The Mathematics of Secrets: Cryptography from Caesar Ciphers to Digital Encryption

by Joshua Holden

Overview

Explaining the mathematics of cryptography

The Mathematics of Secrets takes readers on a fascinating tour of the mathematics behind cryptography—the science of sending secret messages. Using a wide range of historical anecdotes and real-world examples, Joshua Holden shows how mathematical principles underpin the ways that different codes and ciphers work. He focuses on both code making and code breaking and discusses most of the ancient and modern ciphers that are currently known. He begins by looking at substitution ciphers, and then discusses how to introduce flexibility and additional notation. Holden goes on to explore polyalphabetic substitution ciphers, transposition ciphers, connections between ciphers and computer encryption, stream ciphers, public-key ciphers, and ciphers involving exponentiation. He concludes by looking at the future of ciphers and where cryptography might be headed. The Mathematics of Secrets reveals the mathematics working stealthily in the science of coded messages.

A blog describing new developments and historical discoveries in cryptography related to the material in this book is accessible at http://press.princeton.edu/titles/10826.html.

Product Details

ISBN-13:	9780691183312
Publisher:	Princeton University Press
Publication date:	10/02/2018
Pages:	392
Product dimensions:	6.00(w) x 9.20(h) x 1.00(d)

About the Author

Joshua Holden is professor of mathematics at the Rose-Hulman Institute of Technology.

Table of Contents

Preface xi

Acknowledgments xiii

Introduction to Ciphers and Substitution 1

1.1 Alice and Bob and Carl and Julius: Terminology and Caesar Cipher 1

1.2 The Key to the Matter: Generalizing the Caesar Cipher 4

1.3 Multiplicative Ciphers 6

1.4 Affine Ciphers 15

1.5 Attack at Dawn: Cryptanalysis of Sample Substitution Ciphers 18

1.6 Just to Get Up That Hill: Polygraphic Substitution Ciphers 20

1.7 Known-Plaintext Attacks 25

1.8 Looking Forward 26

Polyalphabetic Substitution Ciphers 29

2.1 Homophonic Ciphers 29

2.2 Coincidence or Conspiracy? 31

2.3 Alberti Ciphers 36

2.4 It’s Hip to Be Square: Tabula Recta or Vigenère Square Ciphers 39

2.5 How Many Is Many? Determining the Number of
Alphabets 43

2.6 Superman Is Staying for Dinner: Superimposition and Reduction 52

2.7 Products of Polyalphabetic Ciphers 55

2.8 Pinwheel Machines and Rotor Machines 58

2.9 Looking Forward 73

Transposition Ciphers 75

3.1 This Is Sparta! The Scytale 75

3.2 Rails and Routes: Geometric Transposition Ciphers 78

3.3 Permutations and Permutation Ciphers 81

3.4 Permutation Products 86

3.5 Keyed Columnar Transposition Ciphers 91

Sidebar 3.1 Functional Nihilism 94

3.6 Determining the Width of the Rectangle 97

3.7 Anagramming 101

Sidebar 3.2 But When You Talk about Disruption 104

3.8 Looking Forward 106

Ciphers and Computers 109

4.1 Bringing Home the Bacon: Polyliteral Ciphers and Binary Numerals 109

4.2 Fractionating Ciphers 115

4.3 How to Design a Digital Cipher: SP-Networks and Feistel Networks 119

Sidebar 4.1 Digitizing Plaintext 125

4.4 The Data Encryption Standard 130

4.5 The Advanced Encryption Standard 135

4.6 Looking Forward 143

Stream Ciphers 145

5.1 Running-Key Ciphers 145

Sidebar 5.1 We Have All Been Here Before 150

5.2 One-Time Pads 153

5.3 Baby You Can Drive My Car: Autokey Ciphers 157

5.4 Linear Feedback Shift Registers 167

5.5 Adding Nonlinearity to LFSRs 174

5.6 Looking Forward 178

Ciphers Involving Exponentiation 182

6.1 Encrypting Using Exponentiation 182

6.2 Fermat’s Little Theorem 183

6.3 Decrypting Using Exponentiation 186

6.4 The Discrete Logarithm Problem 188

6.5 Composite Moduli 190

6.6 The Euler Phi Function 192

6.7 Decryption with Composite Moduli 195

Sidebar 6.1 Fee-fi-fo-fum 197

6.8 Looking Forward 199

Public-Key Ciphers 201

7.1 Right out in Public: The Idea of Public-Key Ciphers 201

7.2 Diffie-Hellman Key Agreement 207

7.3 Asymmetric-Key Cryptography 213

7.4 RSA 216

7.5 Priming the Pump: Primality Testing 222

7.6 Why is RSA a (Good) Public-Key System? 226

7.7 Cryptanalysis of RSA 229

7.8 Looking Forward 233

Appendix A The Secret History of Public-Key Cryptography 235

Other Public-Key Systems 241

8.1 The Three-Pass Protocol 241

8.2 ElGamal 247

8.3 Elliptic Curve Cryptography 251

8.4 Digital Signatures 265

8.5 Looking Forward 271

The Future of Cryptography 276

9.1 Quantum Computing 276

9.2 Postquantum Cryptography 281

9.3 Quantum Cryptography 292

9.4 Looking Forward 301

List of Symbols 303

Notes 305

Suggestions for Further Reading 345

Bibliography 349

Index 367

What People are Saying About This

From the Publisher

"Suitable for anyone with a basic understanding of high school math, The Mathematics of Secrets presents the theoretical principles of cryptography, from Julius Caesar's primitive cipher to the intricacies of the modern digital signature. This terrific book is a testament to the almost supernatural power of mathematics."—Paul J. Nahin, author of In Praise of Simple Physics

Editorial Reviews

"For anyone with an interest in cryptography."—-Noel-Ann Bradshaw, Times Higher Education

"Any book on cryptography written for a more-or-less lay audience must inevitably face comparisons to The Code Book . . . by Simon Singh. . . . The Mathematics of Secrets is tilted (and indeed titled) more towards a fuller explanation of the mathematical techniques underlying the various ciphers. . . . [F]or anyone who wants to go a bit deeper than Simon Singh took them."—-Paul Taylor, Aperiodical

"The best book I have seen on this subject and should be useful to students as well as enjoyable to casual readers."—-Phil Dyke, Leonardo Reviews

"This is an extremely clearly, well-written book covering a lot of ground in the mathematics of cyphers."—-Jonathan Shock, Mathemafrica

"This book engaging and very well written, explaining some rather difficult concepts and using innovative (and consistent) approaches to promote this understanding"—-Edward Rochead, Mathematics Today

From the Publisher

"A fascinating tour of the mathematics behind cryptography, showing how its principles underpin the ways that different codes and ciphers operate. . . . While it's all about maths, [The Mathematics of Secrets] is accessible—basic high school algebra is all that’s needed to understand and enjoy it."

Cosmos Magazine