John Augustine
ABSTRACT
We study Byzantine agreement in dynamic networks where topology can change from round to round and nodes can also experience heavy churn (i.e., nodes can join and leave the network continuously over time). Our main contributions are randomized distributed algorithms that achieve almost-everywhere Byzantine agreement with high probability even under a large number of adaptively chosen Byzantine nodes and continuous adversarial churn in a number of rounds that is polylogarithmic in n (where n is the stable network size). We show that our algorithms are essentially optimal (up to polylogarithmic factors) with respect to the amount of Byzantine nodes and churn rate that they can tolerate by showing a lower bound. In particular, we present the following results:
1. An O(log3 n) round randomized algorithmto achieve almost everywhere Byzantine agreement with high probability under a presence of up to O(√n/polylog(n)) Byzantine nodes and up to a churn of O(√n/polylog(n)) nodes per round. We assume that the Byzantine nodes have knowledge about the entire state of network at every round (including random choices made by all the nodes) and can behave arbitrarily. We also assume that an adversary controls the churn - it has complete knowledge and control of what nodes join and leave and at what time and has unlimited computational power (but is oblivious to the topology changes from round to round). Our algorithm requires only polylogarithmic in n bits to be processed and sent (per round) by each node.
2. We also present an O(log3 n) round randomized algorithm that has same guarantees as the above algorithm, but works even when the connectivity of the network is controlled by an adaptive adversary (that can choose the topology based on the current states of the nodes). However, this algorithm requires up to polynomial in n bits to be processed and sent (per round) by each node.
3. We show that the above bounds are essentially the best possible, if one wants fast (i.e., polylogarithmic run time) algorithms, by showing that any (randomized) algorithm to achieve agreement in a dynamic network controlled by an adversary that can churn up to Θ(√n log n) nodes per round should take at least a polynomial number of rounds.
Our algorithms are the first-known, fully distributed, Byzantine agreement algorithms in highly dynamic networks. We view our results as a step towards understanding the possibilities and limitations of highly dynamic networks that are subject to malicious behavior by a large number of nodes.
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Index Terms
- Fast byzantine agreement in dynamic networks
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