If you had C3 on your computer, I presume you already had your key tied to your Origin account. Origin does not require a new registration of a product every time you reinstall your system, games or just the program itself.
Check if you have the game in your Origin library. If you don't, then contact EA support with your problem and be prepared to provide the following information to them: C3 key, picture of reciept, box and disc or a picture of the payment via online banks if you did buy it in electronic way.