Algorithm 1 Iterative Number Theoretic Transform

Require: A polynomial $a\left(x\right)\in {\mathbb{Z}}_q\left[x\right]$ of degree $n-1$ and n-th primitive $\omega \in {\mathbb{Z}}_q$ of unity

Ensure: Polynomial $a\left(x\right)=NTT\left(a\right)\in {\mathbb{Z}}_q\left[x\right]$

1: $a=BitReverse\left(a\right)$

2: for i from 2 by $i=2i$ to n do

3:  ${\omega }_i={\omega }_n^{n/i}$, $\omega =1$

4:  for j from 0 by 1 to $i/2-1$ do

5:   for k from 0 by i to $n-1$ do

6:    $U=a[k+j]$

7:    $V=\omega ·a[k+j+i/2]$

8:    $a[k+j]=U+V$

9:    $a[k+j+i/2]=U-V$

10:   $\omega =\omega ·{\omega }_i$

11: return a