Description

Cryptological Mathematics introduces the reader to elementary yet rigorous mathematics from several different branches of the subject including number theory, abstract algebra, matrix algebra, probability and statistics, all of which play a role in enciphering and deciphering secret messages. And it does so in a light-hearted fashion, thanks to the carryings-on of the book's three main characters, Beth, Stephanie and Molly.

The book can be used at several different levels. Those interested in learning how to encipher messages using a variety of symmetric and asymmetric schemes will learn those techniques in this volume. Those interested in learning the mathematical foundations of these techniques will learn how and why the enciphering schemes actually work. And those who are looking for some challenging computer programming exercises will also find exercises in this book to keep them busy. So, the book may be used profitably by amateur cryptographers, motivated liberal arts students, and mathematics or computer science majors alike.

The book begins by considering simple monoalphabetic substitution ciphers (additive, multiplicative, affine and keyword) and then moves through a polyalphabetic substitution cipher (the Vigenère Square) and on to polygraphic substitution ciphers (including Hill's system). Finally the RSA Public Key Encryption System is introduced along with several of its variations. At each step the mathematics required to understand these systems is introduced in a thorough yet gentle manner.

The book also invites those readers who may be interested to program the algorithms it describes. Some exercises can be done using a simple spreadsheet program; others are best handled by a symbolic algebra system.

In addition, the book contains biographical sketches of some lesser-known yet fascinating personalities who have been involved with cryptography.

Whether a reader is interested in cryptology, mathematics, or computer science, he or she will discover engaging and delightful material in this volume.

TABLE OF CONTENTS

Chapter One - Monoalphabetic Substitution Ciphers

Section 1.1 - Well-Ordering Axiom, Principle of Mathematical Induction
The Well-Ordering Axiom
The Principle of Mathematical Induction
Exercises 1.1

Section 1.2 - Prime Numbers, Division Algorithm, Greatest Common Divisor
Algorithm for Finding gcd (a,b)
Exercises 1.2

Section 1.3 - Relatively Prime Integers, Fundamental Theorem of Arithmetic
The Fundamental Theorem of Arithmetic
Exercises 1.3

Section 1.4 - Modular Arithmetic
Exercises 1.4

Section 1.5 - Simple Ciphers
The Cast Of Characters
Additive Ciphers
Multiplicative Ciphers
Affine Ciphers
Keyword Ciphers
Exercises 1.5

Section 1.6 - Cryptanalysis of Monoalphabetic Substitution Ciphers
Exercises 1.6

Section 1.7 - Personalities (Herbert Yardley)

Chapter Two - Polyalphabetic Substitution Ciphers

Section 2.1 - The Multiplication Principle
Exercises 2.1

Section 2.2 - Permutations and Combinations
Exercises 2.2

Section 2.3 - Probability
Some Definitions
Determining the Probability of an Event
Properties of Probability
Exercises 2.3

Section 2.4 - Independent Events and Expected Value
Independent Events
Expected Number
Exercises 2.4

Section 2.5 - Disguising the Frequencies
Changing Letters to Numbers
The Vigenère Square
Enciphering a Message Using the Vigenère Square
Deciphering a Message Using the Vigenère Square
Decrypting a Message Using the Vigenère Square
The Friedman Test
Determining the Length of the Keyword
Exercises 2.5

Section 2.6 - Personalities (William Friedman)

Chapter Three - Polygraphic Substitution Ciphers

Section 3.1 - Elementary Polygraphic Substitution Ciphers
Exercises 3.1

Section 3.2 - Elementary Matrix Theory
Identities and Inverses
Linear Systems of Equations
Geometrical Transformations
Exercises 3.2

Section 3.3 - Hill's System
Enciphering and Deciphering a Message Using Hill's System
Cryptanalysis of a Message Enciphered Using Hill's System
Exercises 3.3

Section 3.4 - Personalities (Agnes Meyer Driscoll)

Chapter Four - Public Key Cryptography

Section 4.1 - More Number Theory
Exercises 4.1

Section 4.2 - The RSA Algorithm
Generation of Keys
Exchanging Messages
Frequently Asked Questions
Exercises 4.2

Section 4.3 - Two Examples
Mathematica
Maple
Exercises 4.3

Section 4.4 - Other Illustrations of Public Key Cryptography
Signature Authentication
Hybrid Systems
Exercises 4.4

Section 4.5 - Personalities (Frank Rowlett)

Appendices

Appendix A - ASCII Code
Appendix B - Taxonomy of Cryptology
Appendix C - Answers to Even-Numbered Problems

Catalog Code: CPT/W
xxx pp., Paperbound, 2000
ISBN 0-88385-719-7

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