SSH login without password

Sometimes users want to automatic login from host A to Host B without
entering any password.
For scheduling any script, like to scp from one server another server
without entering any password we required ssh login without password.
So to do this we need to create key.

a=user1, b=user2, A=server1, B=server2

Note: User should be SAME on both servers

 
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