Need help understanding this exercise's solution
OK, so this takes a bit of guessing, but I'm assuming the following:


A is the identity of A, which can be used to select the right public key of A;
Kxa() is a signing operation (with message recovery) that signs the random Ra and a tag Ts which is used as proof;
Kya() is an encryption operation with the public key A, so that the session key Ks is kept confidential;
Kxb() is a signing operation (with message recovery) that signs the random Ra together with session key Ks;
Ks(M) is message encryption with the established session key Ks.


The general idea seems to be to perform authentication of both parties and session key establishment for message transport security.