3,199 research outputs found

    Byzantine Vector Consensus in Complete Graphs

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    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

    Relaxed Byzantine Vector Consensus

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    Exact Byzantine consensus problem requires that non-faulty processes reach agreement on a decision (or output) that is in the convex hull of the inputs at the non-faulty processes. It is well-known that exact consensus is impossible in an asynchronous system in presence of faults, and in a synchronous system, n>=3f+1 is tight on the number of processes to achieve exact Byzantine consensus with scalar inputs, in presence of up to f Byzantine faulty processes. Recent work has shown that when the inputs are d-dimensional vectors of reals, n>=max(3f+1,(d+1)f+1) is tight to achieve exact Byzantine consensus in synchronous systems, and n>= (d+2)f+1 for approximate Byzantine consensus in asynchronous systems. Due to the dependence of the lower bound on vector dimension d, the number of processes necessary becomes large when the vector dimension is large. With the hope of reducing the lower bound on n, we consider two relaxed versions of Byzantine vector consensus: k-Relaxed Byzantine vector consensus and (delta,p)-Relaxed Byzantine vector consensus. In k-relaxed consensus, the validity condition requires that the output must be in the convex hull of projection of the inputs onto any subset of k-dimensions of the vectors. For (delta,p)-consensus the validity condition requires that the output must be within distance delta of the convex hull of the inputs of the non-faulty processes, where L_p norm is used as the distance metric. For (delta,p)-consensus, we consider two versions: in one version, delta is a constant, and in the second version, delta is a function of the inputs themselves. We show that for k-relaxed consensus and (delta,p)-consensus with constant delta>=0, the bound on n is identical to the bound stated above for the original vector consensus problem. On the other hand, when delta depends on the inputs, we show that the bound on n is smaller when d>=3

    Consensus in Byzantine asynchronous systems

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    AbstractThis paper presents a consensus protocol resilient to Byzantine failures. It uses signed and certified messages and is based on two underlying failure detection modules. The first is a muteness failure detection module of the class ♢M. The second is a reliable Byzantine behaviour detection module. More precisely, the first module detects processes that stop sending messages, while processes experiencing other non-correct behaviours (i.e., Byzantine) are detected by the second module. The protocol is resilient to F faulty processes, F⩽min(⌊(n−1)/2⌋,C) (where C is the maximum number of faulty processes that can be tolerated by the underlying certification service).The approach used to design the protocol is new. While usual Byzantine consensus protocols are based on failure detectors to detect processes that stop communicating, none of them use a module to detect their Byzantine behaviour (this detection is not isolated from the protocol and makes it difficult to understand and prove correct). In addition to this modular approach and to a consensus protocol for Byzantine systems, the paper presents a finite state automaton-based implementation of the Byzantine behaviour detection module. Finally, the modular approach followed in this paper can be used to solve other problems in asynchronous systems experiencing Byzantine failures

    High-Performance Asynchronous Byzantine Fault Tolerance Consensus Protocol

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    In response to new and innovating blockchain-based systems with Internet of Things (IoT), there is a need for consensus mechanisms that can provide high transaction throughput and security, despite varying network quality. Honeybadger was the first practical, asynchronous Byzantine Fault Tolerance (BFT) consensus protocol, achieving high scalability and robustness without making any timing assumptions regarding the network. To improve the current asynchronous consensus protocols, we designed Asynchronous Byzantine Fault Tolerance (ABFT) consensus protocol through integrating threshold Elliptic Curve Digital Signature Algorithm (ECDSA) signatures and optimization of erasure coding parameters, as well as additional implementation-level optimizations. We implement a prototype of ABFT, and evaluate its performance at scale in a global WAN network and a network affected by asymmetric network degradation. Our results show that ABFT provides considerably higher performance, significantly lower computational overhead, and greater scalability than its predecessors. ABFT can reach up to 38.700 transactions per second in throughput. Furthermore, we empirically show that ABFT is unaffected by asymmetric network degradation within the fault threshold.acceptedVersio

    Fast Multidimensional Asymptotic and Approximate Consensus

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    We study the problems of asymptotic and approximate consensus in which agents have to get their values arbitrarily close to each others\u27 inside the convex hull of initial values, either without or with an explicit decision by the agents. In particular, we are concerned with the case of multidimensional data, i.e., the agents\u27 values are d-dimensional vectors. We introduce two new algorithms for dynamic networks, subsuming classical failure models like asynchronous message passing systems with Byzantine agents. The algorithms are the first to have a contraction rate and time complexity independent of the dimension d. In particular, we improve the time complexity from the previously fastest approximate consensus algorithm in asynchronous message passing systems with Byzantine faults by Mendes et al. [Distrib. Comput. 28] from Omega(d log (d Delta)/epsilon) to O(log Delta/epsilon), where Delta is the initial and epsilon is the terminal diameter of the set of vectors of correct agents
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