a@A:~> ssh-keygen -t rsa
Now use ssh to create a directory ~/.ssh as user b on B. (The directory may already exist, which is fine):
a@A:~> ssh b@B mkdir -p .ssh b@B's password:
Finally append a’s new public key to b@B:.ssh/authorized_keys and enter b’s password one last time:
a@A:~> cat .ssh/id_rsa.pub | ssh b@B 'cat >> .ssh/authorized_keys' b@B's password:
From now on you can log into B as b from A as a without password:
a@A:~> ssh b@B hostname
More info at SSH login without password.