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. Within a round, they may also 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 log n bits per round. For any constant k, our protocol can be modified to take k rounds and be resilient against coalitions of size εn(log^(k)n)³ , where ε is a small enough constant and log(k) denotes the logarithm iterated k times. This is constructive for k⩾3