Klima R.E., Sigmon N., Stitzinger E. Applications of Abstract Algebra with Maple

CRC Press, 1999. — 272 p.The mathematical concepts of abstract algebra may indeed be considered abstract, but its utility is quite concrete and continues to grow in importance. Unfortunately, the practical application of abstract algebra typically involves extensive and cumbersome calculations-often frustrating even the most dedicated attempts to appreciate and employ its intricacies. Now, however, sophisticated mathematical software packages help obviate the need for heavy number-crunching and make fields dependent on the algebra more interesting-and more accessible. Applications of Abstract Algebra with Maple opens the door to cryptography, coding, Polya counting theory, and the many other areas dependent on abstract algebra.Preliminary Mathematics
Permutation Groups
Cosets and Quotient Groups
Rings and Euclidean Domains
Finite Fields
Finite Fields with Maple
The Euclidean Algorithm
Block Designs
General Properties of Block Designs
Hadamard Matrices
Hadamard Matrices with Maple
Difference Sets
Difference Sets with Maple
Error-Correcting Codes
General Properties of Codes
Hadamard Codes
Reed-Muller Codes
Reed-Muller Codes withMapl e
Linear Codes
Hamming Codes withMaple
BCH Codes
Construction of BCH Codes
Error Correction in BCH Codes
BCH Codes withMapl e
Construction of the Generator Polynomial
Error Correction
Reed-Solomon Codes
Construction of Reed-Solomon Codes
Error Correction in Reed-Solomon Codes
Proof of Reed-Solomon Error Correction
Binary Reed-Solomon Codes
Reed-Solomon Codes withMapl e
Construction of the Codewords
Error Correction
Reed-Solomon Codes in Voyager2
Algebraic Cryptography
Some Elementary Cryptosystems
The Hill Cryptosystem
The Hill Cryptosystem with Maple
Generalizations of the Hill Cryptosystem
The Two-Message Problem
The RSA Cryptosystem
Mathematical Prerequisites
RSA Encryption and Decryption
The RSA Cryptosystem with Maple
A Note on Modular Exponentiation
A Note on Primality Testing
A Note on Integer Factorization
A Note on Digital Signatures
The Diffie-Hellman Key Exchange
Elliptic Curve Cryptography
The ElGamal Cryptosystem
The ElGamal Cryptosystem with Maple
Elliptic Curves
Elliptic Curves withMaple
Elliptic Curve Cryptography
Elliptic Curve Cryptography with Maple
Polya Theory
Group Actions
Burnside’s Theorem
The Cycle Index
The Pattern Inventory
The Pattern Inventory with Maple
Switching Functions
Switching Functions with MapleBasic Maple Tutorial
Introduction to Maple
Arithmetic
Defining Variables and Functions
Algebra
Case Sensitivity
Help File
Arrays and Loops
Conditional Statements
Maple Procedures
Some Maple Linear Algebra Commands
User-Written Maple Procedures
Chapter 5 Procedures
Chapter 7 Procedures
Chapter 8 Procedures
Chapter 9 Procedures
Hints and Solutions to Selected Written Exercises