2nd Edition

Applied Algebra Codes, Ciphers and Discrete Algorithms, Second Edition

    424 Pages 32 B/W Illustrations
    by Chapman & Hall

    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

    • Downloadable resources 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.


    Integers and Computer Algebra


    Computer Algebra vs. Numerical Analysis

    Sums and Products

    Mathematical Induction


    Binary and Hexadecimal Codes

    ASCII Code

    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








    Darel W. Hardy (Author) , Fred Richman (Author) , Carol L. Walker (Author)

    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