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144 CHAPTER 8. POLYALPHABETIC CIPHERS Example: Compute Φ for the following message, and then decrypt it. EFBTL QUUEH JMDRV EFBTL CBDPQ UWVEJ GGRQW VWPIK EOOPW FHZMF QFUIU KDUSU CZVGA GBFIK CBGMF HIQGL KCQXZ GMLRV GSGQA TFRVG PSDRG VVHVO JOWSF GRRIK VVHSL JSUYF FCHWL JSLVF CHXVW UVRAW XSUHA HTHVX WBGEE GBWED NMFVQ RHRKJ CDKCA UHKIG TSWMU CZDRV CPVXJ CQWGJ ADWEF CZBWA UWVIE RWUMU CZDRV ECQGJ GHHHS XWGOS JB The frequency count is A B C D E F G H I J K L M N O P Q R S T U V W X Y Z total 7 9 14 9 11 14 19 15 7 10 8 7 7 1 5 5 10 13 11 5 15 20 17 6 1 6 252 Computing, Φ = 0.0446152, which suggests a keyword length of 5. We next use Kasiski’s idea of building up a depth based on a keylength of 5: First letters: EQJECUGVEFQKCGCHKGGTPVJGVJFJCUXHWGNRCUTCCCACURCEGXJ A B C D E F G H I J K L M N O P Q R S T U V W X Y Z total 1 0 10 0 4 2 7 2 0 5 2 0 0 1 0 1 2 2 0 2 4 3 1 2 0 0 51 Φ 1 = 0.0768627. (Here Φ 1 represents the computation for the letters enciphered by the 1st letter of the keyword.) Second letters: FUMFBWGWOHFDZBBICMSFSVORVSCSHVSTBBMHDHSZPQDZWWZCHWB A B C D E F G H I J K L M N O P Q R S T U V W X Y Z total 0 6 3 3 0 4 1 5 1 0 0 0 3 0 2 1 1 1 6 1 1 3 5 0 0 4 51 Φ 2 = 0.0588235. Third letters: BUDBDVRPOZUUVFGQQLGRDHWRHUHLXRUHGWFRKKWDVWWBVUDQHG A B C D E F G H I J K L M N O P Q R S T U V W X Y Z total 0 3 0 5 0 2 4 5 0 0 2 2 0 0 1 1 3 5 0 0 6 4 5 1 0 1 50 Φ 3 = 0.0620408. Fourth letters: TERTPEQIPMISGIMGXRQVRVSISYWVVAHVEEVKCIMRXGEWIMRGHO A B C D E F G H I J K L M N O P Q R S T U V W X Y Z total 1 0 1 0 5 0 4 2 6 0 1 0 4 0 1 2 2 5 3 2 0 6 2 2 1 0 50 Φ 4 = 0.0579592. Fifth letters: LHVLQJWKWFUUAKFLZVAGGOFKLFLFWWAXEDQJAGUVJJFAEUVJSS A B C D E F G H I J K L M N O P Q R S T U V W X Y Z total 5 0 0 1 2 6 3 1 0 5 3 5 0 0 1 0 2 0 2 0 4 4 4 1 0 1 50 Φ 5 = 0.0587755. The values of Φ for each of the depths are larger than 0.052, so within the range of values we’d expect from the Friedman test for a monoalphabetic cipher.
8.4. MULTIPLE ENCIPHERINGS 145 Being confident of our keyword length, we now have five simple Caesar ciphers to break and soon the message is decrypted. 8 To further emphasize the power of Friedman’s Index, we provide the values of Φ that are produced assuming various keylengths. Keylength Φ 1 Φ 2 Φ 3 Φ 4 Φ 5 Φ 6 Φ 7 Average One .045 .045 Two .046 .042 .044 Three .047 .043 .045 .045 Four .048 .044 .052 .036 .045 Five .077 .059 .062 .058 .059 .063 Six .048 .045 .046 .041 .038 .043 .043 Seven .044 .052 .044 .043 .048 .048 .040 .046 The values in the Keylength Five row are clearly the largest. In particular, the average of .063 stands out. Five must indeed be the keylength. Especially when coupled with a computer program, Friedman’s Test provides for the almost routine determination of keylength. ⋄ As a method for computing the keylength for a ciphertext of only a couple hundred letters the Friedman Test is too highly influenced by quirks of the individual ciphertext and keyword. However, as the ciphertext grows, the Friedman Test hones in on the keylength with amazing accuracy. 8.4 Multiple Encipherings Kasiski’s and Friedman’s tests are a very powerful one-two combination to use on polyalphbetic ciphers, especially when used on a computer, enabling the almost automatic detection of a polyalphabetic cipher’s keylength. Is there anything about the polyalphabetic ciphers that might be saved Ever since Section 7.6 we have been working under the (often implicit) assumption that determining the keylength of a polyalphabetic ciphertext was hard, but once we know the keylength breaking the cipher was easy. This is because the polyalphabetic ciphers are really just several Caesar ciphers twisted together – once we could pull the strands apart we were back to Chapter 1 material. Since it appears relatively easy to pull the strands apart, is there a way to make the strands shorter That is, to make the number of letters per Caesar alphabet so small that decryption of the individual Kasiski depths is difficult For example, if our message has 1000 letters in it, but we could use a 100 letter keyword, there would be only 10 letters per alphabet! Even Linquist’s method 8 Keyword is CODES. “Cryptography and cryptanalysis are sometimes called twin or reciprocal sciences, and in function they indeed mirror one another. What one does the other undoes. Their natures, however, differ fundamentally. Cryptography is theoretical and abstract. Cryptanalysis is empirical and concrete.” David Kahn
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Cryptology: An Historical Introduct
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Contents List of Figures 8 1 Caesar
