Provably Secure Password Authenticated Key Exchange Based on RLWE for the Post-QuantumWorld In this paper (Provably Secure Password Authenticated Key Exchange Based on RLWE for the Post-Quantum World), author describe password authenticated key exchange scheme on page 9 and 10 (see Fig. 1 on page number 10).     

I have following doubts in my mind    


  
  How initially new user starts communication with the server. Whether there is any registration process for new user? 
  How server gets the client password, as in the protocol client do not send any password. On the contrary, server verifies it as $\gamma$' = -H1(pwc) where pwc is client password. I think Server may ask for password from client during registration phase and then store it in its database but i am not sure.    
  In Fig. 1 on page number 10, client send $&lt;C,m&gt;$ to server. Is 'C' is client's id or sid or something else. As author do not give any description of it.     
  Similarly, S is in session key formation skc= H4(C,S,m,$\mu$,$\sigma$,$\gamma$') in Fig.1. Is S is server id or sid or something else?    
  


Please Guide.
A password authenticated key exchange (PAKE) scheme assumes that the two parties have some shared secret (the password) with too little entropy. They now want to agree on a key and authenticate each other using this password.

The trick is to do this in such a way that an attacker cannot brute force the password with recorded data. This ensures the password remaining secure in spite of the low entropy because you always have to talk to one of the parties to try a guess.

So 1. and 2. are out of scope for PAKE schemes. The password is just assumed to be pre-shared. 3. and 4.: Yes, $\mathcal{C}, \mathcal{S}$ identify the client and the server. These must usually be included  in authentication protocols to avoid tricky relay attacks. I recommend Boyd's book for details.