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

摘要

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: although 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 log n bits per round. For any constant k, our protocol can be modified to take k rounds and offer resilience against coalitions of size εn/(log(k) n)3, were ε is a small enough constant and log(k) 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|α (1−β) for a constant α>0.