Translation from Complex-FFT to Finite-Field-FFT

Question

Good afternoon!

I am trying to develop an NTT algorithm based on the naive recursive FFT implementation I already have.

Consider the following code (co

Answer 1

You must make sure that roots of unity actually exist. In R there are only 2 roots of unity: 1 and -1, since only for them x^n=1 can be true.

In C you have infinitely many roots of unity: w=exp(2*pi*i/N) is a primitive N-th roots of unity and all w^k for 0<=k

Now to your problem: you have to make sure the ring you're working in offers the same property: enough roots of unity.

Schönhage and Strassen (http://en.wikipedia.org/wiki/Sch%C3%B6nhage%E2%80%93Strassen_algorithm) use integers modulo 2^N+1. This ring has enough roots of unity. 2^N == -1 is a 2nd root of unity, 2^(N/2) is a 4th root of unity and so on. Furthermore, these roots of unity have the advantage that they are powers of two and can be implemented as binary shifts (with a modulo operation afterwards, which comes down to a add/subtract).

I think QuickMul (http://www.cs.nyu.edu/exact/doc/qmul.ps) works modulo 2^N-1.