Q1. I do not understand why e should be public? It may be more secure to keep it private and known only to the sender and receiver.
Q2. I need comments on the following proposed algorithm: Both sender and receiver have their own encryptors and decryptors $(e_1,d_1)$ and $(e_2,d_2)$. Only p is known in public.
i- A message $m$ is sent as $m^{e_1}~mod~p.$
ii- It is returned as $m^{e_1\ast e_2}~mod~p.$
iii- It is sent back as $m^{e_1\ast e_2\ast d_1}=m^{e_2}~mod~p.$
iv It is decrypted as $m^{e_2\ast d_2}~mod~p$
Thank you for your help.