I'm researching on different variants of Number Theoretic Transform (NTT) and some concepts just got me confused.
What's the difference between Cooley-Tukey transform and Gentleman-Sande transform? My understanding is that both CT and GS transform are just a way to calculate NTT/INTT, so they may use different strategy to apply divide and conquer, their outputs however, should be the same. I see lots of implementation use both of them: they use CT for NTT and GS for INTT. So it's a bit confusing.
When you do NTT/INTT on module ring like $\mathbb{Z}_q[X]/(X^N\pm1)$, you can also apply some technique like cyclic convolution (for mod $X^N-1$) or negative wrapped convolution (NWC-NTT, for mod $X^N+1$). What's the difference of them and why they apply to different quotient ring? What's the relationship between these two techniques with CT & GS butterflies?