An Introduction to Number Theory with Cryptography

568 Pages·2013·3.351 MB·English

by  Kraft, James S., Washington, Lawrence C

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Number theory has a rich history. For many years it was one of the purest
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areas of pure mathematics, studied because of the intellectual fascination
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with properties of integers. More recently, it has been an area that also has

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important applications to subjects such as cryptography. An Introduction
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to Number Theory with Cryptography presents number theory along with N
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many interesting applications. Designed for an undergraduate-level course,
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it covers standard number theory topics and gives instructors the option
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of integrating several other topics into their coverage. The “Check Your y R
Understanding” problems aid in learning the basics, and there are numerous O
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exercises, projects, and computer explorations of varying levels of difficulty. D
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Features t C
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• Contains material for a standard course in number theory along with T
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several more advanced topics C O
• Includes more than 500 exercises, projects, and computer explorations r N
of varying levels of difficulty y T
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• Provides full coverage of all traditional number theory topics, along
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with cryptography
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• Presents the building blocks first, gradually increasing the level of
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difficulty
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Jim Kraft received his Ph.D. from the University of Maryland in 1987 and has
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published several research papers in algebraic number theory. His previous
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teaching positions include the University of Rochester, St. Mary’s College of
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California, and Ithaca College, and he has also worked in communications
security. Dr. Kraft currently teaches mathematics at the Gilman School.
Larry Washington received his Ph.D. from Princeton University in 1974 and
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has published extensively in number theory, including books on cryptography
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(with Wade Trappe), cyclotomic fields, and elliptic curves. Dr. Washington is
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currently Professor of Mathematics and Distinguished Scholar-Teacher at
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the University of Maryland. W
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K21751 t
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K21751_Cover.indd 1 8/12/13 10:12 AM
AN INTRODUCTION TO
Number Theory
with
Cryptography
K21751_FM.indd 1 8/5/13 5:00 PM
K21751_FM.indd 2 8/5/13 5:00 PM
AN INTRODUCTION TO
Number Theory
with
Cryptography
James S. Kraft
Gilman School
Baltimore, Maryland, USA
Lawrence C. Washington
University of Maryland
College Park, Maryland, USA
K21751_FM.indd 3 8/5/13 5:00 PM
CRC Press
Taylor & Francis Group
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© 2014 by Taylor & Francis Group, LLC
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Version Date: 20130801
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Dedication
To Kristi, Danny, and Aaron
and to Miriam Kraft and the memory of Norman Kraft
To Susan and Patrick
v
vi
Jim Kraft received his Ph.D. from the University of Maryland in
1987 and has published several research papers in algebraic num-
ber theory. His previous teaching positions include the University
of Rochester, St. Mary’s College of California, and Ithaca College,
and he has also worked in communications security. Dr. Kraft cur-
rently teaches mathematics at the Gilman School.
Larry Washington received his Ph.D. from Princeton University
in 1974 and has published extensively in number theory, includ-
ing books on cryptography (with Wade Trappe), cyclotomic fields,
and elliptic curves. Dr. Washington is currently Professor of Math-
ematics and Distinguished Scholar-Teacher at the University of
Maryland.
Contents
Preface xv
0 Introduction 1
