**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 Member Price: $41.60

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

List Price: $52.00

Individual Member Price: $41.60

#### You may also like

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

Share this page
*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.