  Leader election between n parties is known to be impossible classically. This
work gives a simple algorithm that does it, based on the weak coin flipping
protocol with arbitrarily small bias derived by Mochon in 2007, and recently
published and simplified in Aharonov et al in 2016. A protocol with linear
number of coin flipping rounds is quite simple to achieve; We further provide
an improvement to logarithmic number of coin flipping rounds. This is a much
improved journal version of a preprint posted in 2009: many typos are
corrected, proofs are more rigorous and accurate, and the analysis of
complexity is significantly tightened which gives a stronger result in terms of
efficiency; The first protocol with linear number of rounds, was achieved
independently also by Aharon and Silman in 2010.
