Cyclic convolution of long sequences using number theoretic transform

In this paper, number theoretic transforms (NTT) are examined and expressed in a way that facilitates the computation of the convolution sum of two one-dimensional sequences each of lengthN in two-dimensional (or multidimensional) form. The number of multiplications per point is shown to be always less than two whenN1 andN2 are not mutually prime, and this figure is reduced to one multiplication per point whenN1, andN2 are mutually prime and a nested NTT is used. For the convolution of two-dimensional arrays, the number of multiplications per point turns out to be always less than four whenN1, andN2 are not mutually prime, and the number of multiplications remains as one per point whenN1 andN2 are mutually prime.