I don't know how far you have gotten, but I think it is a good idea to start with FFT-based fast multiplication and read up on whatever unknowns you encounter. In particular I think some abstract algebra (e.g. Dummit-Foote) would be helpful.

There are tons of good expositions on the FFT-way, e.g. check out

"Prime Numbers - A Computational Perspective" by Crandall-Pomerance see ch. 9, Fast algorithms for large-integer arithmetic