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

An $$\alpha $$α-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 $$\alpha $$α. Cleve [STOC 1986] has shown that if half of the parties can be corrupted, then, no $$r$$r-round coin-tossing protocol is $$o1/r$$o1/r-fair. For over two decades the best known m-party protocols, tolerating upi¾?to $${t}\ge m/2$$ti¾?m/2 corrupted parties, were only $$O\left {t}/\sqrt{r} \right $$Ot/r-fair. In a surprising result, Moran, Naor, and Segev [TCC 2009] constructed an $$r$$r-round two-party $$O1/r$$O1/r-fair coin-tossing protocol, i.e., an optimally fair protocol. Beimel, Omri, and Orlov [Crypto 2010] extended the result of Moran et al.i¾?to the multiparty setting where strictly fewer than 2/3 of the parties are corrupted. They constructed a $$2^{2^k}/r$$22k/r-fair r-round m-party protocol, tolerating upi¾?to $$t=\frac{m+k}{2}$$t=m+k2 corrupted parties. Recently, in a breakthrough result, Haitner and Tsfadia [STOC 2014] constructed an $$O\left \log ^3r/r \right $$Olog3r/r-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 $$t=2m/3$$t=2m/3, where $$m>3$$m>3 were $$\theta \left 1/\sqrt{r} \right $$i¾?1/r-fair. We construct an $$O\left \log ^3r/r \right $$Olog3r/r-fair m-party coin-tossing protocol, tolerating upi¾?to t corrupted parties, whenever m is constant and $$t<3m/4$$t<3m/4.