Extra Registration Field – Invite Key

Hello all. I have a different question this time, one that I hope someone can answer fairly quickly.

I want to allow users to register on my wordpress install, but only if they have a valid invite key which I specify.

Basically, the reason I want to do this is as follows:

Currently I sell a premium wordpress theme and directly after purchase the user is sent a user and password by which to login to wordpress with. Upon logging in, they can access the download page for the theme. Obviously, this is not very secure because if one person leaks the user name and password, everyone can easily access the download center. If i was able to set a random invite code and change it every once it in a while, it would make the entire process much more secure.

If anyone knows how I could add a field like this to the registration page I would truly appreciate it.

Thank you!