KeyGen for cryptographic algorithms

After studying cryptography and algebra from their theoretical aspects, I am now trying to implement some algorithms.

More specifically, I am stuck with the KeyGen part: I need to generate Cyclic Groups of order q (where q is prime), Finite Fields of order q, and, more importantly, I need to be able to pick random elements from these groups.

There are many open source programs implementing cyprographic functions. However, about all of them require as input the fields, groups, and random numbers ... which I don't know how to generate.

Since the final scope is an implementation for micro-controllers, I would appreciate any directions in C code.

Thank you in advance,

Best regards