Computational Number Theory and Digital Signal Processing

Fast Algorithms and Error Control Techniques

Hari Krishna, Bal Krishna, Kuo-Yu Lin, Jenn-Dong Sun

• Publisher: CRC Press
• Year: 2026
• eBook ISBN: 9781040892978
• Topic: Technology & Engineering
• Subtopic: Counting & Numeration
• Index: Technology & Engineering

Military service involves exposure to multiple sources of chronic, acute, and potentially traumatic stress, especially during deployment and combat. Notoriously variable, the effects of stress can be subtle to severe, immediate or delayed, impairing individual and group readiness, operational performance, and—ultimately—survival. A comprehensive compilation on the state of the science, Biobehavioral Resilience to Stress identifies key factors and characteristics that are essential to a scientifically useful and behaviorally predictive understanding of resilience to stress.

Contributions from Uniquely Qualified Military and Civilian Experts

Initiated by the Military Operational Medicine Research Directorate of the US Army Medical Research and Material Command (USAMRMC), this seminal volume integrates recent research and experience from military and civilian experts in behavioral and social sciences, human performance, and physiology. Each chapter is grounded in vigorous research with emphasis on relevance to a variety of real-world operations and settings, including extreme environments encountered in modern war.

Logical Progression, Cross-Disciplinary Appeal

Organized into four sections, the text begins with a discussion of the relevant aspects of stress in the context of military life to offer civilian readers a window into contemporary military priorities. Later chapters consider biological, physiological, and genetic factors, psychosocial aspects of resilience, and "community capacity" variables that influence psychological responses to stressful events. This multidisciplinary effort concludes with an overview of emergent themes and related issues to advance the science of resilience toward predictive research, theory, and application for all those—military and civilian—who serve in the national defense.

