SSH login without password

Your aim
You want to use Linux and OpenSSH to automate your tasks.
Therefore you need an automatic login from host
A / user b 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@localhost mkdir -p .ssh
b@localhost'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

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