Suppose you have a possibly compromised system, which you do not trust enough to stores your private keys on.
Is there some process where you can generate unique bitcoin addresses within the system, that does not contain a private key?
You have a web server that clients use to make orders. You want to create a unique bitcoin address per purchase. In addition, you don't want the server to contain your bitcoin private keys - an attacker gaining access to the server shouldn't be able to touch your bitcoins.