An error was encountered while trying to add the item to the cart. Please try again.
Copy To Clipboard
Successfully Copied!
Modern Cryptography and Elliptic Curves: A Beginner’s Guide

Thomas R. Shemanske Dartmouth College, Hanover, NH
Available Formats:
Softcover ISBN: 978-1-4704-3582-0
Product Code: STML/83
List Price: $52.00 Individual Price:$41.60
Electronic ISBN: 978-1-4704-4123-4
Product Code: STML/83.E
List Price: $52.00 Individual Price:$41.60
Bundle Print and Electronic Formats and Save!
This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.
List Price: $78.00 Click above image for expanded view Modern Cryptography and Elliptic Curves: A Beginner’s Guide Thomas R. Shemanske Dartmouth College, Hanover, NH Available Formats:  Softcover ISBN: 978-1-4704-3582-0 Product Code: STML/83  List Price:$52.00 Individual Price: $41.60  Electronic ISBN: 978-1-4704-4123-4 Product Code: STML/83.E  List Price:$52.00 Individual Price: $41.60 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$78.00
• Book Details

Student Mathematical Library
Volume: 832017; 252 pp
MSC: Primary 11; 68; 14; 91; 81;

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.

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

• Chapters
• Three motivating problems
• Back to the beginning
• Some elementary number theory
• A second view of modular arithmetic: $\mathbb {Z}_n$ and $U_n$
• Public-key cryptography and RSA
• A little more algebra
• Curves in affine and projective space
• Applications of elliptic curves
• Deeper results and concluding thoughts

• Reviews

• 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
• Requests

Review Copy – for reviewers who would like to review an AMS book
Desk Copy – for instructors who have adopted an AMS textbook for a course
Examination Copy – for faculty considering an AMS textbook for a course
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Volume: 832017; 252 pp
MSC: Primary 11; 68; 14; 91; 81;

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.

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

• Chapters
• Three motivating problems
• Back to the beginning
• Some elementary number theory
• A second view of modular arithmetic: $\mathbb {Z}_n$ and $U_n$
• Public-key cryptography and RSA
• A little more algebra
• Curves in affine and projective space
• Applications of elliptic curves
• Deeper results and concluding thoughts