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Concurrent Asynchronous Byzantine Agreement in Expected-Constant Rounds, Revisited

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Theory of Cryptography (TCC 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14372))

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Abstract

It is well known that without randomization, Byzantine agreement (BA) requires a linear number of rounds in the synchronous setting, while it is flat out impossible in the asynchronous setting. The primitive which allows to bypass the above limitation is known as oblivious common coin (OCC). It allows parties to agree with constant probability on a random coin, where agreement is oblivious, i.e., players are not aware whether or not agreement has been achieved.

The starting point of our work is the observation that no known protocol exists for information-theoretic multi-valued OCC with optimal resiliency in the asynchronous setting (with eventual message delivery).

This apparent hole in the literature is particularly problematic, as multi-valued OCC is implicitly or explicitly used in several constructions.

In this paper, we present the first information-theoretic multi-valued OCC protocol in the asynchronous setting with optimal resiliency, i.e., tolerating \(t<n/3\) corruptions, thereby filling this important gap. Further, our protocol efficiently implements OCC with an exponential-size domain, a property which is not even achieved by known constructions in the simpler, synchronous setting.

We then turn to the problem of round-preserving parallel composition of asynchronous BA. A protocol for this task was proposed by Ben-Or and El-Yaniv [Distributed Computing ’03]. Their construction, however, is flawed in several ways. Thus, as a second contribution, we provide a simpler, more modular protocol for the above task. Finally, and as a contribution of independent interest, we provide proofs in Canetti’s Universal Composability framework; this makes our work the first one offering composability guarantees, which are important as BA is a core building block of secure multi-party computation protocols.

The full version of this paper can be found at the IACR Cryptology ePrint Archive, report 2023/1003.

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Notes

  1. 1.

    That is, without setup assumptions and without imposing resource restrictions on the adversary.

  2. 2.

    The bound \(t<n/2\) is tight for BA [49]; however, under the same setup assumptions, broadcast can be solved for any number of corruptions.

  3. 3.

    Fitzi and Garay [50] devised expected-constant-round BA for \(t<n/2\) in the PKI model under number-theoretic assumptions. Katz and Koo [61] established a similar result from the minimal assumption of digital signatures, which yields an information-theoretic variant using pseudo-signatures [75].

  4. 4.

    This primitive is sometimes known as a “weak” common coin in the literature.

  5. 5.

    Feldman’s A-VSS suffers from a negligible error probability. An errorless A-VSS scheme for \(t<n/4\) is given in [10] and used to construct a perfectly secure asynchronous MPC protocol with resiliency \(t<n/4\).

  6. 6.

    This is true even if one is interested only in binary concurrent BA (i.e., when the input vectors consist of bits). Multi-valued BA is needed to agree on the leader’s output vector.

  7. 7.

    They are also concerned with obtaining \(O(n^2)\) message complexity. The novelty of their result, even without this more stringent requirement, does not seem to be acknowledged in the paper.

  8. 8.

    This claim no longer appears in the ICALP [43] or journal [45] versions of the paper, or in Feldman’s thesis [44].

  9. 9.

    Ben-Or and El-Yaniv [12] introduced and used a strengthened property for (A-)BA without naming it, which was later called “non-intrusion” validity in [70]. Non-intrusion validity lies between standard validity and “strong” validity [50], as it requires that a value decided by an honest party is either an honest party’s input or a special symbol \(\bot \) (i.e., the adversary cannot intrude malicious values into the output).

  10. 10.

    Recall that while (concurrent) A-BA is not a private task, secure channels are needed to construct an OCC.

  11. 11.

    It is important to note that the term “oblivious” in this context refers to the fact that parties do not learn whether an agreement on a random coin value has been achieved or not, while the adversary does.

  12. 12.

    Feldman calculated the size of the overlap, denoted as x, based on the number of participants n and the maximum number of corruptions t. The general relation is \(x \ge n-t-\frac{t^2}{n-2t}\), which yields \(x \ge n/3\) and \(x \ge 5n/8\) when \(t \le n/3\) and \(t \le n/4\), respectively. This argument was later used in [24] to achieve optimal resiliency.

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Acknowledgements

Our original motivation for this project was to provide a simulation-based treatment of concurrent A-BA protocols, such as Ben-Or and El-Yaniv’s [12], but the search for building blocks, in particular of an optimally resilient asynchronous OCC protocol became a bit of a “detective story,” as many references pointed to an unpublished manuscript by Feldman [41], which was nowhere to be found. We thank Michael Ben-Or for providing it to us, which corroborated its in-existence.

Ran Cohen’s research is supported in part by NSF grant no. 2055568. Juan Garay’s research is supported in part by NSF grants no. 2001082 and 2055694. Vassilis Zikas’s research is supported in part by NSF grant no. 2055599 and by Sunday Group. The authors were also supported by the Algorand Centres of Excellence programme managed by Algorand Foundation. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of Algorand Foundation.

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Authors and Affiliations

  1. Reichman University, Herzliya, Israel

    Ran Cohen

  2. Texas A &M University, College Station, USA

    Pouyan Forghani, Juan Garay & Rutvik Patel

  3. Purdue University, West Lafayette, USA

    Vassilis Zikas

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Cohen, R., Forghani, P., Garay, J., Patel, R., Zikas, V. (2023). Concurrent Asynchronous Byzantine Agreement in Expected-Constant Rounds, Revisited. In: Rothblum, G., Wee, H. (eds) Theory of Cryptography. TCC 2023. Lecture Notes in Computer Science, vol 14372. Springer, Cham. https://doi.org/10.1007/978-3-031-48624-1_16

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-48623-4

  • Online ISBN: 978-3-031-48624-1

  • eBook Packages: Computer ScienceComputer Science (R0)

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