SSH 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.
 Yo ur 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: 

If you run ssh on a different port, add it to the ~/.ssh/config file

Host *

 AddKeysToAgent yes
 UseKeychain yes
 IdentityFile ~/.ssh/id_ed25519
 Port 2046

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