Optimal Resilience Asynchronous Approximate Agreement

Abraham, Ittai; Amit, Yonatan; Dolev, Danny

doi:10.1007/11516798_17

Cited by 53 publications

(120 citation statements)

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“…Dolev et al [3] were the first to consider approximate consensus in presence of Byzantine faults in asynchronous systems. Subsequently, for complete graphs, Abraham, Amit and Dolev [1] established that approximate Byzantine consensus is possible in asynchronous systems if n ≥ 3f + 1. Other algorithms for approximate consensus in presence of Byzantine faults have also been proposed (e.g., [4]).…”

Section: Introductionmentioning

confidence: 98%

Iterative Byzantine Vector Consensus in Incomplete Graphs

Vaidya

2014

Lecture Notes in Computer Science

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Abstract. This work addresses Byzantine vector consensus, wherein the input at each process is a d-dimensional vector of reals, and each process is required to decide on a decision vector that is in the convex hull of the input vectors at the fault-free processes [9,12]. The input vector at each process may also be viewed as a point in the d-dimensional Euclidean space R d , where d > 0 is a finite integer. Recent work [9,12] has addressed Byzantine vector consensus, and presented algorithms with optimal fault tolerance in complete graphs. This paper considers Byzantine vector consensus in incomplete graphs using a restricted class of iterative algorithms that maintain only a small amount of memory across iterations. For such algorithms, we prove a necessary condition, and a sufficient condition, for the graphs to be able to solve the vector consensus problem iteratively. We present an iterative Byzantine vector consensus algorithm, and prove it correct under the sufficient condition. The necessary condition presented in this paper for vector consensus does not match with the sufficient condition for d > 1; thus, a weaker condition may potentially suffice for Byzantine vector consensus.

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Section: Introductionmentioning

confidence: 98%

Iterative Byzantine Vector Consensus in Incomplete Graphs

Vaidya

2014

Lecture Notes in Computer Science

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“…When d = 1, in a complete graph, we have that 3 f + 1 processes are sufficient for exact consensus on synchronous systems, as well as approximate consensus on asynchronous systems [1]. For d > 1, the lower bound for asynchronous systems is larger than the lower bound for synchronous systems.…”

Section: Multidimensional Byzantine Approximate Agreementmentioning

confidence: 99%

“…In order to circumvent this impossibility result, Dolev et al [9] introduced the notion of approximate agreement over scalars, and provided an algorithm for asynchronous systems where n > 5 f . Subsequently, Abraham et al [1] showed that approximate agreement over scalars is possible on asynchronous systems where n > 3 f . Other algorithms for approximate consensus over scalars have also been proposed (e.g., [4,12]).…”

Section: Multidimensional Byzantine Approximate Agreementmentioning

confidence: 99%

“…With the witness technique, originally presented by Abraham et al [1], we can make non-faulty processes have n − f common values, which is essential for our correctness and optimality arguments. The witness technique will only wait for messages that are certain to be delivered.…”

Section: Witness Techniquementioning

confidence: 99%

“…So, they receive at least n − f values in common. As formally shown in [1]: Fact 1 If n > 3 f , any two non-faulty processes that obtain messages through the witness technique in a single round r obtain n − f messages in common.…”

Section: Witness Techniquementioning

confidence: 99%

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Multidimensional agreement in Byzantine systems

et al. 2015

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Consider a network of n processes, where each process inputs a d-dimensional vector of reals. All processes can communicate directly with others via reliable FIFO channels. We discuss two problems. The multidimensional Byzantine consensus problem, for synchronous systems, requires processes to decide on a single d-dimensional vector v ∈ R d , inside the convex hull of d-dimensional vectors that were input by the non-faulty processes. Also, the multidimensional Byzantine approximate agreement (MBAA) problem, for asynchronous systems, requires processes to decide on multiple d-dimensional vectors in R d , all within a fixed Euclidean distance of each other, and inside the convex hull of d-dimensional vectors that were input by the non-faulty processes. We obtain the following results for the problems (M. Herlihy) Supported by NSF 0830491. (N. Vaidya and V. K. Garg) above, while tolerating up to f Byzantine failures in systems with complete communication graphs: (1) In synchronous systems, n > max{3 f, (d + 1) f } is necessary and sufficient to solve the multidimensional consensus problem. (2) In asynchronous systems, n > (d + 2) f is necessary and sufficient to solve the multidimensional approximate agreement problem. Our sufficiency proofs are constructive, giving explicit protocols for the problems. In particular, for the MBAA problem, we give two protocols with strictly different properties and applications.

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