Random selection with an adversarial majority

Version History: Full version published in ECCC TR 06-026, February 2006. Updated full version published June 2006.

We consider the problem of random selection, where \(p\) players follow a protocol to jointly select a random element of a universe of size \(n\). However, some of the players may be adversarial and collude to force the output to lie in a small subset of the universe. We describe essentially the first protocols that solve this problem in the presence of a dishonest majority in the full-information model (where the adversary is computationally unbounded and all communication is via non-simultaneous broadcast). Our protocols are nearly optimal in several parameters, including the round complexity (as a function of \(n\)), the randomness complexity, the communication complexity, and the tradeoffs between the fraction of honest players, the probability that the output lies in a small subset of the universe, and the density of this subset.