An Optimally Fair Coin Toss

We address one of the foundational problems in cryptography: the bias of coin-flipping protocols. Coin-flipping protocols allow mutually distrustful parties to generate a common unbiased random bit, guaranteeing that even if one of the parties is malicious, it cannot significantly bias the output of the honest party. A classical result by Cleve (Proceedings of the 18th annual ACM symposium on theory of computing, pp 364–369, 1986) showed that for any two-party $$r$$r-round coin-flipping protocol there exists an efficient adversary that can bias the output of the honest party by $$\varOmega (1/r)$$Ω(1/r). However, the best previously known protocol only guarantees $$O(1/\sqrt{r})$$O(1/r) bias, and the question of whether Cleve’s bound is tight has remained open for more than 20 years. In this paper, we establish the optimal trade-off between the round complexity and the bias of two-party coin-flipping protocols. Under standard assumptions (the existence of oblivious transfer), we show that Cleve’s lower bound is tight: We construct an $$r$$r-round protocol with bias $$O(1/r)$$O(1/r).