Consider we want to login from a MacBook Pro (mbp) to a remote Linux machine (tsingyi), where both computers have OpenSSH installed. In order to make tsingyi trust mbp, we use RSA cryptographic method to generate a public and a private key for mbp, which will be used to identify mbp during login.
To generate the pair of keys, on mbp, type
ssh-keygen -t rsa
Accept all default answers, and we get two files:
~/.ssh/id_rsa --- the private key
~/.ssh/id_rsa.pub --- the public key
Now, copy the public key file to tsingyi by typing following command on mbp:
scp ~/.ssh/id_rsa.pub wyi@tsingyi:/home/wyi/.ssh/id_rsa-mbp.pub
and add the public key of mbp to ~/.ssh/authorized_keys of tsingyi by typing following command on tsingyi:
cat ~/.ssh/id_rsa-mbp.pub >> ~/.ssh/authorized_keys
Here we are. We should be able to ssh to tsingyi from mbp without typing password now.