Fast fourier transform multiplication

1. How is FFT used in multiplication?
2. How many multiplies in an FFT?
3. What is the fastest multiplication algorithm?

How is FFT used in multiplication?

This algorithm is known as Fast Fourier Transform.

To multiply two polynomials in value representation, we just do a pairwise multiplication of the function evaluations on each point, where pairwise multiplication means multiplying the vectors in pairs, element by element, which is very cheap computationally speaking.

How many multiplies in an FFT?

Each pair requires 4 additions and 4 multiplications, giving a total number of computations equaling 8N4=N2. This number of computations does not change from stage to stage. Because the number of stages, the number of times the length can be divided by two, equals log2N, the complexity of the FFT is O(NlogN).

What is the fastest multiplication algorithm?

Karatsuba's algorithm was the first known algorithm for multiplication that is asymptotically faster than long multiplication, and can thus be viewed as the starting point for the theory of fast multiplications.