systems)

   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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5. Byzantine Consensus in Abstract MAC Layer

   https://doi.org/10.4230/LIPICS.OPODIS.2023.9  

   Tseng, Lewis ; Sardina, Callie ( January 2024 , International Conference on Principles of Distributed Systems)

   Bessani, Alysson ; Défago, Xavier ; Nakamura, Junya ; Wada, Koichi ; Yamauchi, Yukiko (Ed.)

   This paper studies the design of Byzantine consensus algorithms in an asynchronous single-hop network equipped with the "abstract MAC layer" [DISC09], which captures core properties of modern wireless MAC protocols. Newport [PODC14], Newport and Robinson [DISC18], and Tseng and Zhang [PODC22] study crash-tolerant consensus in the model. In our setting, a Byzantine faulty node may behave arbitrarily, but it cannot break the guarantees provided by the underlying abstract MAC layer. To our knowledge, we are the first to study Byzantine faults in this model. We harness the power of the abstract MAC layer to develop a Byzantine approximate consensus algorithm and a Byzantine randomized binary consensus algorithm. Both of our algorithms require only the knowledge of the upper bound on the number of faulty nodes f, and do not require the knowledge of the number of nodes n. This demonstrates the "power" of the abstract MAC layer, as consensus algorithms in traditional message-passing models require the knowledge of both n and f. Additionally, we show that it is necessary to know f in order to reach consensus. Hence, from this perspective, our algorithms require the minimal knowledge. The lack of knowledge of n brings the challenge of identifying a quorum explicitly, which is a common technique in traditional message-passing algorithms. A key technical novelty of our algorithms is to identify "implicit quorums" which have the necessary information for reaching consensus. The quorums are implicit because nodes do not know the identity of the quorums - such notion is only used in the analysis. 

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