SSH Public Key Format

As already mentioned by pulegium, any generic configuration management software like Puppet, Chef, Bcfg2 or cfengine could accomplish the task.

Since the authorized_keys file is not that complicated, you could also use rsync or a (D)SCM like git or hg to manage this file. You have the "master" file on one of your servers and serve it via rsync/git/hg/…. On every other server you run a cron job which periodically retrieves the master copy (if it was changed) and copies it to the correct local location. Heck, this would even work with pure HTTP or FTP.

The bottom line is: Have one "master" copy of your authorized_keys file and update it. Let the "clients" (the computers, which should have the current authorized_keys file) fetch it from your master server and deploy it locally.

in order to connect to an SSH server and authenticate with your public/private keypair you have to first share your public key with the server.

this is done by copying the public key for your private key to the server, and adding it to ~/ssh/authorized_keys either by copy/paste, copying id_rsa.pub to ~/.ssh/authorized_keys on the server or with cat id_rsa.pub >> ~/.ssh/authorized_keys, appending it to the list.

when you connect, the server uses your public key to sign a challenge, and your client uses your private key id_rsa to decrypt the challenge, re-encrypt it with the server's public host key and send it back.

the host verifies that you decrypted the challenge properly, by decrypting your response with its private key, and the client/host establish an encrypted connection, based on the shared data, not on your public/private keys.

at NO POINT in the exchange is your private key, or the host's private key exchanged or revealed to one another. your public key IS stored on the server, but that's why it is a PUBLIC key.