0.1 Diophantine Equations . . . . . . . . . . . . . . . 2
0.2 Modular Arithmetic . . . . . . . . . . . . . . . . . 4
0.3 Primes and the Distribution of Primes . . . . . . . 5
0.4 Cryptography . . . . . . . . . . . . . . . . . . . . . 7
1 Divisibility 9
1.1 Divisibility . . . . . . . . . . . . . . . . . . . . . . 9
1.2 Euclid’s Theorem . . . . . . . . . . . . . . . . . . . 11
1.3 Euclid’s Original Proof . . . . . . . . . . . . . . . 13
1.4 The Sieve of Eratosthenes . . . . . . . . . . . . . . 15
1.5 The Division Algorithm . . . . . . . . . . . . . . . 17
1.5.1 A Cryptographic Application . . . . . . . . 19
1.6 The Greatest Common Divisor . . . . . . . . . . . 20
1.7 The Euclidean Algorithm . . . . . . . . . . . . . . 22
1.7.1 The Extended Euclidean Algorithm . . . . . 25
1.8 Other Bases . . . . . . . . . . . . . . . . . . . . . 30
1.9 Linear Diophantine Equations . . . . . . . . . . . . 32
1.10 The Postage Stamp Problem . . . . . . . . . . . . 38
1.11 Fermat and Mersenne Numbers . . . . . . . . . . . 41
1.12 Chapter Highlights . . . . . . . . . . . . . . . . . . 46
1.13 Problems . . . . . . . . . . . . . . . . . . . . . . . 46
1.13.1 Exercises . . . . . . . . . . . . . . . . . . . . 46
1.13.2 Projects . . . . . . . . . . . . . . . . . . . . 53
1.13.3 Computer Explorations . . . . . . . . . . . . 55
vii
viii Contents
1.13.4 Answers to “Check Your Understanding” . . 57
2 Unique Factorization 59
2.1 Preliminary Results . . . . . . . . . . . . . . . . . 59
2.2 The Fundamental Theorem of Arithmetic . . . . . 61
2.3 Euclid and the Fundamental Theorem of Arithmetic 66
2.4 Chapter Highlights . . . . . . . . . . . . . . . . . . 67
2.5 Problems . . . . . . . . . . . . . . . . . . . . . . . 67
2.5.1 Exercises . . . . . . . . . . . . . . . . . . . . 67
2.5.2 Projects . . . . . . . . . . . . . . . . . . . . 68
2.5.3 Answers to “Check Your Understanding” . . 70
3 Applications of Unique Factorization 71
3.1 A Puzzle . . . . . . . . . . . . . . . . . . . . . . . 71
3.2 Irrationality Proofs . . . . . . . . . . . . . . . . . . 73
√
3.2.1 Four More Proofs That 2 Is Irrational . . 75
3.3 The Rational Root Theorem . . . . . . . . . . . . 77
3.4 Pythagorean Triples . . . . . . . . . . . . . . . . . 80
3.5 Differences of Squares . . . . . . . . . . . . . . . . 86
3.6 Prime Factorization of Factorials . . . . . . . . . . 88
3.7 The Riemann Zeta Function . . . . . . . . . . . . . 90
3.8 Chapter Highlights . . . . . . . . . . . . . . . . . . 96
3.9 Problems . . . . . . . . . . . . . . . . . . . . . . . 96
3.9.1 Exercises . . . . . . . . . . . . . . . . . . . . 96
3.9.2 Projects . . . . . . . . . . . . . . . . . . . . 100
3.9.3 Computer Explorations . . . . . . . . . . . . 104
3.9.4 Answers to “Check Your Understanding” . . 105
4 Congruences 107
4.1 Definitions and Examples . . . . . . . . . . . . . . 107
4.2 Modular Exponentiation . . . . . . . . . . . . . . . 115
4.3 Divisibility Tests . . . . . . . . . . . . . . . . . . . 116
4.4 Linear Congruences . . . . . . . . . . . . . . . . . 120
4.5 The Chinese Remainder Theorem . . . . . . . . . . 127
ix
4.6 Fractions mod m . . . . . . . . . . . . . . . . . . . 132
4.7 Fermat’s Theorem . . . . . . . . . . . . . . . . . . 134
4.8 Euler’s Theorem . . . . . . . . . . . . . . . . . . . 139
4.9 Wilson’s Theorem . . . . . . . . . . . . . . . . . . 147
4.10 Queens on a Chessboard . . . . . . . . . . . . . . . 149
4.11 Chapter Highlights . . . . . . . . . . . . . . . . . . 151
4.12 Problems . . . . . . . . . . . . . . . . . . . . . . . 151
4.12.1 Exercises . . . . . . . . . . . . . . . . . . . . 151
4.12.2 Projects . . . . . . . . . . . . . . . . . . . . 159
4.12.3 Computer Explorations . . . . . . . . . . . . 163
4.12.4 Answers to “Check Your Understanding” . . 164
5 Cryptographic Applications 167
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . 167
5.2 Shift and Affine Ciphers . . . . . . . . . . . . . . . 170
5.3 Secret Sharing . . . . . . . . . . . . . . . . . . . . 175
5.4 RSA . . . . . . . . . . . . . . . . . . . . . . . . . . 177
5.5 Chapter Highlights . . . . . . . . . . . . . . . . . . 184
5.6 Problems . . . . . . . . . . . . . . . . . . . . . . . 184
5.6.1 Exercises . . . . . . . . . . . . . . . . . . . . 184
5.6.2 Projects . . . . . . . . . . . . . . . . . . . . 188
5.6.3 Computer Explorations . . . . . . . . . . . . 191
5.6.4 Answers to “Check Your Understanding” . . 192
6 Polynomial Congruences 193
6.1 Polynomials Mod Primes . . . . . . . . . . . . . . 193
6.2 Solutions Modulo Prime Powers. . . . . . . . . . . 196
6.3 Composite Moduli . . . . . . . . . . . . . . . . . . 202
6.4 Chapter Highlights . . . . . . . . . . . . . . . . . . 203
6.5 Problems . . . . . . . . . . . . . . . . . . . . . . . 203
6.5.1 Exercises . . . . . . . . . . . . . . . . . . . . 203
6.5.2 Projects . . . . . . . . . . . . . . . . . . . . 204
6.5.3 Computer Explorations . . . . . . . . . . . . 205