Applications of Abstract Algebra with MAPLE - R. Klima, N. Sigmon

250 Pages·2002·1.56 MB·English

Applications of
Abstract Algebra
with MAPLE
(cid:1)c 1999byCRCPressLLC
Applications of
Abstract Algebra
with MAPLE
Richard E. Klima
Neil Sigmon
Ernest Stitzinger
CRC Press
Boca Raton London New York Washington, D.C.
Library of Congress Cataloging-in-Publication Data
Klima, Richard E.
Applications of abstract algebra with Maple / Richard E. Klima,
Neil P. Sigmon, Ernest Stitzinger.
p. cm. - (Discrete mathematics and its applications)
Includes bibliographical references and index.
ISBN 0-8493-8170-3 (alk. paper)
1. Algebra, Abstract—Data processing. 2. Maple (Computer file)
I. Stitzinger, Ernest. II. Sigmon, Neil P. III. Title.
IV. Series.
QAl62.K65 1999
5 12´.02´02855369—dc2 1 99-37392
CIP
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International Standard Book Number 0-8493-8170-3
Library of Congress Card Number 99-37392
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Printed on acid-free paper
Preface
In 1990 we introduced a one-semester applications of algebra course at
North Carolina State University for students who had successfully com-
pletedsemestersoflinearandabstractalgebra. Weintendedforthecourse
to give students more exposure to basic algebraic concepts, and to show
students some practical uses of these concepts. The course was received
enthusiastically by both students and faculty and has become one of the
most popular mathematics electives at NC State.
When we were originally deciding on material for the course, we knew
that we wanted to include several topics from coding theory, cryptography,
and counting (what we call Polya theory). With this in mind, at the sug-
gestion of Michael Singer, we used George Mackiw’s book Applications of
Abstract Algebra for the first few years, and supplemented as we saw fit.
After several years, Mackiw’s book went out of print temporarily. Rather
thansearchforanewbookforthecourse,wedecidedtowriteourownnotes
and teach the course from a coursepack. About the same time, NC State
incorporatedthemathematicssoftwarepackageMapleVTM1 intoitscalcu-
lus sequence, and we decided to incorporate it into our course as well. The
use of Maple played a central role in the recent development of the course
becauseitprovidesawayforstudentstoseerealisticexamplesofthetopics
discussed without having to struggle with extensive computations. With
additional notes regarding the use of Maple in the course, our coursepack
evolved into this book. In addition to the topics discussed in this book, we
haveincludedanumberofothertopicsinthecourse. However, thepresent
material has become the constant core for the course.
Our philosophy concerning the use of technology in the course is that
it be a useful tool and not present new problems or frustrations. Conse-
quently, we have included very detailed instructions regarding the use of
1MapleVisaregisteredtrademarkofWaterlooMaple,Inc.,57ErbSt. W,Waterloo,
CanadaN2L6C2,www.maplesoft.com.
(cid:1)c 1999byCRCPressLLC
Maple in this book. It is our hope that the Maple discussions are thorough
enough to allow it to be used without much alternative aid. As alterna-
tive aids, we have included a basic Maple tutorial in Appendix A, and an
introduction to some of Maple’s linear algebra commands in Appendix B.
Although we do not require students to produce the Maple code used in
the course, we do require that they obtain a level of proficiency such that
theycanmakebasicchangestoprovidedworksheetstocompletenumerous
Maple exercises. So that this book can be used for applications of algebra
courses in which Maple is not incorporated, we have separated all Maple
materialintosectionsthatareclearlylabeled, andseparatedallMapleand
non-Maple exercises.
When teaching the course, we discuss the material in Chapter 1 as
needed rather than review it all at once. More specifically, we discuss the
materialinChapter1throughexamplesthefirsttimeitisneededintheap-
plicationsthatfollow. SomeofthematerialinChapter1isreviewmaterial
that does not apply specifically to the applications that follow. However,
for students with weak backgrounds, Chapter 1 provides a comprehensive
review of all necessary prerequisite mathematics.
