Paper 2018/901

On the Complexity of Fair Coin Flipping

Iftach Haitner, Nikolaos Makriyannis, and Eran Omri

Abstract

A two-party coin-flipping protocol is $ϵ$-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 $ϵ$. Cleve [STOC '86] showed that $r$-round $o(1/r)$-fair coin-flipping protocols do not exist. Awerbuch et al. [Manuscript '85] constructed a $\mathrm{\Theta }(1/\sqrt{r})$-fair coin-flipping protocol, assuming the existence of one-way functions. Moran et al. [Journal of Cryptology '16] constructed an $r$-round coin-flipping protocol that is $\mathrm{\Theta }(1/r)$-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 $$-fair coin flipping. This question was partially answered by Dachman-Soled et al. [TCC '11] and Dachman-Soled et al. [TCC '14], who showed that restricted types of fully black-box reductions cannot establish $$-fair coin-flipping protocols from one-way functions. In particular, for constant-round coin-flipping protocols, Dachman-Soled et al. showed that black-box techniques from one-way functions can only guarantee fairness of order $$. We make progress towards answering the above question by showing that, for any constant $$, the existence of an $$-fair, $$-round coin-flipping protocol implies the existence of an infinitely-often key-agreement protocol, where $$ denotes some universal constant (independent of $$). 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 recent attack of Beimel et al. [FOCS '18] on multi-party coin-flipping protocols.