  Mutual learning of a pair of tree parity machines with continuous and
discrete weight vectors is studied analytically. The analysis is based on a
mapping procedure that maps the mutual learning in tree parity machines onto
mutual learning in noisy perceptrons. The stationary solution of the mutual
learning in the case of continuous tree parity machines depends on the learning
rate where a phase transition from partial to full synchronization is observed.
In the discrete case the learning process is based on a finite increment and a
full synchronized state is achieved in a finite number of steps. The
synchronization of discrete parity machines is introduced in order to construct
an ephemeral key-exchange protocol. The dynamic learning of a third tree parity
machine (an attacker) that tries to imitate one of the two machines while the
two still update their weight vectors is also analyzed. In particular, the
synchronization times of the naive attacker and the flipping attacker recently
introduced in [1] are analyzed. All analytical results are found to be in good
agreement with simulation results.
