Some Historical Notes on Number Theoretic Transform

Modulo arithmetic modulo a prime integer have many interesting properties. Such properties are found in standard books on number theory. Some properties are especially of interest to the signal processing application. It was observed analogy exists between some of them and that cyclic convolution of two sequences modulo a prime integer of two sequences could be computed in integer domain as can be done by Fast Fourier Transform using complex real numbers, leading to exactness of the final result (i.e., free of any roundoff errors). These methods, appropriately named as Number Theoretic Transform, are associated with both advantages and disadvantages. These developments in signal processing algorithms took place following the footsteps of developments of Fast Fourier Transform techniques. This paper traverses some of the developments of the Number Theoretic Transform techniques over time and discusses mostly the initial contributions and efforts made by various researchers.