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118 CHAPTER 7. VIGENÈRE CIPHERS unbreakable. So how finally was the Vigenère Cipher broken By turning the power of the keyword against the cipher. Let’s start by backing up. What enabled us to decrypt a Caesar cipher We knew that every letter in the message was enciphered using the same key letter. And we could detect this key letter from the frequency habits of normal English. Why can’t we do this with a Vigenère cipher Because not all of the letters are enciphered using the same key letter. In other words, a Caesar cipher is monoalphabetic, a Vigenère cipher is polyalphabetic. Now suppose we did know which letters were enciphered by the same letters of the keyword. Then we could split up the ciphertext into groups in such a way that all the letters in each group would be enciphered with the same keyletter. Each group of letters would be like a (different) Caesar cipher. And we know how to break Caesar ciphers. Since we could decrypt each of these groups of letters, then we could decrypt all of the letters in the ciphertext, thus breaking the message. So (perhaps) we don’t actually need to know the keyword, but we do need to determine which ciphertext letters have been enciphered with the same letter of the keyword. How can we do this Imagine rewriting the ciphertext downward into columns, with as many letters per column as there are letters in the keyword. Of course, we’d need to know how many letters there are in the keyword, but pretend for a moment we know that. Then the letters in the first row of our re-written ciphertext would all have been enciphered using the first letter of the keyword. And the letters in the second row of the re-written ciphertext would all have been enciphered using the second letter of the keyword. Similarly for the other rows. We would be left with several Caesar ciphers to break. To make this a bit less abstract, let’s suppose the keyword contained 5 letters, and we number them k 1 k 2 k 3 k 4 k 5 , and similarly number the ciphertext letters ct 1 ct 2 ct 3 · · · . Then the complex of rows and columns from the previous paragraph takes the form k 1 ct 1 ct 6 ct 11 ct 16 . . . k 2 ct 2 ct 7 ct 12 ct 17 . . . k 3 ct 3 ct 8 ct 13 ct 18 . . . k 4 ct 4 ct 9 ct 14 ct 19 . . . k 5 ct 5 ct 0 ct 15 ct 10 . . . The letters ct 1 , ct 6 , ct 11 , ct 16 , . . . all have been enciphered by the key k 1 . So we can treat ct 1 ct 6 ct 11 ct 16 . . . as a Caesar cipher with unknown key k 1 . Likewise, ct 2 , ct 7 , ct 12 , ct 17 , . . . all have been enciphered by k 2 , so ct 2 ct 7 ct 12 ct 17 . . . can be treated as a Caesar cipher. So to decrypt a Vigenère-enciphered message, assuming we know the length of the keyword, first rewrite the ciphertext in columns with as many letters per column as letters in the keyword. Then the letters of each row will constitute a Caesar cipher that can be broken with our techniques from Chapter 1. To summarize, from what does the Vigenère cipher get its security Ob-
7.6. HOW TO BREAK VIGENÈRE CIPHERS 119 viously from its keyword. But, surprisingly, the important thing about the keyword, at least from a security standpoint, is not which letters are in it, but rather how many! Let’s illustrate the method with the example we considered earlier. Example: QNTQH PIECN SCPOE QXPZC FFILF ACNMY FRRPW XUTKK RROLV CVEQX KASFS IMGSY XYRRM POEFH MYROI OEEUO EVSET EVPIX KAEFC ZGKEQ AHTAK KXTEI EGRNA DZPKM MDCAC ZGPVT QTFHZ OXZDE RFEWX YFWVN ARIGW AANUG RNQAE FXSMT FHCZG RWTMP ZDJII YQRRN QLDCV NKTHI VTKP Now suppose somehow, by hook or by crook, we learned that this is indeed a Vigenère cipher with a keyword consisting of 5 letters. This would mean that every fifth letter of the message is enciphered by the same letter of the keyword. So the letters appearing first in each five-letter grouping are enciphered, as a Caesar cipher, using the same keyletter. Similarly, the letters occurring second in each five-letter grouping are enciphered using the same keyletter. Likewise for the third, the fourth and the fifth letters. So if we group together the letters that are enciphered by the same keyletter, then we can decipher each of these groups as a Caesar Cipher. First, we regroup and make new frequency counts. First letters: QPSQFAFXRCKIXPMOEEKZAKEDMZQORYAARFFRZYQNV 4 0 1 1 3 4 0 0 1 0 3 0 2 1 2 2 4 4 1 0 0 1 0 2 2 3 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 Second letters: NICXFCRURVAMYOYEVVAGHXGZDGTXFFRANXHWDQLKT 3 0 2 2 1 3 3 2 1 0 1 1 1 2 1 0 1 3 0 2 1 3 1 4 2 1 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 Third letters: TEPPINRTOESGRERESPEKTTRPCPFZEWINQSCTJRDTK 0 0 2 1 6 1 1 0 2 1 2 0 0 2 1 5 1 5 3 6 0 0 1 0 0 1 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 Fourth letters: QCOZLMPKLQFSRFOUEIFEAENKAVHDWVGUAMZMIRCHP 3 0 2 1 3 3 1 2 2 0 2 2 3 1 2 2 2 2 1 0 2 2 1 0 0 2 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 Fifth letters: HNECFYWKVXSYMHIOTXCQKIAMCTZEXNWGETGPINVI 1 0 3 0 3 1 2 2 4 0 2 0 2 3 1 1 1 0 1 3 0 2 2 3 2 1 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
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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
Page 67 and 68: 4.6. SUMMARY 67 To put it more blun
Page 69 and 70: 4.8. EXERCISES 69 5. Here we work w
Page 71 and 72: Chapter 5 Monoalphabetic Ciphers He
Page 73 and 74: 5.2. KEYWORD MIXED CIPHERS 73 Examp
Page 75 and 76: 5.4. INTERRUPTED KEYWORD CIPHERS 75
Page 77 and 78: 5.6. BASIC LETTER CHARACTERISTICS 7
Page 79 and 80: 5.7. ARISTOCRATS 79 the 69971 or 42
Page 81 and 82: 5.9. TOPICS AND TECHNIQUES 81 5.9 T
Page 83 and 84: 5.10. EXERCISES 83 (h) DYVTP KKIQ.
Page 85 and 86: 5.10. EXERCISES 85 13. Ciphers in w
Page 87 and 88: 5.10. EXERCISES 87 17. General Pier
Page 89 and 90: Chapter 6 Decrypting Monoalphabetic
Page 91 and 92: 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: 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 and 144: 8.3. THE FRIEDMAN TEST 143 The Frie
Page 145 and 146: 8.4. MULTIPLE ENCIPHERINGS 145 Bein
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
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9.1. POLYGRAPHIC CIPHERS 169 f a g
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9.2. HILL CIPHERS 171 th 2.96 es 1.
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9.2. HILL CIPHERS 173 We will be us
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9.3. RECOGNIZING AND BREAKING POLYG
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9.4. PLAYFAIR 177 Playfair Ciphers:
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9.5. SUMMARY 179 9.5 Summary Polygr
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9.7. EXERCISES 181 (b) The use of
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9.7. EXERCISES 183 10. Just as we c
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9.7. EXERCISES 185 (d) (For those(
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9.7. EXERCISES 187 The rules that W
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Chapter 10 Transposition Ciphers In
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10.3. TURNING GRILLES 191 1 2 3 7 4
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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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