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Almost-Optimally Fair Multiparty Coin-Tossing with Nearly Three-Quarters Malicious

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Abstract

An \(\alpha \)-fair coin-tossing protocol allows a set of mutually distrustful parties to generate a uniform bit, such that no efficient adversary can bias the output bit by more than \(\alpha \). Cleve (in: Proceedings of the 18th annual ACM symposium on theory of computing (STOC), 1986) has shown that if half of the parties can be corrupted, then no \(r\)-round coin-tossing protocol is \(o(1/r)\)-fair. For over two decades, the best-known m-party protocols, tolerating up to \({t}\ge m/2\) corrupted parties, were only \(O\left( {t}/\sqrt{r} \right) \)-fair. In a surprising result, Moran et al. (in: Theory of cryptography, sixth theory of cryptography conference, TCC, 2009) constructed an \(r\)-round two-party \(O(1/r)\)-fair coin-tossing protocol, i.e., an optimally fair protocol. Beimel et al. (in: Rabin (ed) Advances in cryptology—CRYPTO 2010, volume 6223 of lecture notes in computer science, Springer, 2010) extended the result of Moran et al. to the multiparty setting where strictly fewer than 2/3 of the parties are corrupted. They constructed a \(2^{2^k}/r\)-fair r-round m-party protocol, tolerating up to \(t=\frac{m+k}{2}\) corrupted parties. In a breakthrough result, Haitner and Tsfadia (in: Symposium on theory of computing, STOC, 2014) constructed an \(O\left( \log ^3(r)/r \right) \)-fair (almost optimal) three-party coin-tossing protocol. Their work brought forth a combination of novel techniques for coping with the difficulties of constructing fair coin-tossing protocols. Still, the best coin-tossing protocols for the case where more than 2/3 of the parties may be corrupted (and even when \(t=2m/3\), where \(m>3\)) were \(\theta \left( 1/\sqrt{r} \right) \)-fair. We construct an \(O\left( \log ^3(r)/r \right) \)-fair m-party coin-tossing protocol, tolerating up to t corrupted parties, whenever m is constant and \(t<3m/4\).

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Notes

  1. This was generalized to include all functions that imply non-trivial sampling [1, 40], where the two parties wish to sample correlated values.

  2. The idea is to randomly and secretly choose a special round in which the parties unknowingly get the output of the computation.

  3. Note that computing \(\varepsilon \) might take super-polynomial time. However, as noted by [30], \(\varepsilon \) can be efficiently approximated without a significant loss in security.

  4. Beimel et al. [9] use a slightly more involved technique to distribute defense values to the different subsets of parties, allowing several subsets to be assigned the same output bit, while maintaining the guarantee that the adversary cannot bias the output of the honest parties without guessing the value of the special round \(i^{*}\).

  5. Actually, in our construction, we only call subsets \(\mathcal {J}\), such that \(2h-1\le \left| \mathcal {J}\right| \le t\), protected. This suffices since, if a smaller subset of active parties is left, they have an honest majority and thus can use the defense value of its lexicographically first superset of size \(2h-1\).

  6. Here we let \(\Delta _{r+1}\left( y \right) =1\) if \(Y_r\ge 0\) and 0 otherwise, and we let \(\Delta _{r+1}\left( y,d \right) =\Delta _{r+1}\left( y \right) \)

  7. Note that in the case where \(|\mathcal {J}|\le 2h-1\), there is an honest majority, and so, in \({\text {MultipartyShareGen}}_{<3/4}\) we could have given them a common bit to reconstruct with full security. We decided to instruct the parties to execute Protocol 3.2 for the sake of simplicity.

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Acknowledgements

We are grateful to Iftach Haitner and Amos Beimel for useful conversations.

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  1. Department of Computer Science, Ariel University, Ariel, Israel

    Bar Alon & Eran Omri

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  1. Bar Alon
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Correspondence to Eran Omri.

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Communicated by Rafail Ostrovsky.

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This work was supported by ISF Grant 544/13 and by the Ariel Cyber Innovation Center in conjunction with the Israel National Cyber directorate in the Prime Minister’s Office. A preliminary version of this work appeared in [2].

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Alon, B., Omri, E. Almost-Optimally Fair Multiparty Coin-Tossing with Nearly Three-Quarters Malicious. J Cryptol 36, 24 (2023). https://doi.org/10.1007/s00145-023-09466-2

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