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Mathematics of Secrets: Cryptography from Caesar Ciphers to Digital Encryption

by Joshua Holden Princeton University Press
Pub Date:
Hbk 392 pages
AU$76.99 NZ$79.12
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The Mathematics of Secrets takes readers on a fascinating tour of the mathematics behind cryptography - the science of sending secret messages. Most books about cryptography are organized historically, or around how codes and ciphers have been used, such as in government and military intelligence or bank transactions. Joshua Holden instead shows how mathematical principles underpin the ways that different codes and ciphers operate. Holden focuses on both code making and code breaking and he discusses the majority of ancient and modern ciphers currently known.


Holden begins by looking at substitution ciphers, built by substituting one letter or block of letters for another. Explaining one of the simplest and historically well-known ciphers, the Caesar cipher, Holden establishes the key mathematical idea behind the cipher and discusses how to introduce flexibility and additional notation. Holden goes on to explore polyalphabetic substitution ciphers, transposition ciphers, including one developed by the Spartans, connections between ciphers and computer encryption, stream ciphers, and ciphers involving exponentiation. He also examines public-key ciphers, where the methods used to encrypt messages are public knowledge, and yet, intended recipients are still the only ones who are able to read the message. He concludes with a look at the future of ciphers and where cryptography might be headed. Only basic mathematics up to high school algebra is needed to understand and enjoy the book.


With a plethora of historical anecdotes and real-world examples, The Mathematics of Secrets reveals the mathematics working stealthily in the science of coded messages.




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

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