Improved Bounds for Coin Flipping, Leader Election, and Random Selection

Random selection, leader election, and collective coin flipping are fundamental tasks in fault-tolerant distributed computing. We study these problems in the full-information model where despite decades of study, key gaps remain in our understanding of the trade-offs between round complexity, communication per player in each round, and adversarial resilience. We make progress by proving improved bounds for these problems. We first show that any $k$-round coin flipping protocol over $\ell$ players, each player sending one bit per round, can be biased by $O(\ell/\log^{(k)}(\ell))$ bad players. We obtain a similar lower bound for leader election. This strengthens prior best bounds [RSZ, SICOMP 2002] of $O(\ell/\log^{(2k-1)}(\ell))$ for coin flipping protocols and $O(\ell/\log^{(2k+1)}(\ell))$ for leader election protocols. Our result implies that any (1-bit per player) protocol tolerating linear fraction of bad players requires at least $\log^* \ell$ rounds, showing existing protocols [RZ, JCSS 2001; F, FOCS 1999] are near-optimal. We next initiate the study of one-round, (1-bit per player) random selection. We construct a protocol resilient to $\ell / (\log \ell)^2$ bad players that outputs $(\log \ell)^2 / (\log \log \ell)^2$ uniform random bits. This implies a one-round leader election protocol resilient to $\ell / (\log \ell)^2$ bad players, improving the prior best protocol [RZ, JCSS 2001] which was resilient to $\ell / (\log \ell)^3$ bad players. Our resilience matches that of the best one-round coin flipping protocol by Ajtai & Linial. We also obtain an almost matching lower bound: any protocol outputting $(\log \ell)^2 / (\log \log \ell)^2$ bits can be corrupted by $\ell (\log \log \ell)^2 / (\log \ell)^2$ bad players. To obtain our lower bound, we introduce multi-output influence, an extension of influence of boolean functions to the multi-output setting.