Darel W. Hardy, Fred Richman, Carol L. Walker
Chapman and Hall/CRC
Published February 17, 2009
Textbook - 424 Pages - 32 B/W Illustrations
ISBN 9781420071429 - CAT# C7142
Series: Discrete Mathematics and Its Applications
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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 error-control 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 error-control 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.
Integers and Computer Algebra
Computer Algebra vs. Numerical Analysis
Sums and Products
Binary and Hexadecimal Codes
The Mod Function
Greatest Common Divisors
Extended Euclidean Algorithm
The Fundamental Theorem of Arithmetic
Substitution and Permutation Ciphers
The Playfair Cipher
Weights and Hamming Distance
Bar Codes Based on Two-out-of-Five 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
Theorems of Fermat and Euler
Powers Modulo n
Fermat’s Little Theorem
Rabin’s Probabilistic Primality Test
Public Key Ciphers
The Rivest–Shamir–Adleman Cipher System
A System for Exchanging Messages
Digital Signature Standard
The Galois Field GFp
The Ring GFp[x] of Polynomials
The Galois Field GF4
The Galois Fields GF8 and GF16
The Galois Field GFpn
The Multiplicative Group of GFpn
Random Number Generators
A BCH Decoder
Advanced Encryption Standard
Data Encryption Standard
The Galois Field GF256
The Rijndael Block Cipher
Polynomial Algorithms and Fast Fourier Transforms
Lagrange Interpolation Formula
Neville’s Iterated Interpolation Algorithm
Secure Multiparty Protocols
Discrete Fourier Transforms
Fast Fourier Interpolation
Appendix A: Topics in Algebra and Number Theory
Rings and Polynomials
Linear Algebra and Matrices
Solutions to Odd Problems
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 CD-ROM 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
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