^June July – 2013

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 which returns the output of the functionality to all parties. In particular, in the ideal model such computation is fair – if the corrupt 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 construct $1/p$-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 corrupt parties provided that the number of parties is constant. 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. Joint work with Amos Beimel, Yehuda Lindell, and Eran Omri.