Softcover ISBN:  9780821837306 
Product Code:  MAWRLD/25 
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AMS Member Price:  $27.20 
Electronic ISBN:  9781470411954 
Product Code:  MAWRLD/25.E 
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Book DetailsMathematical WorldVolume: 25; 2006; 159 ppMSC: Primary 11; 94;
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 Ciphers begins. 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 oneofakind, 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.ReadershipUndergraduate students interested in number theory, cryptology, and discrete mathematics.

Table of Contents

Chapters

Chapter 1. Introduction

Chapter 2. Caesar cipher

Chapter 3. Terminology and results from number theory

Chapter 4. Modular arithmetic

Chapter 5. Describing the Caesar cipher mathematically

Chapter 6. Cryptanalysis for the Caesar cipher

Chapter 7. Multiplication cipher

Chapter 8. Cryptanalysis for the multiplication cipher

Chapter 9. Multiplicationshift cipher

Chapter 10. Cryptanalysis for the multiplicationshift cipher

Chapter 11. Nonmathematical substitution ciphers

Chapter 12. Preparing to generalize

Chapter 13. Finding inverses modulo $n$

Chapter 14. General multiplicationshift cipher

Chapter 15. Security of the general multiplicationshift cipher

Chapter 16. Introduction to the exponential cipher

Chapter 17. Deciphering the exponential cipher

Chapter 18. Cryptanalysis for the exponential cipher

Chapter 19. Mathematical basis for the exponential cipher

Chapter 20. Public key ciphers

Chapter 21. RSA cipher

Chapter 22. Signatures

Chapter 23. Security and implementation of the RSA cipher

Chapter 24. Computer programs

Chapter 25. Further reading

Chapter 26. Answers to selected exercises

27. Index


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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 Ciphers begins. 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 oneofakind, 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.
Undergraduate students interested in number theory, cryptology, and discrete mathematics.

Chapters

Chapter 1. Introduction

Chapter 2. Caesar cipher

Chapter 3. Terminology and results from number theory

Chapter 4. Modular arithmetic

Chapter 5. Describing the Caesar cipher mathematically

Chapter 6. Cryptanalysis for the Caesar cipher

Chapter 7. Multiplication cipher

Chapter 8. Cryptanalysis for the multiplication cipher

Chapter 9. Multiplicationshift cipher

Chapter 10. Cryptanalysis for the multiplicationshift cipher

Chapter 11. Nonmathematical substitution ciphers

Chapter 12. Preparing to generalize

Chapter 13. Finding inverses modulo $n$

Chapter 14. General multiplicationshift cipher

Chapter 15. Security of the general multiplicationshift cipher

Chapter 16. Introduction to the exponential cipher

Chapter 17. Deciphering the exponential cipher

Chapter 18. Cryptanalysis for the exponential cipher

Chapter 19. Mathematical basis for the exponential cipher

Chapter 20. Public key ciphers

Chapter 21. RSA cipher

Chapter 22. Signatures

Chapter 23. Security and implementation of the RSA cipher

Chapter 24. Computer programs

Chapter 25. Further reading

Chapter 26. Answers to selected exercises

27. Index