A cipher is a scheme for creating coded messages for the secure exchange of information. Throughout history, many different coding schemes have been devised. One of the oldest and simplest mathematical systems was used by Julius Caesar. This is where Mathematical Ciphersbegins. Building on that simple system, Young moves on to more complicated schemes, ultimately ending with the RSA cipher, which is used to provide security for the Internet.
This book is structured differently from most mathematics texts. It does not begin with a mathematical topic, but rather with a cipher. The mathematics is developed as it is needed; the applications motivate the mathematics. As is typical in mathematics textbooks, most chapters end with exercises. Many of these problems are similar to solved examples and are designed to assist the reader in mastering the basic material. A few of the exercises are one-of-a-kind, intended to challenge the interested reader.
Implementing encryption schemes is considerably easier with the use of the computer. For all the ciphers introduced in this book, JavaScript programs are available from the Web.
In addition to developing various encryption schemes, this book also introduces the reader to number theory. Here, the study of integers and their properties is placed in the exciting and modern context of cryptology. Mathematical Ciphers can be used as a textbook for an introductory course in mathematics for all majors. The only prerequisite is high school mathematics.
Anne L. Young: Loyola College in Maryland, Baltimore, MD
• Preface 8 • Chapter 1. Introduction 10 • Chapter 2. Caesar Cipher 12 • Chapter 3. Terminology and Results from Number Theory 18 • Chapter 4. Modular Arithmetic 26 • Chapter 5. Describing the Caesar Cipher Mathematically 36 • Chapter 6. Cryptanalysis for the Caesar Cipher 40 • Chapter 7. Multiplication Cipher 46 • Chapter 8. Cryptanalysis for the Multiplication Cipher 56 • Chapter 9. Multiplication-Shift Cipher 60 • Chapter 10. Cryptanalysis for the Multiplication-Shift Cipher 68 • Chapter 11. Non-Mathematical Substitution Ciphers 78 • Chapter 12. Preparing to Generalize 84 • Chapter 13. Finding Inverses Modulo n 90 • Chapter 14. General Multiplication-Shift Cipher 98 • Chapter 15. Security of the General Multiplication-Shift Cipher 102 • Chapter 16. Introduction to the Exponential Cipher 112 • Chapter 17. Deciphering the Exponential Cipher 120 • Chapter 18. Cryptanalysis for the Exponential Cipher 128 • Chapter 19. Mathematical Basis for the Exponential Cipher 132 • Chapter 20. Public Key Ciphers 136 • Chapter 21. RSA Cipher 140 • Chapter 22. Signatures 142 • Chapter 23. Security and Implementation of the RSA Cipher 152 • Chapter 24. Computer Programs 158 • Chapter 25. Further Reading 160 • Chapter 26. Answers to Selected Exercises 162 • Index 166