Is there a best-known implementation for a Number Theoretic Transform on the 257 (2^8 + 1) finite field?

Question

I'm a bit of a novice when it comes to implementing FFTs in general, but I have most of the basic ideas down I think. In this specific case, I've an implementation of the number theoretic transform on the 257 finite field. It's basically your typical Radix-2 Cooley-Tukey FFT. What Id like to know is either: is there a good alternative to the Cooley-Tukey Radix-2 that's better suited to doing this particular NTT efficiently (if the answer is an unqualified yes or a yes conditional on something not entirely within the scope of this question, I'm interested in hearing about either), or are there things specific to a Mersenne NTT that allow for a more efficient implementation than a more general case?

Answer 1

I'd say that for dyadic length FFT there is nothing better than Cooley-Tukey.

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This has nothing directly to to with Mersenne numbers, any number field with modulus 2^(m*2^n)+1 qualifies. I=2^(m*2^(n-1)) is the complex unit, I^2=2^(m*2^n)=-1 mod (2^(m*2^n)+1), and q=2^(2*m) is a primitive 2^n-th root of unity.

For inspiration for the second point see Section 1 of Schönhage: Asymtotically fast algorithms for the numerical multiplication ..., with overall summary of fast multiplications