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More About This Textbook
Overview
Using mathematical tools from number theory and finite fields, Applied Algebra: Codes, Ciphers, and Discrete Algorithms, Second Edition presents practical methods for solving problems in data security and data integrity. It is designed for an applied algebra course for students who have had prior classes in abstract or linear algebra. While the content has been reworked and improved, this edition continues to cover many algorithms that arise in cryptography and errorcontrol codes.
New to the Second Edition
Instead of a general study on finite groups, the book considers finite groups of permutations and develops just enough of the theory of finite fields to facilitate construction of the fields used for errorcontrol codes and the Advanced Encryption Standard. It also deals with integers and polynomials. Explaining the mathematics as needed, this text thoroughly explores how mathematical techniques can be used to solve practical problems.
About the Authors
Darel W. Hardy is Professor Emeritus in the Department of Mathematics at Colorado State University. His research interests include applied algebra and semigroups.
Fred Richman is a professor in the Department of Mathematical Sciences at Florida Atlantic University. His research interests include Abelian group theory and constructive mathematics.
Carol L. Walker is Associate Dean Emeritus in the Department of Mathematical Sciences at New Mexico State University. Her research interests include Abelian group theory, applications of homological algebra and category theory, and the mathematics of fuzzy sets and fuzzy logic.
Editorial Reviews
From the Publisher
This book attempts to show the power of algebra in a relatively simple setting.—Mathematical Reviews, 2010
… The book supports learning by doing. In each section we can find many examples which clarify the mathematics introduced in the section and each section is followed by a series of exercises of which approximately half are solved in the end of the book. Additional the book comes with a CDROM containing an interactive version of the book powered by the computer algebra system Scientific Notebook. … the mathematics in the book are developed as needed and the focus of the book lies clearly on learning by examples and exercises. … the book gives good insight on how algebra can be used in coding and cryptography … The strength of the book is clearly the number of examples …
—IACR book reviews, January 2010
Product Details
Related Subjects
Table of Contents
Preface
Integers and Computer Algebra
Integers
Computer Algebra vs. Numerical Analysis
Sums and Products
Mathematical Induction
Codes
Binary and Hexadecimal Codes
ASCII Code
Morse Code
Braille
TwooutofFive Code
Hollerith Codes
Euclidean Algorithm
The Mod Function
Greatest Common Divisors
Extended Euclidean Algorithm
The Fundamental Theorem of Arithmetic
Modular Arithmetic
Ciphers
Cryptography
Cryptanalysis
Substitution and Permutation Ciphers
Block Ciphers
The Playfair Cipher
Unbreakable Ciphers
Enigma Machine
ErrorControl Codes
Weights and Hamming Distance
Bar Codes Based on TwooutofFive Code
Other Commercial Codes
Hamming (7, 4) Code
Chinese Remainder Theorem
Systems of Linear Equations Modulo n
Chinese Remainder Theorem
Extended Precision Arithmetic
Greatest Common Divisor of Polynomials
Hilbert Matrix
Theorems of Fermat and Euler
Wilson’s Theorem
Powers Modulo n
Fermat’s Little Theorem
Rabin’s Probabilistic Primality Test
Exponential Ciphers
Euler’s Theorem
Public Key Ciphers
The Rivest–Shamir–Adleman Cipher System
Electronic Signatures
A System for Exchanging Messages
Knapsack Ciphers
Digital Signature Standard
Finite Fields
The Galois Field GF_{p}
The Ring GF_{p}[x] of Polynomials
The Galois Field GF_{4}
The Galois Fields GF_{8} and GF_{16}
The Galois Field GF_{p}^{n}
The Multiplicative Group of GF_{p}^{n}
Random Number Generators
ErrorCorrecting Codes
BCH Codes
A BCH Decoder
Reed–Solomon Codes
Advanced Encryption Standard
Data Encryption Standard
The Galois Field GF_{256}
The Rijndael Block Cipher
Polynomial Algorithms and Fast Fourier Transforms
Lagrange Interpolation Formula
Kronecker’s Algorithm
Neville’s Iterated Interpolation Algorithm
Secure Multiparty Protocols
Discrete Fourier Transforms
Fast Fourier Interpolation
Appendix A: Topics in Algebra and Number Theory
Number Theory
Groups
Rings and Polynomials
Fields
Linear Algebra and Matrices
Solutions to Odd Problems
Bibliography
Notation
Algorithms
Figures
Tables
Index