  Mutual learning process between two parity feed-forward networks with
discrete and continuous weights is studied analytically, and we find that the
number of steps required to achieve full synchronization between the two
networks in the case of discrete weights is finite. The synchronization process
is shown to be non-self-averaging and the analytical solution is based on
random auxiliary variables. The learning time of an attacker that is trying to
imitate one of the networks is examined analytically and is found to be much
longer than the synchronization time. Analytical results are found to be in
agreement with simulations.
