Over 50 years ago, the mathematician Leonard Dickson said, “Thank God that number theory is unsullied by any application.” As it turns out, much later, number theory formed the basis of many, many applications. In particular, it underlies cryptography and, more specifically, public key cryptography. It is the technology that allows us to connect to sites via the Internet and to use our credit cards in a way that stops eavesdroppers from stealing our information.

Until now, as far as I know, books on number theory have only briefly mentioned cryptography, and books on cryptography have mentioned number theory only to introduce the basics needed to understand the protocols being discussed. While I would say the main focus of this book is number theory, it provides a deeper look at cryptography than many other number theory books.

I learned on page 1 about the Babylonian clay tablet called Plimpton 322 and that over 3,000 years ago it showed a set of Pythagorean triplets. This was just the first of many surprises and bits of trivia that I picked up from the book. But it is not just trivia; it also adds a depth to the material that the authors present. It offers a refreshing approach to the topic, which not only demonstrates that the authors appear to have a genuine interest in the topic, but also that they were interested in writing a textbook that was not dry.

The book can be divided into roughly two parts. The first part is an introduction to number theory that should be able to be understood by anyone with a basic foundation in mathematics and an interest in the topic. It starts with division, moves on to factorization and primes, and explores congruences and polynomial congruences. It then looks at some cryptographic applications and continues with primality and factorization. The second half of the book is more complex and examines topics like the geometry of numbers, continued fractions, Gaussian integers, and algebraic integers.

Each chapter ends with a summary (called “Chapter Highlights”), followed by several exercises. The exercises are usually arranged neatly by section. These are followed by “projects,” which are slightly longer and more challenging exercises. “Computer explorations”--exercises that require a program to solve--follow. While first-year university students should be able to pick up the topics easily, I think the projects and computer explorations would definitely test their limits. Each chapter ends with a “Check Your Understanding” section that contains a set of answers to problems that are scattered throughout the chapter.

While the main focus of the book is number theory, cryptography is well covered. There is the standard text explaining plaintext and ciphertext. The book then goes on to describe RSA and explain how it works. Diffie-Hellman key exchange is covered in a later chapter, as is ElGamal. Chapter 17, “Epilogue: Fermat’s Last Theorem,” which is the last chapter, mentions elliptic curves, but does not discuss their application to cryptography.

Overall, I think this is a fine book for someone with a reasonable background in mathematics who wants to do some self study and gain an understanding of number theory. The book is rigorous, but not dry. I also think it would make a great textbook. The exercises are good and plentiful. There is enough explanation that students will find it a valuable reference.

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