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• 1 Introduction
• 1.1 Outline
• 1.2 The Organization
• PART I: COMPUTATIONAL NUMBER THEORY
• 2 Computational Number Theory
• 2.1 Groups, Rings, and Fields
• 2.2 Elements of Number Theory
• 2.2.1 Integer Rings and Fields
• 2.3 Linear Congruences Over Z(M)
• 2.3.1 Linear Independence and Vector Spaces Over Z(M)
• 2.3.2 Orthogonality and Null Spaces Over Z(M)
• 2.3.3 Rank of Matrices Over Z(M)
• 2.4 The Chinese Remainder Theorem for Integers
• 2.5 Residue Number Systems
• 2.5.1 Other Residue Operations
• 2.6 The Problem of Error Control
• 2.7 Bibliography
• 3 Polynomial Algebra
• 3.1 Algebra of Polynomials Over a Field
• 3.2 Roots of a Polynomial
• 3.3 Polynomial Fields and Rings
• 3.4 The CRT for Polynomials
• 3.5 Polynomial Algebra Over a Ring
• 3.5.1 Roots of a Polynomial Over Z^M)
• 3.6 Computing Convolution Over a Field
• 3.6.1 Algorithms for Computing Convolutions Over a Field
• 3.7 On Factorization of (un — 1) Over Finite Fields
• 3.7.1 Factorization of (un — 1) Over GF(p)
• 3.7.2 Primitive Polynomial Over GF(p)
• 3.8 Convolution Algorithms Over Z(M) and NTTs
• 3.8.1 Mersenne and Fermat Number Transforms
• 3.9 The Problem of Error Control
• 3.10 Bibliography
• PART II: DIGITAL SIGNAL PROCESSING
• 4 New Algorithms Over Integer Rings
• 4.1 Monic Polynomial Factorization
• 4.1.1 Monic Polynomial Factorization Over Z(pα)
• 4.1.2 A Systematic Procedure
• 4.1.3 Monic Polynomial Factorization over Z(M)
• 4.2 The AIC Extension of the CRT
• 4.2.1 The AICE-CRT Reconstruction
• 4.2.2 AICE-CRT in the Matrix Form
• 4.3 Convolution Algorithms Over Z(M)
• 4.3.1 AICE-CRT Based Algorithms for Cyclic Convolution
• 4.3.2 AICE-CRT Based Algorithms for Acyclic Convolution
• 4.4 Computational Complexity Analysis
• 4.5 Bibliography
• 5 AICE-CRT: The Complex Case
• 5.1 Multilinear Forms Over Complex Integer Rings
• 5.1.1 Factorization of j2 + 1 Over GF(p)
• 5.1.2 Algorithms for Convolutions: Special Cases
• 5.2 Factorization Over a Complex Integer Ring
• 5.3 The AICE-CRT Over Complex Integer Rings
• 5.4 Algorithms for Convolution
• 5.4.1 Algorithms for Cyclic Convolution
• 5.4.2 Algorithms for Acyclic Convolution
• 5.5 Discussion
• 5.6 Bibliography
• 6 Fault Tolerance for Integer Sequences
• 6.1 Introduction and Mathematical Preliminaries
• 6.2 A Framework for Fault Tolerance
• 6.2.1 Fault Detection and Correction
• 6.2.2 Mathematical Structure of C Over Z(M)
• 6.2.3 Decoding Algorithms Over Z(M)
• 6.3 Coding Techniques Over Z(q)
• 6.3.1 Decoding Algorithms Over Z(q)
• 6.3.2 A Fast Algorithm For Decoding Over Z(q)
• 6.3.3 Some General Remarks
• 6.4 Examples and SFC-DFD Codes
• 6.5 NTT Based Cyclic Codes
• 6.6 Bibliography
• PART III: ERROR CONTROL TECHNIQUES IN RESIDUE NUMBER SYSTEMS
• 7 Fault Control in Residue Number Systems
• 7.1 Background and Terminology
• 7.2 A Coding Theory Framework for RRNS
• 7.2.1 Minimum Distance of RRNS
• 7.2.2 Error Detection and Correction in RRNS
• 7.2.3 Weight Distribution of the MDS-RRNS Code
• 7.3 Consistency Checking for RRNS
• 7.4 A Coding Theory Framework for RNS-PC
• 7.4.1 Error Detection and Correction in RNS-PC
• 7.5 Consistency Checking for RNS-PC
• 7.6 Bibliography
• 8 Single Error Correction in RNS
• 8.1 Single Error Correction in RRN
• 8.2 Computational Analysis and Examples
• 8.3 Single-Error Correction in RNS-PC
• 8.4 A Superfast Algorithm for RNS
• 8.4.1 A Superfast Algorithm for Single Error Correction
• 8.5 A Procedure for Single Error Correction
• 8.6 A Hardware Design for the Algorithms
• 8.6.1 A Hardware Design for the Fast Algorithm
• 8.6.2 A Hardware Design for the Superfast Algorithm
• 8.6.3 Comparison and Discussion
• 8.7 Bibliography
• 9 Multiple Error Control in RRNS
• 9.1 Errors and Consistency Checking
• 9.2 Single Error Correction: Continued
• 9.3 Double Error Correction
• 9.4 Single-Burst Error Correction
• 9.5 General Fault Detection and Correction
• 9.6 Extensions of Previous Algorithms
• 9.7 Computational Complexity Analysis
• 9.8 Bibliography
• 10 Erasure and Error Control in RRNS
• 10.1 Erasures in RRNS
• 10.2 Consistency Checking
• 10.3 Multiple-Erasure Correction
• 10.4 Erasure Correction and Error Detection
• 10.5 Error and Erasure Correction
• 10.6 Extensions of Previous Algorithms
• 10.7 Computational Complexity Analysis
• 10.8 Bibliography
• 11 Multiple Error Control in RNS-PC
• 11.1 Single Error Correction: Continued
• 11.2 Double Error Correction
• 11.3 Multiple Error Correction
• 11.4 Extensions of Previous Algorithms
• 11.5 Computational Complexity Analysis
• 11.6 Bibliography
• A Computational Complexity Analysis
• A.1 Yau and Liu [1973] algorithm
• A.2 Jenkins and Altman [1988] algorithm
• A.3 New algorithm
• B Complexity Analysis: Continued
• B.1 Extended Jenkins and Altman [1988] algorithm
• B.2 New algorithm
• Index