SSH login without password

You want to use Linux and OpenSSH to automate 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@B:~> ssh-copy-id b@A
b@A's password: 

From now on you can log into A as a from B as a without password:

a@B:~> ssh b@A