Applications of Abstract Algebra with MAPLE

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CRC Press, Sep 28, 1999 - Mathematics - 272 pages
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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. The authors have carefully integrated Maple V throughout the text, enabling readers to see realistic examples of the topics discussed without struggling with the computations. But the book stands well on its own if the reader does not have access to the software.

The text includes a first-chapter review of the mathematics required-groups, rings, and finite fields-and a Maple tutorial in the appendix along with detailed treatments of coding, cryptography, and Polya theory applications.

Applications of Abstract Algebra with Maple packs a double punch for those interested in beginning-or advancing-careers related to the applications of abstract algebra. It not only provides an in-depth introduction to the fascinating, real-world problems to which the algebra applies, it offers readers the opportunity to gain experience in using one of the leading and most respected mathematical software packages available.
 

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Contents

Preliminary Mathematics
1
12 Cosets and Quotient Groups
6
13 Rings and Euclidean Domains
9
14 Finite Fields
13
15 Finite Fields with Maple
16
16 The Euclidean Algorithm
18
Block Designs
27
22 Hadamard Matrices
31
73 The RSA Cryptosystem with Maple
147
74 A Note on Modular Exponentiation
150
75 A Note on Primality Testing
152
76 A Note on Integer Factorization
153
77 A Note on Digital Signatures
154
78 The DiffieHellman Key Exchange
155
Elliptic Curve Cryptography
163
82 The ElGamal Cryptosystem with Maple
166

23 Hadamard Matrices with Maple
33
24 Difference Sets
36
25 Difference Sets with Maple
39
ErrorCorrecting Codes
43
32 Hadamard Codes
46
33 ReedMuller Codes
48
35 Linear Codes
53
36 Hamming Codes with Maple
60
BCH Codes
67
42 Error Correction in BCH Codes
70
43 BCH Codes with Maple
77
431 Construction of the Generator Polynomial
78
432 Error Correction
80
ReedSolomon Codes
91
52 Error Correction in ReedSolomon Codes
93
53 Proof of ReedSolomon Error Correction
97
54 Binary ReedSolomon Codes
101
55 ReedSolomon Codes with Maple
102
551 Construction of the Codewords
103
552 Error Correction
105
56 ReedSolomon Codes in Voyager 2
111
Algebraic Cryptography
115
62 The Hill Cryptosystem
119
63 The Hill Cryptosystem with Maple
124
64 Generalizations of the Hill Cryptosystem
129
65 The TwoMessage Problem
131
The RSA Cryptosystem
139
71 Mathematical Prerequisites
140
72 RSA Encryption and Decryption
142
83 Elliptic Curves
168
84 Elliptic Curves with Maple
175
85 Elliptic Curve Cryptography
178
86 Elliptic Curve Cryptography with Maple
182
Polya Theory
189
91 Group Actions
190
92 Burnsides Theorem
192
93 The Cycle Index
195
94 The Pattern Inventory
198
95 The Pattern Inventory with Maple
203
96 Switching Functions
205
97 Switching Functions with Maple
208
Basic Maple Tutorial
213
A2 Arithmetic
214
A3 Defining Variables and Functions
216
A4 Algebra
217
A5 Case Sensitivity
218
A6 Help File
219
A8 Conditional Statements
221
A9 Maple Procedures
223
Some Maple Linear Algebra Commands
225
UserWritten Maple Procedures
231
C2 Chapter 7 Procedures
234
C3 Chapter 8 Procedures
235
C4 Chapter 9 Procedures
236
Hints and Solutions to Selected Written Exercises
243
Index
248
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