Cryptology: An Historical Introduction DRAFT Jim Sauerberg February 5, 2013 2 Copyright 2013 All rights reserved Jim Sauerberg Saint Mary's College Contents List of Figures 8 1 Caesar Ciphers 9 1.1 Saint Cyr Slide . 12 1.2 Running Down the Alphabet . 14 1.3 Frequency Analysis . 15 1.4 Linquist's Method . 20 1.5 Summary . 22 1.6 Topics and Techniques . 22 1.7 Exercises . 23 2 Cryptologic Terms 29 3 The Introduction of Numbers 31 3.1 The Remainder Operator . 33 3.2 Modular Arithmetic . 38 3.3 Decimation Ciphers . 40 3.4 Deciphering Decimation Ciphers . 42 3.5 Multiplication vs. Addition . 44 3.6 Koblitz's Kid-RSA and Public Key Codes . 44 3.7 Summary . 48 3.8 Topics and Techniques . 48 3.9 Exercises . 49 4 The Euclidean Algorithm 55 4.1 Linear Ciphers . 55 4.2 GCD's and the Euclidean Algorithm . 56 4.3 Multiplicative Inverses . 59 4.4 Deciphering Decimation and Linear Ciphers . 63 4.5 Breaking Decimation and Linear Ciphers . 65 4.6 Summary . 67 4.7 Topics and Techniques . 67 4.8 Exercises . 68 3 4 CONTENTS 5 Monoalphabetic Ciphers 71 5.1 Keyword Ciphers . 72 5.2 Keyword Mixed Ciphers . 73 5.3 Keyword Transposed Ciphers . 74 5.4 Interrupted Keyword Ciphers . 75 5.5 Frequency Counts and Exhaustion . 76 5.6 Basic Letter Characteristics . 77 5.7 Aristocrats . 78 5.8 Summary . 80 5.9 Topics and Techniques . 81 5.10 Exercises . 81 6 Decrypting Monoalphabetic Ciphers 89 6.1 Letter Interactions . 90 6.2 Decrypting Monoalphabetic Ciphers . 91 6.3 Sukhotin's Method for Finding Vowels . 97 6.4 Final Monoalphabetic Tricks . 99 6.5 Summary . 101 6.6 Topics and Techniques . 102 6.7 Exercises . 102 7 Vigen`ereCiphers 109 7.1 Alberti, the Father of Western Cryptology . 110 7.2 Trithemius, the Father of Bibliography . 111 7.3 Belaso, the Unknown and Porta, the Great . 113 7.4 Vigen`ereCiphers . 114 7.5 Variants and Beaufort . 116 7.6 How to Break Vigen`ereCiphers . 117 7.7 The Kasiski Test . 120 7.8 Summary . 123 7.9 Topics and Techniques . 124 7.10 Exercises . 124 8 Polyalphabetic Ciphers 135 8.1 Coincidences . 135 8.2 The Measure of Roughness . 138 8.3 The Friedman Test . 142 8.4 Multiple Encipherings . 145 8.5 Vigen`ere'sAuto Key Cipher . 149 8.6 Perfect Secrecy . 152 8.7 Summary . 154 8.8 Terms and Topics . 155 8.9 Exercises . 156 CONTENTS 5 9 Digraphic Ciphers 167 9.1 Polygraphic Ciphers . 167 9.2 Hill Ciphers . 170 9.3 Recognizing and Breaking Polygraphic Ciphers . 174 9.4 Playfair . 176 9.5 Summary . 179 9.6 Topics and Techniques . 179 9.7 Exercises . 180 10 Transposition Ciphers 189 10.1 Route Ciphers . 189 10.2 Geometrical Ciphers . 190 10.3 Turning Grilles . 190 10.4 Columnar Transposition . 192 10.5 Transposition vs. Substitution . 195 10.6 Letter Connections . 196 10.7 Breaking the Columnar Transposition Cipher . 198 10.8 Double Transposition . 201 10.9 Transposition during the Civil War . 202 10.10 The Battle of the Civil War Ciphers . 207 10.11 Summary . 207 10.12 Topics and Techniques . 208 10.13 Exercises . 209 11 Knapsack Ciphers 219 11.1 The Knapsack Problem . 219 11.2 A Related Knapsack Problem . 220 11.3 An Easy Knapsack Problem . 221 11.4 The Knapsack Cipher System . 223 11.5 Public Key Cipher . 227 11.6 Summary . 228 11.7 Topics and Techniques . 228 11.8 Exercises . 228 12 RSA 231 12.1 Fermat's Theorem . 232 12.2 Complication I: a small one . 234 12.3 Complication II: a substantial one . 235 12.4 Complication III: a mini one . 238 12.5 Complication IV: the last one . 239 12.6 Putting It All Together . 241 12.7 Exponential Problems (and answers) . 241 12.8 RSA . 242 12.9 RSA and Public Keys . 245 12.10 How to break RSA . 245 12.11 Authenticity { Proof of Authorship . 248 6 CONTENTS 12.12 Summary . 249 12.13 Topics and Techniques . 250 12.14 Exercises . 251 Bibliography 253 Index 257 List of Figures 1.1 Saint Cyr Slide . 12 1.2 Decrypting a Caesar cipher by running down the alphabet . 15 1.3 Letter Frequency, in percent. From Sinkov. 16 1.4 English Letter Frequency Chart . 16 1.5 Letter Frequency Charts for Several Languages . 18 2.1 Alice, Bob and Eve: the three names of Cryptography . 30 3.1 The Standard Translation of Letters into Numbers . 31 3.2 Enciphering/Deciphering pairs modulo 26. 43 5.1 Letter Frequencies { Anywhere. 77 5.2 Letter Frequencies { Initial Letters. 77 5.3 Letter Frequencies { Final Letters. 78 5.4 Characteristics of etaoinshr..................... 78 5.5 Most Common Short Words . 78 5.6 The 100 Most Common Words in English . 79 6.1 Some Basic Letter Behaviors . 91 6.2 Digraph Table . 93 6.3 Digraph Table . 96 6.4 Letter Behaviors . 97 7.1 Trithemius' tabula recta . 112 7.2 Beaufort's Tableaux . 116 7.3 A Kasiski Table . 123 8.1 Frequency Counts: Same quote, different keylengths. 138 8.2 Larrabee's Cipher Code . 163 9.1 A Simple Digraphic Substitution Chart . 169 9.2 A More Complicated.