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Encyclopedia of Algorithms pp 604–606Cite as

Optimal Probabilistic Synchronous Byzantine Agreement

1988; Feldman, Micali

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Keywords and Synonyms

Distributed consensus; Byzantine generals problem          

Problem Definition

The Byzantine agreement problem (BA) is concerned with multiple processors (parties, “players”) all starting with some initial value, agreeing on a common value, despite the possible disruptive or even malicious behavior of some them. BA is a fundamental problem in fault-tolerant distributed computing and secure multi-party computation.

The problem was introduced by Pease, Shostak and Lamport in [18], who showed that the number of faulty processors must be less than a third of the total number of processors for the problem to have a solution. They also presented a protocol matching this bound, which requires a number of communication rounds proportional to the number of faulty processors—exactly \( { t+1 } \), where t is the number of faulty processors. Fischer and Lynch [10] later showed that this number of rounds is necessary in the worst-case run of any deterministic BA protocol. Furthermore,...

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Notes

  1. 1.

    Karlin and Yao [14] showed that the maximum number of faulty processors for probabilistic BA is also \( { t < \frac{n}{3} } \), where n is the total number of processors.

  2. 2.

    Indeed, it was shown by Dwork and Moses [7] that at least \( { t+1 } \) rounds are necessary for simultaneous termination. In [13], Goldreich and Petrank combine “the best of both worlds” by showing a BA protocol running in expected constant number of rounds which always terminates within \( { t+O(\log t) } \) rounds.

Recommended Reading

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  2. Ben-Or, M., El-Yaniv, R.: Optimally-resilient interactive consistency in constant time. Distrib. Comput. 16(4), 249–262 (2003)

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  3. Bracha, G.: An \( { {O}(\log n) } \) expected rounds randomized Byzantine generals protocol. J. Assoc. Comput. Mach. 34(4), 910–920 (1987)

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  4. Canetti, R., Rabin, T.: Fast asynchronous Byzantine agreement with optimal resilience. In: Proceedings of the 25th Annual ACM Symposium on the Theory of Computing, San Diego, California, 16–18 May 1993, pp. 42–51

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  5. Chor, B., Coan, B.: A simple and efficient randomized Byzantine agreement algorithm. IEEE Trans. Softw. Eng. SE-11(6), 531–539 (1985)

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  6. Chor, B., Dwork, C.: Randomization in Byzantine Agreement. Adv. Comput. Res. 5, 443–497 (1989)

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  7. Dwork, C., Moses, Y.: Knowledge and Common Knowledge in a Byzantine Environment: Crash Failures. Inf. Comput. 88(2), 156–186 (1990). Preliminary version in TARK'86

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  8. Feldman, P.: Optimal Algorithms for Byzantine Agreement. Ph. D. thesis, MIT (1988)

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  9. Feldman, P., Micali, S.: An optimal probabilistic protocol for synchronous Byzantine agreement. SIAM J. Comput. 26(4), 873–933 (1997). Preliminary version in STOC'88

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  14. Karlin, A., Yao, A.C.: Probabilistic lower bounds for the byzantine generals problem. Unpublished manuscript

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Author information

Authors and Affiliations

  1. Bell Labs, Murray Hill, NJ, USA

    Juan Garay

Authors
  1. Juan Garay
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Editor information

Editors and Affiliations

  1. Department of Electrical Engineering and Computer ScienceMcCormick School of Engineering and Applied Science, Northwestern University, Evanston, IL, 60208, USA

    Ming-Yang Kao Professor of Computer Science

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Garay, J. (2008). Optimal Probabilistic Synchronous Byzantine Agreement. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-30162-4_269

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