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I am trying to create a system where to unlock the application one needs to enter...

I am trying to create a system where to unlock the application one needs to enter a serial code. I have read many articles on the theme but there are two problems bugging me. One is, If I have a keygen and I can generate as many keys as I like then anyone else can do that too, but solving this by requiring a long "private key" there occurs another problem.

Usually serial codes are supposed to be short and easy to type but even the shortest Elliptical curve signature algorithm has a signature size of 160-bits which could not possibly fit into the 100 bits I have. Is it possible to use a threshold approach?

I have seen many similar questions (please do not mark as duplicate) but none have answered my main question as to how can you treat the serial code as a signed document if the length of signature is at least 60 bits more than the maximum allowable size of a serial code?

Could you please give a deeper insight on my question and bare with my lack of knowledge in many areas? Thank you.

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