You want to auto login Host B from Host A
Host A Look in your ~/.ssh directory. There should be two files, id_rsa and id_rsa.pub. If not, create them using ssh-keygen -t rsa.
Host B Append local id_rsa.pub to the Host B ~/.ssh/authorized_keys.
The file mode of ~/.ssh/authorized_keys must be 644. You can assure this with chmod 644 ~/.ssh/authorized_keys
You might have to change your /etc/sshd_config and add the following lines: