An Introduction to Number Theory with Cryptography

Building on the success of the first edition, An Introduction to Number Theory with Cryptography, Second Edition, increases coverage of the popular and important topic of cryptography, integrating it with traditional topics in number theory.

The authors have written the text in an engaging style to reflect number theory’s increasing popularity. The book is designed to be used by sophomore, junior, and senior undergraduates, but it is also accessible to advanced high school students and is appropriate for independent study. It includes a few more advanced topics for students who wish to explore beyond the traditional curriculum.

Features of the second edition include

• Over 800 exercises, projects, and computer explorations
• Increased coverage of cryptography, including Vigenere, Stream, Transposition, and Block ciphers, along with RSA and discrete log-based systems
• “Check Your Understanding” questions for instant feedback to students
• New Appendices on “What is a proof?” and on Matrices
• Select basic (pre-RSA) cryptography now placed in an earlier chapter so that the topic can be covered right after the basic material on congruences
• Answers and hints for odd-numbered problems

Table of Contents

1: Introduction

2: Divisibility

3: Linear Diophantine Equations

4: Unique Factorization

5: Applications of Unique Factorization

6: Congruences

7: Classical Cryptosystems

8: Fermat, Euler, and 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: The Distribution of Primes

21: Epilogue: Fermat’s Last Theorem

A: Supplementary Topics

B: Answers and Hints for Odd-Numbered Exercises

About the Authors

James S. Kraft received his Ph.D. from the University of Maryland in 1987 and has published several research papers in algebraic number theory. His previous teaching positions include the University of Rochester, St. Mary’s College of California, and Ithaca College, and he has also worked in communications security. Dr. Kraft currently teaches mathematics at the Gilman School.

Lawrence C. Washington received his Ph.D. from Princeton University in 1974 and has published extensively in number theory, including books on cryptography (with Wade Trappe), cyclotomic fields, and elliptic curves. Dr. Washington is currently Professor of Mathematics and Distinguished Scholar-Teacher at the University of Maryland.