**Student Mathematical Library**

Volume: 83;
2017;
252 pp;
Softcover

MSC: Primary 11; 68; 14; 91; 81;

Print ISBN: 978-1-4704-3582-0

Product Code: STML/83

List Price: $52.00

Individual Price: $41.60

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**Electronic ISBN: 978-1-4704-4123-4
Product Code: STML/83.E**

List Price: $52.00

Individual Price: $41.60

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#### Supplemental Materials

# Modern Cryptography and Elliptic Curves: A Beginner’s Guide

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*Thomas R. Shemanske*

This book offers the beginning undergraduate
student some of the vista of modern mathematics by developing and
presenting the tools needed to gain an understanding of the arithmetic
of elliptic curves over finite fields and their applications to modern
cryptography. This gradual introduction also makes a significant
effort to teach students how to produce or discover a proof by
presenting mathematics as an exploration, and at the same time, it
provides the necessary mathematical underpinnings to investigate the
practical and implementation side of elliptic curve cryptography
(ECC).

Elements of abstract algebra, number theory, and affine and
projective geometry are introduced and developed, and their interplay
is exploited. Algebra and geometry combine to characterize congruent
numbers via rational points on the unit circle, and group law for the
set of points on an elliptic curve arises from geometric intuition
provided by Bézout's theorem as well as the construction of projective
space. The structure of the unit group of the integers modulo a prime
explains RSA encryption, Pollard's method of factorization,
Diffie–Hellman key exchange, and ElGamal encryption, while the group
of points of an elliptic curve over a finite field motivates Lenstra's
elliptic curve factorization method and ECC.

The only real prerequisite for this book is a course on
one-variable calculus; other necessary mathematical topics are
introduced on-the-fly. Numerous exercises further guide the
exploration.

#### Readership

Undergraduate and graduate students interested in elliptic curves with applications to cryptography.

#### Reviews & Endorsements

The main objective of this book, which is mainly aimed at undergraduate students, is to explain the arithmetic of elliptic curves defined over finite fields and to show how those curves can be used in cryptography. In order to do that, the author purposely avoids complex mathematical demonstrations and, instead, presents the concepts in a more descriptive way, suggesting some topics for further exploration by the reader.

-- Victor Gayoso Martíinez, Mathematical Reviews

#### Table of Contents

# Table of Contents

## Modern Cryptography and Elliptic Curves: A Beginner's Guide

- Cover Cover11
- Title page i2
- Contents iii4
- Preface vii8
- Chapter 1. Three Motivating Problems 114
- Chapter 2. Back to the Beginning 922
- 2.1. The Unit Circle: Real vs. Rational Points 1023
- 2.2. Parametrizing the Rational Points on the Unit Circle 1225
- 2.3. Finding all Pythagorean Triples 1629
- 2.4. Looking for Underlying Structure: Geometry vs. Algebra 2740
- 2.5. More about Points on Curves 3447
- 2.6. Gathering Some Insight about Plane Curves 3851
- 2.7. Additional Exercises 4356

- Chapter 3. Some Elementary Number Theory 4558
- Chapter 4. A Second View of Modular Arithmetic: \Z_{𝑛} and 𝑈_{𝑛} 7386
- Chapter 5. Public-Key Cryptography and RSA 101114
- Chapter 6. A Little More Algebra 127140
- 6.1. Towards a Classification of Groups 128141
- 6.2. Cayley Tables 128141
- 6.3. A Couple of Non-abelian Groups 131144
- 6.4. Cyclic Groups and Direct Products 134147
- 6.5. Fundamental Theorem of Finite Abelian Groups 138151
- 6.6. Primitive Roots 141154
- 6.7. Diffie–Hellman Key Exchange 143156
- 6.8. ElGamal Encryption 144157

- Chapter 7. Curves in Affine and Projective Space 147160
- 7.1. Affine and Projective Space 147160
- 7.2. Curves in the Affine and Projective Plane 153166
- 7.3. Rational Points on Curves 156169
- 7.4. The Group Law for Points on an Elliptic Curve 159172
- 7.5. A Formula for the Group Law on an Elliptic Curve 179192
- 7.6. The Number of Points on an Elliptic Curve 185198

- Chapter 8. Applications of Elliptic Curves 189202
- Appendix A. Deeper Results and Concluding Thoughts 203216
- Appendix B. Answers to Selected Exercises 219232
- Bibliography 245258
- Index 249262
- Back Cover Back Cover1266