Your Goal

You want to use Linux and OpenSSH to automize your tasks. Therefore you need an automatic login from Host-1/user-A to Host-2/user-A. You don't want to enter any passwords, because you want to call ssh from within a shell script.

How To Accomplish It

First log in onto Host-1 as user-A and generate a pair of authentication keys. Do not enter a passphrase:

ssh-keygen -t rsa

Generating public/private rsa key pair.

Enter file in which to save the key (/home/a/.ssh/id_rsa): 

Created directory '/home/a/.ssh'.

Enter passphrase (empty for no passphrase): 

Enter same passphrase again: 

Your identification has been saved in /home/a/.ssh/id_rsa.

Your public key has been saved in /home/a/.ssh/id_rsa.pub.

The key fingerprint is:

3e:4f:05:79:3a:9f:96:7c:3b:ad:e9:58:37:bc:37:e4 a@A

Now use ssh to create a directory ~/.ssh as user-A on Host-2. (The directory may already exist, which is fine):

ssh user-A@localhost mkdir -p .ssh

user-A@localhost's password:

Finally append user-A's new public key to user-A@Host-2:.ssh/authorized_keys and enter user-A's password one last time:

cat .ssh/id_rsa.pub | ssh user-A@Host-2 'cat >> .ssh/authorized_keys'

user-A@Host-2's password:

From now on you can log into Host-2 as user-A from Host-1 as user-A without a password:

ssh user-A@Host-2 hostname

user-A@Host-2:~>