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CONTENTS 5 9 Digraphic Ciphers 167
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List of Figures 1.1 Saint Cyr Slide
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Chapter 1 Caesar Ciphers There are
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11 Examples: Encipher or decipher t
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1.1. SAINT CYR SLIDE 13 paper turne
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1.3. FREQUENCY ANALYSIS 15 PX PBEE
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1.3. FREQUENCY ANALYSIS 17 A bit mo
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1.3. FREQUENCY ANALYSIS 19 Examples
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1.4. LINQUIST’S METHOD 21 the cip
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1.7. EXERCISES 23 12. What is the m
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1.7. EXERCISES 25 (d) VTXLTKBTG LXV
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1.7. EXERCISES 27 (d) IKSUT JKIOT K
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Chapter 2 Cryptologic Terms Cryptog
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Chapter 3 The Introduction of Numbe
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3.1. THE REMAINDER OPERATOR 33 Exam
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3.1. THE REMAINDER OPERATOR 35 Perf
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3.1. THE REMAINDER OPERATOR 37 As a
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3.2. MODULAR ARITHMETIC 39 While %
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3.3. DECIMATION CIPHERS 41 previous
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3.4. DECIPHERING DECIMATION CIPHERS
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3.6. KOBLITZ’S KID-RSA AND PUBLIC
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3.6. KOBLITZ’S KID-RSA AND PUBLIC
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3.9. EXERCISES 49 8. How do we enci
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3.9. EXERCISES 51 (c) Encipher 54 4
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3.9. EXERCISES 53 (b) Encipher secr
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Chapter 4 The Euclidean Algorithm I
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4.2. GCD’S AND THE EUCLIDEAN ALGO
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4.3. MULTIPLICATIVE INVERSES 59 (2)
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4.3. MULTIPLICATIVE INVERSES 61 (2)
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4.4. DECIPHERING DECIMATION AND LIN
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4.5. BREAKING DECIMATION AND LINEAR
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4.6. SUMMARY 67 To put it more blun
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4.8. EXERCISES 69 5. Here we work w
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Chapter 5 Monoalphabetic Ciphers He
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5.2. KEYWORD MIXED CIPHERS 73 Examp
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5.4. INTERRUPTED KEYWORD CIPHERS 75
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5.6. BASIC LETTER CHARACTERISTICS 7
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5.7. ARISTOCRATS 79 the 69971 or 42
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5.9. TOPICS AND TECHNIQUES 81 5.9 T
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5.10. EXERCISES 83 (h) DYVTP KKIQ.
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5.10. EXERCISES 85 13. Ciphers in w
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5.10. EXERCISES 87 17. General Pier
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Chapter 6 Decrypting Monoalphabetic
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6.2. DECRYPTING MONOALPHABETIC CIPH
Page 93 and 94: 6.2. DECRYPTING MONOALPHABETIC CIPH
Page 95 and 96: 6.2. DECRYPTING MONOALPHABETIC CIPH
Page 97 and 98: 6.3. SUKHOTIN’S METHOD FOR FINDIN
Page 99 and 100: 6.4. FINAL MONOALPHABETIC TRICKS 99
Page 101 and 102: 6.5. SUMMARY 101 our ciphertext muc
Page 103 and 104: 6.7. EXERCISES 103 Frequency count:
Page 105 and 106: 6.7. EXERCISES 105 53++!305))6*;482
Page 107 and 108: 6.7. EXERCISES 107 93 93 82 48 39 6
Page 109 and 110: Chapter 7 Vigenère Ciphers It was
Page 111 and 112: 7.2. TRITHEMIUS, THE FATHER OF BIBL
Page 113 and 114: 7.3. BELASO, THE UNKNOWN AND PORTA,
Page 115 and 116: 7.4. VIGENÈRE CIPHERS 115 to encip
Page 117 and 118: 7.6. HOW TO BREAK VIGENÈRE CIPHERS
Page 119 and 120: 7.6. HOW TO BREAK VIGENÈRE CIPHERS
Page 121 and 122: 7.7. THE KASISKI TEST 121 Kasiski
Page 123 and 124: 7.8. SUMMARY 123 Repetition Start P
