Fast algorithms for 2-D circular convolutions and number theoretic transforms based on polynomial transforms over finite rings

We develop new fast algorithms for 2-D integer circular convolutions and 2-D number theoretic transforms (NTT). These new algorithms, which offer improved computational complexity, are constructed based on polynomial transforms over Z_p; these transforms are Fourier-like transforms over Z_p, which is the integral domain of polynomial forms over Z_p[x]. Having defined such polynomial transforms over Z_p we prove several necessary and sufficient conditions for their existence. We then apply the existence conditions to recognize two applicable polynomial transforms over Z_p. One is for p equal to Mersenne numbers and the other for Fermat numbers. Based on these two transforms, referred to as Mersenne number polynomial transforms (MNPT) and Fermat number polynomial transforms (FNPT), we develop fast algorithms for 2-D integer circular convolutions, 2-D Mersenne number transforms, and 2-D Fermat number transforms. As compared to the conventional row-column computation of 2-D NTT for 2-D integer circular convolutions and 2-D NTT, the new algorithms give rise to reduced computational complexities by saving more than 25 or 42% in numbers of operations for multiplying 2 ⁱ, i⩾1; these percentages of savings also grow with the size of the 2-D integer circular convolutions or the 2-D NTT