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An Introduction to Number Theory with Cryptography

James S. Kraft and Lawrence C. Washington
Chapman & Hall/CRC
Publication Date: 
Number of Pages: 
Textbooks in Mathematics
[Reviewed by
John D. Cook
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A lot of books on number theory with cryptography are heavy on number theory and light on cryptography. A page or two on RSA encryption is enough for a number theory book to say that it includes applications to cryptography. The same is true of books on number theory “with applications”, because the applications in such books are often limited to cryptography.

An Introduction to Number Theory with Cryptography by James Kraft and Lawrence Washington includes more cryptography than many books with similar titles. The book is still primarily about number theory—after all the title isn’t An introduction to Cryptography with Number Theory — but there are cryptographical applications throughout the book.

The authors say in the preface that the book can be used for undergraduate classes and has even been used in high schools. The book is indeed elementary in its tone, minimizing prerequisites and explaining steps thoroughly, and yet it at least mentions more advanced results in passing fairly often. It seems slightly paradoxical that the book moves at a slow pace and yet covers a lot of ground. There’s plenty of material here for a course in number theory, even without going into cryptography.

While this is not a book devoted to cryptography per se, it’s clear that the authors have actual experience in the area. The terms “in practice” and “practical” appear 27 times in the book, speaking of implementation details that go beyond naive application of theory. The reasons for these practical matters cannot always be explained — there’s only so much that can fit into any book, especially an introductory one — but the reader comes away with the impression that number theory actually plays an important role in cryptography.

John D. Cook is an independent consultant working in data privacy.

1. Introduction
2 Divisibility
3. Linear Diophantine Equations
4. Unique Factorization
5. Applications of Unique Factorization
6. Congruences
7. Classsical Cryposystems
8. Fermat, Euler, Wilson
9. RSA
10. Polynomial Congruences
11. Order and Primitive Roots
12. More Cryptographic Applications
13. Quadratic Reciprocity
14. Primality and Factorization
15. Geometry of Numbers
16. Arithmetic Functions
17. Continued Fractions
18. Gaussian Integers
19. Algebraic Integers
20. Analytic Methods
21. Epilogue: Fermat's Last Theorem
Answers and Hints for Odd-Numbered Exercises