Page 125 and 126: 7.10. EXERCISES 125 2. Little Miss
Page 127 and 128: 7.10. EXERCISES 127 to Gen. E. K. S
Page 129 and 130: 7.10. EXERCISES 129 MEJMY JKXCD LCN
Page 131 and 132: 7.10. EXERCISES 131 22. (a) Use Kas
Page 133 and 134: 7.10. EXERCISES 133 (a) Construct a
Page 135 and 136: Chapter 8 Polyalphabetic Ciphers Th
Page 137 and 138: 8.1. COINCIDENCES 137 (being born o
Page 139 and 140: 8.2. THE MEASURE OF ROUGHNESS 139 I
Page 141 and 142: 8.2. THE MEASURE OF ROUGHNESS 141 P
Page 143: 8.3. THE FRIEDMAN TEST 143 The Frie
Page 147 and 148: 8.4. MULTIPLE ENCIPHERINGS 147 Just
Page 149 and 150: 8.5. VIGENÈRE’S AUTO KEY CIPHER
Page 151 and 152: 8.5. VIGENÈRE’S AUTO KEY CIPHER
Page 153 and 154: 8.6. PERFECT SECRECY 153 invent now
Page 155 and 156: 8.8. TERMS AND TOPICS 155 multiple
Page 157 and 158: 8.9. EXERCISES 157 6. As is this on
Page 159 and 160: 8.9. EXERCISES 159 13. It has been
Page 161 and 162: 8.9. EXERCISES 161 This is Equation
Page 163 and 164: 8.9. EXERCISES 163 A: AB CDE FGH I
Page 165 and 166: 8.9. EXERCISES 165 26. Decipher SVQ
Page 167 and 168: Chapter 9 Digraphic Ciphers Althoug
Page 169 and 170: 9.1. POLYGRAPHIC CIPHERS 169 f a g
Page 171 and 172: 9.2. HILL CIPHERS 171 th 2.96 es 1.
Page 173 and 174: 9.2. HILL CIPHERS 173 We will be us
Page 175 and 176: 9.3. RECOGNIZING AND BREAKING POLYG
Page 177 and 178: 9.4. PLAYFAIR 177 Playfair Ciphers:
Page 179 and 180: 9.5. SUMMARY 179 9.5 Summary Polygr
Page 181 and 182: 9.7. EXERCISES 181 (b) The use of
Page 183 and 184: 9.7. EXERCISES 183 10. Just as we c
Page 185 and 186: 9.7. EXERCISES 185 (d) (For those(
Page 187 and 188: 9.7. EXERCISES 187 The rules that W
Page 189 and 190: Chapter 10 Transposition Ciphers In
Page 191 and 192: 10.3. TURNING GRILLES 191 1 2 3 7 4
Page 193 and 194: 10.4. COLUMNAR TRANSPOSITION 193 co
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10.5. TRANSPOSITION VS. SUBSTITUTIO
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10.6. LETTER CONNECTIONS 197 F 2 G
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10.7. BREAKING THE COLUMNAR TRANSPO
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10.8. DOUBLE TRANSPOSITION 201 Sinc
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10.9. TRANSPOSITION DURING THE CIVI
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10.9. TRANSPOSITION DURING THE CIVI
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10.10. THE BATTLE OF THE CIVIL WAR
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10.13. EXERCISES 209 14. What is a
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10.13. EXERCISES 211 and the second
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10.13. EXERCISES 213 evening Adam h
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10.13. EXERCISES 215 Several codewo
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10.13. EXERCISES 217 4. (which is t
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Chapter 11 Knapsack Ciphers Merkle
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11.3. AN EASY KNAPSACK PROBLEM 221
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11.4. THE KNAPSACK CIPHER SYSTEM 22
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11.4. THE KNAPSACK CIPHER SYSTEM 22
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11.5. PUBLIC KEY CIPHER 227 Even wi
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11.8. EXERCISES 229 2. Given the we
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Chapter 12 RSA Take two large prime
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12.1. FERMAT’S THEOREM 233 Exampl
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12.3. COMPLICATION II: A SUBSTANTIA
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12.3. COMPLICATION II: A SUBSTANTIA
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12.5. COMPLICATION IV: THE LAST ONE
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12.6. PUTTING IT ALL TOGETHER 241 1
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12.8. RSA 243 The RSA Algorithm Set
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12.9. RSA AND PUBLIC KEYS 245 There
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12.10. HOW TO BREAK RSA 247 had to
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12.12. SUMMARY 249 cannot read the
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12.14. EXERCISES 251 6. What is a
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12.14. EXERCISES 253 Bob is going t
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Bibliography [Antonucci] Michael An
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BIBLIOGRAPHY 257 [Shamir] [SRA] A.
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