On the Complexity of Fair Coin Flipping

A two-party coin-flipping protocol is epsilon-fair if no efficient adversary can bias the output of the honest party (who always outputs a bit, even if the other party aborts) by more than epsilon. Cleve [STOC '86] showed that r-round o(1/r)-fair coin-flipping protocols do not exist. Awerbuch et al. [Manuscript '85] constructed a circle minus(1/root r)-fair coinflipping protocol, assuming the existence of one-way functions. Moran et al. [Journal of Cryptology '16] constructed an r-round coin-flipping protocol that is circle minus(1/r)-fair (thus matching the aforementioned lower bound of Cleve [STOC '86]), assuming the existence of oblivious transfer. The above gives rise to the intriguing question of whether oblivious transfer, or more generally "public-key primitives", is required for an o(1/root r)-fair coin flipping. This question was partially answered by Dachman-Soled et al. [TCC '11] and Dachman-Soled et al. [TCC '14], who showed that restricted types of fully black-box reductions cannot establish o(1/root r)-fair coin-flipping protocols from one-way functions. In particular, for constant-round coin-flipping protocols, [10] yields that black-box techniques from one-way functions can only guarantee fairness of order 1/root r. We make progress towards answering the above question by showing that, for any constant r is an element of N, the existence of an 1/(c center dot root r)-fair, r-round coin-flipping protocol implies the existence of an infinitely-often key-agreement protocol, where c denotes some universal constant (independent of r). Our reduction is non black-box and makes a novel use of the recent dichotomy for two-party protocols of Haitner et al. [FOCS '18] to facilitate a two-party variant of the attack of Beimel et al. [FOCS '18] on multi-party coin-flipping protocols.