#
Applied Algebra

Codes, Ciphers and Discrete Algorithms, Second Edition

## Preview

## Book Description

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.

## 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

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

**Ciphers**

Cryptography

Cryptanalysis

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 *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

**Error-Correcting 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**

## 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