Lower bounds for leader election and collective coin-flipping in the perfect information model

Collective coin-flipping is the problem of producing common random bits in a distributed computing environment with adversarial faults. We consider the perfect information model: all communication is by broadcast and corrupt players are computationally unbounded. Protocols in this model may involve many asynchronous rounds. We assume that honest players communicate only uniformly random bits. We demonstrate that any n-player coin-flipping protocol that is resilient against corrupt coalitions of linear size must use either at least [1/2 - o(1)]log* n communication rounds or at least [log(2k-1) n]1-o(1) communication bits in the kth round, where log(j) denotes the logarithm iterated j times. In particular, protocols using one bit per round require [1/2 - o(1)]log* n rounds. These bounds also apply to the leader election problem. The primary component of this result is a new bound on the influence of random sets of variables on Boolean functions. Finally, in the one-round case, using other methods we prove a new bound on the influence of sets of variables of size $\beta n$ for $\beta > 1/3$.