Fast algorithms for computing one-and two-dimensional convolution in integer polynomial rings

Abstract

In a recent work, the factorization properties of polynomials defined over finite integer polynomial rings were analyzed. These properties, along with other results pertaining to polymomial theory, led to the direct sum property and the American-Indian-Chinese extension of the Chinese remainder theorem over such integer rings. The objective of this paper is to describe algorithms for computing the one- and two-dimensional convolution of data sequences defined over finite integer rings. For one-dimensional convolution, algorithms for computing acyclic and cyclic convolution are described. For two-dimensional convolution, only the cyclic case is analyzed. Computational and other relevant aspects associated with the structure of these algorithms are also studied.