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