Towards a Restrained Use of Non-Equivocation for Achieving Iterative Approximate Byzantine Consensus

Abstract

We consider the approximate consensus problem in a partially connected network of n nodes where at most f nodes may suffer from Byzantine faults. We study under which conditions this problem can be solved using an iterative algorithm. A Byzantine node can equivocate: it may provide different values to its neighbors. To restrict the possibilities of equivocation, the 3-partial multicast primitive is considered. When a (correct or faulty) node uses this communication primitive, it provides necessarily the same value to the two identified receivers. Based on this communication primitive, a novel condition called f-resilient is proposed and proved to be necessary and sufficient to solve the approximate Byzantine consensus problem in a synchronous network. This condition takes into account two different communication primitives: unicast and 3-partial multicast. It expresses a trade-off between the two known approaches that make the problem solvable (increasing the number of neighbors or/and increasing the power of the communication primitives). The condition fresilient does not require to eliminate all the possibilities of equivocation. Furthermore, it can be satisfied when there is just a majority of correct nodes. The relationships between the condition f-resilient and the condition h-disjoint (proposed by Alexander Jaffe et al. in 2012 to solve another problem, namely exact Byzantine consensus) are investigated. Two preliminary conclusions are obtained. When a network does not satisfy hdisjoint, it also does not satisfy f-resilient. But when a network satisfies h-disjoint, f-resilient is not necessarily satisfied. Finally, the condition is extended to cope with asynchronous networks.

Exact Byzantine Consensus on Undirected Graphs under Local Broadcast Model

“…al. [18] extended this line of work to a network consisting of 3-hyperedges and 2-hyperedges. Vaidya et.…”

“…This paper obtains tight necessary and sufficient conditions on the underlying communication network to be able to achieve Byzantine consensus under the local broadcast model. As summarized in Section 2, although there has been significant work [1,3,5,6,9,10,11,12,14,15,18,22,24,25,26,28,32] that uses either the local broadcast model or other models that restrict equivocation, tight necessary and sufficient conditions for Byzantine consensus under the local broadcast model have not been obtained previously. In particular, this paper makes the following contributions, some of which have been documented elsewhere [13,20,21] as well:…”

Byzantine Consensus under Local Broadcast Model: Tight Sufficient Condition

“…[2] looked at non-equivocation in complete graphs for asynchronous communication. Different works [5,6,9,16] investigate the impact of limiting equivocation via partial broadcast channels modeled as broadcast over hyperedges.…”