Applied Algebra

Applied Algebra: Codes, Ciphers and Discrete Algorithms, Second Edition

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  • Covers topics from algebra, cryptography, and number theory
  • Includes algorithms that offer common-sense approaches to problems, such as computing large powers
  • Provides complete coverage on error-correcting codes
  • Explains the Rijndael algorithm to help students understand the data encryption standard
    Solutions manual available for qualifying instructors

CD-ROM Features

  • Links that make it easy to find topics and navigate page-by-page, chapter-by-chapter, or by keywords
  • Interactive examples
  • Computing hints
  • Self-tests
  • Some details of problem solutions beyond those in the printed text


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

  • A CD-ROM containing an interactive version of the book that is powered by Scientific Notebook®, a mathematical word processor and easy-to-use computer algebra system
  • New appendix that reviews prerequisite topics in algebra and number theory
  • Double the number of exercises

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.

Table of Contents


Integers and Computer Algebra


Computer Algebra vs. Numerical Analysis

Sums and Products

Mathematical Induction


Binary and Hexadecimal Codes


Morse Code


Two-out-of-Five Code

Hollerith Codes

Euclidean Algorithm

The Mod Function

Greatest Common Divisors

Extended Euclidean Algorithm

The Fundamental Theorem of Arithmetic

Modular Arithmetic




Substitution and Permutation Ciphers

Block Ciphers

The Playfair Cipher

Unbreakable Ciphers

Enigma Machine

Error-Control Codes

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

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 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

Error-Correcting Codes

BCH Codes

A BCH Decoder

Reed–Solomon Codes

Advanced Encryption Standard

Data Encryption Standard

The Galois Field GF256

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


Rings and Polynomials


Linear Algebra and Matrices

Solutions to Odd Problems







Editorial Reviews

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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Resource OS Platform Updated Description Instructions Cross Platform March 18, 2014

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