RINGS, FIELDS, THE CHINESE REMAINDER THEOREM AND AN EXTENSION .1. THEORY

The much celebrated Chinese Remainder Theorem has been widely employed in designing fast computationally efficient algorithms in the field o f digital signal processing. It has two versions. One is over a ring o f integers and the second is over a ring of polynomials with coefficie nts defined over a field. In this research work, we extend the Chinese Remainder Theorem to the case of a ring of polynomials with coefficie nts defined over a finite ring of integers. The entire work is closely related to the already established results on finite fields. This ext ension is expected to serve as a keystone in the future design of numb er-theoretic algorithms for performing some of the most computationall y intensive tasks. This approach is superior to the number-theoretic-t ransforms in the sense that the limitations on both the word length an d the sequence length are completely removed. In fact, the number-theo retic-transforms may be considered as a very special case of our gener al approach. Furthermore, the computations required in this work, whic h inherits all the merits of the Chinese Remainder Theorem, can be per formed in parallel.