Hardy / Richman / Walker Applied Algebra

Codes, Ciphers and Discrete Algorithms
2. Auflage 2013
ISBN: 978-1-4398-9185-8
Verlag: Taylor & Francis
Format: EPUB
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 410 Seiten

Reihe: Discrete Mathematics and Its Applications

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.

Autoren/Hrsg.

Hardy, Darel W.

Richman, Fred

Walker, Carol L.

Fachgebiete

• Mathematik | Informatik Mathematik Mathematik Allgemein Diskrete Mathematik, Kombinatorik
• Mathematik | Informatik Mathematik Algebra

Inhaltsverzeichnis

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 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

Groups

Rings and Polynomials

Fields

Linear Algebra and Matrices
Solutions to Odd Problems
Bibliography

Notation

Algorithms

Figures

Tables

Index