## Abstract

A protocol for computing a functionality is secure if an adversary in this protocol cannot cause more harm than in an ideal computation, where parties give their inputs to a trusted party that returns the output of the functionality to all parties. In particular, in the ideal model, such computation is fair—if the corrupted parties get the output, then the honest parties get the output. Cleve (STOC 1986) proved that, in general, fairness is not possible without an honest majority. To overcome this impossibility, Gordon and Katz (Eurocrypt 2010) suggested a relaxed definition—1/*p*-secure computation—which guarantees partial fairness. For two parties, they constructed 1/*p*-secure protocols for functionalities for which the size of either their domain or their range is polynomial (in the security parameter). Gordon and Katz ask whether their results can be extended to multiparty protocols. We study 1/*p*-secure protocols in the multiparty setting for general functionalities. Our main result is constructions of 1/*p*-secure protocols that are resilient against *any* number of corrupted parties provided that the number of parties is constant and the size of the range of the functionality is at most polynomial (in the security parameter \({n}\)). If fewer than 2/3 of the parties are corrupted, the size of the domain of each party is constant, and the functionality is deterministic, then our protocols are efficient even when the number of parties is \(\log \log {n}\). On the negative side, we show that when the number of parties is super-constant, 1/*p*-secure protocols are not possible when the size of the domain of each party is polynomial. Thus, our feasibility results for 1/*p*-secure computation are essentially tight. We further motivate our results by constructing protocols with stronger guarantees: If in the execution of the protocol there is a majority of honest parties, then our protocols provide full security. However, if only a minority of the parties are honest, then our protocols are 1/*p*-secure. Thus, our protocols provide the best of both worlds, where the 1/*p*-security is only a fall-back option if there is no honest majority.

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## Notes

- 1.
A non-rushing adversary decides upon its action in each round, only given its view in all previous rounds. A rushing adversary is one that in each round of the protocol can wait for all honest parties to send their messages and decide upon its reaction, depending also on these messages.

- 2.
Cohen et al. [21] showed that broadcast is necessary, even for the elementary task of coin-tossing with non-trivial bias and even when up to two-thirds of the parties are honest.

- 3.
For the simplicity of the presentation of our protocols, we present a slightly different ideal world than the traditional one. In our model there is no default input in the case of an “abort.” However, the protocol can be presented in the traditional model, where a predefined default input is used if a party aborts.

- 4.
- 5.
Furthermore, the adversary might have some auxiliary information on the inputs of the honest parties; thus, the adversary might be able to deduce that a round is not \(i^\star \) even if all the values that it gets are equal, however they are not equal to a “correct” output.

- 6.
For a randomized functionality, this probability also depends on the size of the range.

- 7.
In [11], the number of parties may be polynomial in the security parameter. Thus, to keep the preprocessing phase constant round, there, the compilation into a secure with identifiable abort preprocessing protocol follows through using the zero knowledge proofs of [45]. This requires assuming the existence of collision resistant hash functions on top of the assumption that enhanced trapdoor permutations exist.

- 8.
These shares are temporary and will later be opened for the actual values during the interaction rounds using the properties of Shamir’s secret-sharing scheme.

- 9.
In Steps (2)–(5), the simulator \({{\mathcal {S}}}\) constructs the messages of the honest parties in order to allow the corrupted parties in each \({L}\in {{\mathcal {J}}}\) to reconstruct \({ \tau _{{L}}^{i} }\).

- 10.
For example, there might not be possible inputs of the corrupted parties causing the honest parties to output such output.

- 11.
For example, there might not be possible inputs of the corrupted parties that together with inputs of the honest parties result in such output.

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Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

A preliminary version of this work appeared in CRYPTO 2011 [9].

Amos Beimel: Generously supported by ISF Grant 938/09 and by the Frankel Center for Computer Science.

Yehuda Lindell: Generously supported by the European Research Council as part of the ERC project LAST, and by the Israel science foundation (Grant No. 781/07).

Eran Omri: Ariel Cyber Innovation Center. Generously supported by the European Research Council as part of the ERC project LAST, and by the Israel science foundation (Grant No. 781/07).

Ilan Orlov: Generously supported by ISF Grant 938/09 and by the Frankel Center for Computer Science.

Communicated by Jonathan Katz.

## Appendix A: Proof of Lemma 2.6

### Appendix A: Proof of Lemma 2.6

### Proof

Fix \(D_1,D_2\) satisfying Inequality (1). We prove the lemma by induction on \({r}\). When \({r}=1\) the lemma is trivially true; Assume \({\text {win}}({r})\le 1/\alpha {r}+ \beta \); we upper-bound \({\text {win}}({r}+1)\). As \({{\mathcal {A}}}\) is unbounded, we can assume without loss of generality that \({{\mathcal {A}}}\) is deterministic. Let *S* be the set in the support of \(D_2\) such that \({{\mathcal {A}}}\) aborts in the first iteration if and only if \(a_1\in S\). We define \(S_h\) as all the elements \(z\in S\) s.t. \(\Pr _{a\leftarrow D_1}[a=z] \ge \alpha \Pr _{a\leftarrow D_2}[a=z]\) holds for them and \(S_\ell = S {\setminus } S_h\). Observe that \(\Pr _{a\leftarrow D_2}[a_1\in S_\ell ]\le \beta \). If \({{\mathcal {A}}}\) does *not* abort in the first iteration, and the game does not end, then the conditional distribution of \(i^\star \) is uniform in \(\left\{ 2,\ldots ,{r}\right\} \) and the game \(\Gamma ({r}+1)\) from this point forward is exactly equivalent to the game \(\Gamma ({r})\). In particular, conditioned on the game \(\Gamma ({r}+1)\) not ending after the first iteration, the probability that \({{\mathcal {A}}}\) wins is at most \({\text {win}}({r})\). We thus have

\(\square \)

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Beimel, A., Lindell, Y., Omri, E. *et al.* \({\varvec{1/p}}\)-Secure Multiparty Computation without an Honest Majority and the Best of Both Worlds.
*J Cryptol* **33, **1659–1731 (2020). https://doi.org/10.1007/s00145-020-09354-z

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### Keywords

- Secure multiparty computation
- 1/p-Security
- Partial-fairness
- Best-of-both-worlds