Chapter 2 is a short chapter on block designs. In Chapters 3, 4, and
5 we discuss some topics from coding theory. In Chapter 3 we introduce
error-correctingcodes,andpresentHadamard,Reed-Muller,andHamming
codes. In Chapters 4 and 5, we present BCH codes and Reed-Solomon
codes. Eachofthesechaptersaredependentinpartontheprecedingchap-
ters. ThedependencyofChapter3onChapter2canbeavoidedbyomitting
Sections 3.2, 3.3, and 3.4 on Hadamard and Reed-Muller codes. In Chap-
ters 6, 7, and 8 we discuss some topics from cryptography. In Chapter 6
we introduce algebraic cryptography, and present several variations of the
Hill cryptosystem. In Chapter 7 we present the RSA cryptosystem and
discuss some related topics, including the Diffie-Hellman key exchange. In
Chapter 8 we present the ElGamal cryptosystem, and describe how elliptic
curves can be incorporated into the system naturally. There is a slight de-
pendency of Chapters 7 and 8 on Chapter 6, and of Chapter 8 on Chapter
7. Chapter 9 is a stand-alone chapter in which we discuss the Polya count-
ing techniques, including Burnside’s Theorem and the Polya Enumeration
Theorem.
We wish to thank all those who have been involved in the develop-
ment of this course and book. Pete Hardy taught from the coursepack and
improved it with his suggestions. Also, Michael Singer suggested various
topics and wrote notes on some of them. Many students have written on
this material for various projects. Of these, the recent master’s project by
KarenKleinonellipticcurveswasespeciallyinteresting. Finally,wewishto
(cid:1)c 1999byCRCPressLLC
thankourmentor,JackLevine,forhisinterestinourprojects,hisguidance
as we learned about applications of algebra, and his many contributions to
the subject, especially in cryptography.
(cid:1)c 1999byCRCPressLLC
Contents
Preface
1 Preliminary Mathematics
1.1 Permutation Groups
1.2 Cosets and Quotient Groups
1.3 Rings and Euclidean Domains
1.4 Finite Fields
1.5 Finite Fields with Maple
1.6 The Euclidean Algorithm
2 Block Designs
2.1 General Properties of Block Designs
2.2 Hadamard Matrices
2.3 Hadamard Matrices with Maple
2.4 Difference Sets
2.5 Difference Sets with Maple
3 Error-Correcting Codes
3.1 General Properties of Codes
3.2 Hadamard Codes
3.3 Reed-Muller Codes
(cid:1)c 1999byCRCPressLLC
3.4 Reed-Muller Codes with Maple
3.5 Linear Codes
3.6 Hamming Codes with Maple
4 BCH Codes
4.1 Construction of BCH Codes
4.2 Error Correction in BCH Codes
4.3 BCH Codes with Maple
4.3.1 Construction of the Generator Polynomial
4.3.2 Error Correction
5 Reed-Solomon Codes
5.1 Construction of Reed-Solomon Codes
5.2 Error Correction in Reed-Solomon Codes
5.3 Proof of Reed-Solomon Error Correction
5.4 Binary Reed-Solomon Codes
5.5 Reed-Solomon Codes with Maple
5.5.1 Construction of the Codewords
5.5.2 Error Correction
5.6 Reed-Solomon Codes in Voyager 2
6 Algebraic Cryptography
6.1 Some Elementary Cryptosystems
6.2 The Hill Cryptosystem
6.3 The Hill Cryptosystem with Maple
6.4 Generalizations of the Hill Cryptosystem
6.5 The Two-Message Problem
7 The RSA Cryptosystem
7.1 Mathematical Prerequisites
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7.2 RSA Encryption and Decryption
7.3 The RSA Cryptosystem with Maple
7.4 A Note on Modular Exponentiation
7.5 A Note on Primality Testing
7.6 A Note on Integer Factorization
7.7 A Note on Digital Signatures
7.8 The Diffie-Hellman Key Exchange
8 Elliptic Curve Cryptography
8.1 The ElGamal Cryptosystem
8.2 The ElGamal Cryptosystem with Maple
8.3 Elliptic Curves
8.4 Elliptic Curves with Maple
8.5 Elliptic Curve Cryptography
8.6 Elliptic Curve Cryptography with Maple
9 Polya Theory
9.1 Group Actions
9.2 Burnside’s Theorem
9.3 The Cycle Index
9.4 The Pattern Inventory
9.5 The Pattern Inventory with Maple
9.6 Switching Functions
9.7 Switching Functions with Maple
Appendices
A Basic Maple Tutorial
A.1 Introduction to Maple
A.2 Arithmetic
(cid:1)c 1999byCRCPressLLC
A.3 Defining Variables and Functions
A.4 Algebra
A.5 Case Sensitivity
A.6 Help File
A.7 Arrays and Loops
A.8 Conditional Statements
A.9 Maple Procedures
B Some Maple Linear Algebra Commands
C User-Written Maple Procedures
C.1 Chapter 5 Procedures
C.2 Chapter 7 Procedures
C.3 Chapter 8 Procedures
C.4 Chapter 9 Procedures
Hints and Solutions to Selected Written Exercises
(cid:1)c 1999byCRCPressLLC