SSH login without password

Your aim

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

How to do it

First log in on A as user a and generate a pair of authentication keys. Do not enter a passphrase:

a@A:~> 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 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
B
A

note from one of our readers: Depending on your version of SSH
you might also have to do the following changes:

Put the public key in .ssh/authorized_keys2  
Change the permissions of .ssh to 700  
Change the permissions of .ssh/authorized_keys2 to 640