Rings, fields, the Chinese remainder theorem and an extension-PartI: theory

摘要

The much celebrated Chinese Remainder Theorem has been widelynemployed in designing fast computationally efficient algorithms in thenfield of digital signal processing. It has two versions. One is over anring of integers and the second is over a ring of polynomials withnCoefficients defined over a field. In this research work, we extend thenChinese Remainder Theorem to the case of a ring of polynomials withncoefficients defined over a finite ring of integers. The entire work isnclosely related to the already established results on finite fields.nThis extension is expected to serve as a keystone in the future designnof number-theoretic algorithms for performing some of the mostncomputationally intensive tasks. This approach is superior to thennumber-theoretic-transforms in the sense that the limitations on bothnthe word length and the sequence length are completely removed. In fact,nthe number-theoretic-transforms may be considered as a very special casenof our general approach. Furthermore, the computations required in thisnwork. Which inherits all the merits of the Chinese Remainder Theorem,ncan be performed in parallel