I like the fast fourier transform method for fast integer multiplication. Only applicable on really big inputs, but the idea is that it computes the fft of both integers, does point-wise multiplication of the resulting vectors, then does the inverse fft to recover the product. More about this (https://en.wikipedia.org/wiki/Multiplication_algorithm#Fouri...).

Another interesting but asymptotically slower integer/polynomial multiplication algorithm is Karatsuba's.