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

To turn Cooley-Tukey (complex) FFT into modular arithmetic approach, i.e. NTT, you must replace complex definition for omega. For the approach to be purely recursive, you also need to recalculate omega for each level based on current signal size. This is possible because min. suitable modulus decreases as we move down in the call tree, so modulus used for root is suitable for lower layers. Additionally, as we are using same modulus, the same generator may be used as we move down the call tree. Also, for inverse transform, you should take additional step to take recalculated omega a and instead use as omega: b = a ^ -1 (via using inverse modulo operation). Specifically, b = invMod(a, N) s.t. b * a == 1 (mod N), where N is the chosen prime modulus.

Rewriting an expression involving omega by exploiting periodicity still works in modular arithmetic realm. You also need to find a way to determine the modulus (a prime) for the problem and a valid generator.

We note that your code works, though it is not a MWE. We extended it using common sense, and got correct result for a polynomial multiplication application. You just have to provide correct values of omega raised to certain powers.

While your code works, though, like from many other sources, you double spacing for each level. This does not lead to recursion that is as clean, though; this turns out to be identical to recalculating omega based on current signal size because the power for omega definition is inversely proportional to signal size. To reiterate: halving signal size is like squaring omega, which is like giving doubled powers for omega (which is what one would do for doubling of spacing). The nice thing about the approach that deals with recalculating of omega is that each subproblem is more cleanly complete in its own right.

There is a paper that shows some of the math for modular approach; it is a paper by Baktir and Sunar from 2006. See the paper at the end of this post.

You do not need to extend the cycle from n / 2 to n.

So, yes, some sources which say to just drop in a different omega definition for modular arithmetic approach are sweeping under the rug many details.

Another issue is that it is important to acknowledge that the signal size must be large enough if we are to not have overflow for result time-domain signal if we are performing convolution. Additionally, it may be useful to find certain implementations for exponentiation subject to modulus exist that are fast, even if the power is quite large.

References

• Baktir and Sunar - Achieving efficient polynomial multiplication in Fermat fields using the fast Fourier transform (2006)