  In this paper we study population protocols governed by the {\em random
scheduler}, which uniformly at random selects pairwise interactions between $n$
agents. The main result of this paper is the first time and space optimal {\em
exact majority population protocol} which also works with high probability. The
new protocol operates in the optimal {\em parallel time} $O(\log n),$ which is
equivalent to $O(n\log n)$ sequential {\em pairwise interactions}, where each
agent utilises the optimal number of $O(\log n)$ states.
  The time optimality of the new majority protocol is possible thanks to the
novel concept of fixed-resolution phase clocks introduced and analysed in this
paper. The new phase clock allows to count approximately constant parallel time
in population protocols.
