Perfect Information Leader Election in log*n + O(1) Rounds

In the leader election problem, n players wish to elect a random leader. The difficulty is that some coalition of players may conspire to elect one of its own members. We adopt the perfect information model: all communication is by broadcast, and the bad players have unlimited computational power. Protocols proceed in rounds: though players are synchronized between rounds, within each round the bad players may wait to see the inputs of the good players. A protocol is called resilient if a good leader is elected with probability bounded away from 0. We give a simple, constructive leader election protocol that is resilient against coalitions of size βn, for any β < 1/2. Our protocol takes log∗ n+O(1) rounds, each player sending at most logn bits per round. For any constant k, our protocol can be modified to take k rounds and offer resilience against coalitions of size n/(log n), where is a small enough constant and log denotes the logarithm iterated k times. This is constructive for k ≥ 3. The primary component of the above protocols is a new collective sampling protocol: for a set S of large enough (polynomial) size, this protocol generates an element s ∈ S in a single round so that for any subset T ⊂ S, Pr[s ∈ T ] ≤ |T | |S| for a constant α > 0.