Elementary number theory and elliptic curves (web draft, Sept. 2004)

The book presents a comprehensive overview of the interplay between elementary number theory and elliptic curves. It begins by building a foundation with classical number theory topics such as divisibility, prime numbers, and modular arithmetic. These concepts then seamlessly lead into more advanced topics, emphasizing their application in understanding elliptic curves.

The core of the book lies in exploring the rich structure and properties of elliptic curves over various fields. Readers are introduced to essential concepts like group law, the Mordell-Weil theorem, and the role of elliptic curves in solving Diophantine equations. The book also touches on algorithms related to elliptic curves, including those used for factorization and primality testing.

Each chapter is designed to progressively increase in complexity, helping readers from diverse backgrounds gradually assimilate the intricate details of elliptic curves and their profound mathematical significance.

• Understand the fundamental concepts of number theory, including modular arithmetic and prime numbers.
• Discover the mathematical elegance of elliptic curves and their applications.
• Learn about algorithms used in modern cryptography, rooted in number theory and elliptic curves.
• Gain insights into solving complex Diophantine equations with the aid of elliptic curves.

"The beauty of number theory lies in its simplicity and its profound implications."

"Elliptic curves are the bridge between the ancient world of number theory and the cutting-edge field of cryptography."

"To understand elliptic curves is to embrace the delicate balance between theorem and application."

This book matters not only to students and professionals in the field of mathematics but also to those in computer science and cryptography. In a world increasingly reliant on secure digital communication, understanding the principles outlined in this text has practical, real-world applications. By breaking down complex theorems and providing a framework for understanding these concepts, the book serves as an essential resource for anyone looking to grasp the mathematical foundations that underpin modern technology.

The meticulous approach to blending historical context with contemporary applications ensures that readers can appreciate the significance of number theory and its evolution. It’s a crucial read for those who aspire to contribute to the ever-growing fields of cryptography and algorithm development.