James Aspnes. Lower bounds for distributed coin-flipping and randomized consensus. Journal of the Association for Computing Machinery 45(3):415–450, May 1998. An earlier version appeared in Twenty-Ninth Annual ACM Symposium on Theory of Computing, May 1997, pp. 559–568.
We examine a class of collective coin-flipping games that arises from randomized distributed algorithms with halting failures. In these games, a sequence of local coin flips is generated, which must be combined to form a single global coin flip. An adversary monitors the game and may attempt to bias its outcome by hiding the result of up to t local coin flips. We show that to guarantee at most constant bias, Ω(t²) local coins are needed, even if (a) the local coins can have arbitrary distributions and ranges, (b) the adversary is required to decide immediately whether to hide or reveal each local coin, and (c) the game can detect which local coins have been hidden. If the adversary is permitted to control the outcome of the coin except for cases whose probability is polynomial in t, Ω(t²/log² t) local coins are needed. Combining this fact with an extended version of the well-known Fischer-Lynch-Paterson impossibility proof of deterministic consensus, we show that given an adaptive adversary, any t-resilient asynchronous consensus protocol requires Ω(t²/log² t) local coin flips in any model that can be simulated deterministically using atomic registers. This gives the first non-trivial lower bound on the total work required by wait-free consensus and is tight to within logarithmic factors.
@article(Aspnes1998coin,
title="Lower bounds for distributed coin-flipping and randomized consensus",
author="James Aspnes",
journal={Journal of the ACM},
volume=45,
number=3,
pages={415--450},
month=may,
year=1998
)