Cryptological Mathematics (Classroom Resource Materials Series) / Edition 1

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Brand new. We distribute directly for the publisher. 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 Vignere 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 had been involved with cryptography. Read more Show Less

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Overview

This is an introduction to the mathematics involved in the intriguing field of cryptology, the science of writing and reading secret messages which are designed to be read only by their intended recipients. It is written at an elementary level, suitable for beginning undergraduates, with careful explanations of all the concepts used. The basic branches of mathematics required, including number theory, abstract algebra and probability, are used to show how to encipher and decipher messages, and why this works, giving a practical as well as theoretical basis to the subject. Challenging computer programming exercises are also included. The book is written in an engaging style which will appeal to all, and also includes historical background on some of the founders of the subject. It will be of interest both to students wishing to learn cryptology per se, and also to those searching for practical applications of seemingly abstract mathematics.

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Editorial Reviews

Neal Koblitz
This book is written with careful attention to the challenge of motivating a humanities or business major. An impressive amount of the mathematics that might make a standard "math for poets" course-mathematical induction, elementary number theory, probabilities, matrices-is presented through the vehicle of cryptography.
American Mathematical Monthly
Booknews
A textbook for a one-semester undergraduate course for students who will or will not go on to more advanced mathematics. Lewant (Groucher College, Baltimore) draws together elementary mathematics from number theory, abstract algebra, matrix algebra, probability, and statistics. The point is not so much to learn to encode and decode, but to use cryptology as a fun way to learn the math itself. Annotation c. Book News, Inc., Portland, OR (booknews.com)
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Product Details

  • ISBN-13: 9780883857199
  • Publisher: Mathematical Association of America
  • Publication date: 11/1/2000
  • Series: Classroom Resource Materials Series
  • Edition description: New Edition
  • Edition number: 1
  • Pages: 220
  • Product dimensions: 7.02 (w) x 9.96 (h) x 0.46 (d)

Table of Contents

Preface xi
1 Monoalphabetic Substitution Ciphers 1
1.1 Well-Ordering Axiom, Principle of Mathematical Induction 2
The Well-Ordering Axiom 2
The Principle of Mathematical Induction 3
Exercises 1.1 7
1.2 Prime Numbers, Division Algorithm, Greatest Common Divisor 7
Exercises 1.2 15
1.3 Relatively Prime Integers, Fundamental Theorem of Arithmetic 16
The Fundamental Theorem of Arithmetic 17
Exercises 1.3 18
1.4 Modular Arithmetic 19
Exercises 1.4 26
1.5 Simple Ciphers 27
The Cast of Characters 27
Additive Ciphers 28
Multiplicative Ciphers 30
Affine Ciphers 32
Keyword Ciphers 33
Exercises 1.5 34
1.6 Cryptanalysis of Monoalphabetic Substitution Ciphers 35
Exercises 1.6 41
1.7 Personalities 42
2 Polyalphabetic Substitution Ciphers 45
2.1 The Multiplication Principle 46
Exercises 2.1 50
2.2 Permutations and Combinations 51
Exercises 2.2 59
2.3 Probability 61
Some Definitions 62
Determining the Probability of an Event 63
Properties of Probability 67
Exercises 2.3 70
2.4 Independent Events and Expected Number 72
Independent Events 72
Expected Number 74
Exercises 2.4 75
2.5 Disguising the Frequencies 76
Changing Letters to Numbers 77
The Vigenere Square 79
Enciphering a Message Using the Vigenere Square 79
Deciphering a Message Using the Vigenere Square 81
Decrypting a Message Using the Vigenere Square 82
The Friedman Test 86
Determining the Length of the Keyword 89
Exercises 2.5 97
2.6 Personalities 99
3 Polygraphic Substitution Ciphers 103
3.1 Elementary Polygraphic Substitution Ciphers 104
Exercises 3.1 108
3.2 Elementary Matrix Theory 108
Identities and Inverses 113
Linear Systems of Equations 116
Geometrical Transformations 118
Exercises 3.2 121
3.3 Hill's System 124
Enciphering and Deciphering a Message Using Hill's System 124
Cryptanalysis of a Message Enciphered Using Hill's System 128
Exercises 3.3 137
3.4 Personalities 138
4 Public Key Cryptography 141
4.1 More Number Theory 142
Exercises 4.1 150
4.2 The RSA Algorithm 150
Generation of Keys 151
Exchanging Messages 151
Frequently Asked Questions 154
Exercises 4.2 159
4.3 Two Examples 160
Exercises 4.3 170
4.4 Other Illustrations of Public Key Cryptography 171
Signature Authentication 171
Hybrid Systems 172
Use RSA to Transmit the Key 172
The Diffie-Hellman Key Exchange System 173
The Massey-Omura System 174
Exercises 4.4 177
4.5 Personalities 178
Appendix A ASCII Code 181
Appendix B Taxonomy of Cryptology 183
Appendix C Answers to Even-Numbered Problems 185
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