This is a followup to this question.

The accepted answer is generally sufficient, but requires user to supply personal information (e.g. name) for generating the key.

I'm wondering if it's possible to generate different keys based on a common seed, in a way that program would be able to validate if those keys belong to particular product, but without making this process obvious to the end user.

I mean it could be a hash of product ID plus some random sequence of characters, but that would allow user to guess potential new keys. There should be some sort of algorithm difficult to guess.