In this paper we develop a leader election protocol P with the following features: 1. The protocol runs in the perfect information model: Every step taken by a player is visible to all others. 2. It has linear immunity: If P is run by n players and a coalition of c(1)n players deviates from the protocol, attempting to have one of them elected, their probability of success is < 1 - c(2), where c(1), c(2) > 0 are absolute constants. 3. It is fast: The running time of P is polylogarithmic in n, the number of players. A previous protocol by Alon and Naor achieving linear immunity in the perfect information model has a linear time complexity. The main ingredient of our protocol is a reduction subprotocol. This is a way for n players to elect a subset of themselves which has the following property. Assume that up to epsilon n of the players are bad and try to have as many of them elected to the subset. Then with high probability, the fraction of bad players among the elected ones will not exceed epsilon in a significant way. The existence of such a reduction protocol is first established by a probabilistic argument. Later an explicit construction is provided which is based on the spectral properties of Ramanujan graphs.
