The authors explore an intersection of number theory, digital signal processing and coding theory. Focusing on the design of number theory based on computationally efficient digital signal processing algorithms, error control techniques and their relationships over a ring of integers, this is the first book to focus on algorithms for processing of data sequences defined over finite integer rings, rather than defined over a field.The authors have developed a deep understanding of the fundamental role that number theory plays in the design of fast algorithms for performing computationally intensive tasks in digital signal processing. This understanding led them to derive a significant body of new research results in number theory, digital signal processing, residue number systems, and error control coding techniques. They present these results-with necessary background theory-for mathematicians, computer scientists, and engineers working in number theory, computer arithmetic, fault-tolerant computing, digital signal processing, VLSI design, and coding theory.