Byzantine Vector Consensus in Complete Graphs

Consider a network of n processes each of which has a d-dimensional vector of reals as its input. Each process can communicate directly with all the processes in the system; thus the communication network is a complete graph. All the communication channels are reliable and FIFO (first-in-first-out). The problem of Byzantine vector consensus (BVC) requires agreement on a d-dimensional vector that is in the convex hull of the d-dimensional input vectors at the non-faulty processes. We obtain the following results for Byzantine vector consensus in complete graphs while tolerating up to f Byzantine failures: * We prove that in a synchronous system, n >= max(3f+1, (d+1)f+1) is necessary and sufficient for achieving Byzantine vector consensus. * In an asynchronous system, it is known that exact consensus is impossible in presence of faulty processes. For an asynchronous system, we prove that n >= (d+2)f+1 is necessary and sufficient to achieve approximate Byzantine vector consensus. Our sufficiency proofs are constructive. We show sufficiency by providing explicit algorithms that solve exact BVC in synchronous systems, and approximate BVC in asynchronous systems. We also obtain tight bounds on the number of processes for achieving BVC using algorithms that are restricted to a simpler communication pattern

Iterative Approximate Consensus in the presence of Byzantine Link Failures

This paper explores the problem of reaching approximate consensus in synchronous point-to-point networks, where each directed link of the underlying communication graph represents a communication channel between a pair of nodes. We adopt the transient Byzantine link failure model [15, 16], where an omniscient adversary controls a subset of the directed communication links, but the nodes are assumed to be fault-free. Recent work has addressed the problem of reaching approximate consen- sus in incomplete graphs with Byzantine nodes using a restricted class of iterative algorithms that maintain only a small amount of memory across iterations [22, 21, 23, 12]. However, to the best of our knowledge, we are the first to consider approximate consensus in the presence of Byzan- tine links. We extend our past work that provided exact characterization of graphs in which the iterative approximate consensus problem in the presence of Byzantine node failures is solvable [22, 21]. In particular, we prove a tight necessary and sufficient condition on the underlying com- munication graph for the existence of iterative approximate consensus algorithms under transient Byzantine link model. The condition answers (part of) the open problem stated in [16].Comment: arXiv admin note: text overlap with arXiv:1202.609

Matrix Representation of Iterative Approximate Byzantine Consensus in Directed Graphs

This paper presents a proof of correctness of an iterative approximate Byzantine consensus (IABC) algorithm for directed graphs. The iterative algorithm allows fault- free nodes to reach approximate conensus despite the presence of up to f Byzantine faults. Necessary conditions on the underlying network graph for the existence of a correct IABC algorithm were shown in our recent work [15, 16]. [15] also analyzed a specific IABC algorithm and showed that it performs correctly in any network graph that satisfies the necessary condition, proving that the necessary condition is also sufficient. In this paper, we present an alternate proof of correctness of the IABC algorithm, using a familiar technique based on transition matrices [9, 3, 17, 19]. The key contribution of this paper is to exploit the following observation: for a given evolution of the state vector corresponding to the state of the fault-free nodes, many alternate state transition matrices may be chosen to model that evolution cor- rectly. For a given state evolution, we identify one approach to suitably "design" the transition matrices so that the standard tools for proving convergence can be applied to the Byzantine fault-tolerant algorithm as well. In particular, the transition matrix for each iteration is designed such that each row of the matrix contains a large enough number of elements that are bounded away from 0

Exact Byzantine Consensus on Arbitrary Directed Graphs Under Local Broadcast Model

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, effectively depriving the faulty nodes of the ability to equivocate. 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 links and the ability to achieve consensus

Asynchronous Convex Consensus in the Presence of Crash Faults

This paper defines a new consensus problem, convex consensus. Similar to vector consensus [13, 20, 19], the input at each process is a d-dimensional vector of reals (or, equivalently, a point in the d-dimensional Euclidean space). However, for convex consensus, the output at each process is a convex polytope contained within the convex hull of the inputs at the fault-free processes. We explore the convex consensus problem under crash faults with incorrect inputs, and present an asynchronous approximate convex consensus algorithm with optimal fault tolerance that reaches consensus on an optimal output polytope. Convex consensus can be used to solve other related problems. For instance, a solution for convex consensus trivially yields a solution for vector consensus. More importantly, convex consensus can potentially be used to solve other more interesting problems, such as convex function optimization [5, 4].Comment: A version of this work is published in PODC 201

Reaching Approximate Byzantine Consensus with Multi-hop Communication

We address the problem of reaching consensus in the presence of Byzantine faults. In particular, we are interested in investigating the impact of messages relay on the network connectivity for a correct iterative approximate Byzantine consensus algorithm to exist. The network is modeled by a simple directed graph. We assume a node can send messages to another node that is up to $l$ hops away via forwarding by the intermediate nodes on the routes, where $l\in \mathbb{N}$ is a natural number. We characterize the necessary and sufficient topological conditions on the network structure. The tight conditions we found are consistent with the tight conditions identified for $l=1$, where only local communication is allowed, and are strictly weaker for $l>1$. Let $l^*$ denote the length of a longest path in the given network. For $l\ge l^*$ and undirected graphs, our conditions hold if and only if $n\ge 3f+1$ and the node-connectivity of the given graph is at least $2f+1$ , where $n$ is the total number of nodes and $f$ is the maximal number of Byzantine nodes; and for $l\ge l^*$ and directed graphs, our conditions is equivalent to the tight condition found for exact Byzantine consensus. Our sufficiency is shown by constructing a correct algorithm, wherein the trim function is constructed based on investigating a newly introduced minimal messages cover property. The trim function proposed also works over multi-graphs.Comment: 24 pages, 1 figure. arXiv admin note: text overlap with arXiv:1203.188