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160 CHAPTER 8. POLYALPHABETIC CIPHERS 18. (a) Finish the computation of Φ for a text consisting of English with the normal distribution started in section 8.3. (Use Figure 1.3. So there are 8.2 A’s, 1.5 B’s, and N = 100.) (b) Compute the value of Φ, if the plaintext consisted of English with the normal distribution that was enciphered with a Caesar cipher with key B. (c) Compute the value of Φ, if the plaintext consisted of English with the normal distribution that was enciphered with a monoalphabetic cipher. (d) Compute Φ for the ciphertext that has all 26 letters occurring an equal number of times. 19. In this exercise we work out the formula for Φ given in Equation 8.3. Assume we have a ciphertext of N letters enciphered with a Vigenère cipher with keylength k. As we did in Section 7.7, lets assume we’ve written the ciphertext in an array, so there are k rows, each containing all the ciphtertext letters that were enciphered with the same keyletter. Now Φ is the probability that two letters, chosen at random from the text, are the same. Either these letters were chosen from the same row, or from different rows. Same row: There are N total letters and k rows, so about N/k letters per row. Hence there are N k ( N k − 1)/2 ways to choose two from any particular row. There are k rows and so k N k ( N k − 1)/2 ways to choose two letters a row. Finally, the likelihood that two letters are the same in any of these rows is 0.0656. So the contribution here is .0656 ∗ k N k ( N k − 1) N(N − k) = .0656 2 2k Different rows: There are k(k − 1)/2 ways to pick to two rows. There are N/k letters in each row, so there are N/k ∗ N/k ways to pick one letter from each. Finally, we have no way of relating the keyletters from the different rows, so we can only approximate the likelihood that these two letters are the same with 1/26. So the contribution here is 1 26 ∗ N N k(k − 1) ∗ = 1 N 2 (k − 1) k k 2 26 k Adding the two probabilities, and simplifying,
8.9. EXERCISES 161 This is Equation 8.3. N(N − k) Φ ≈ .0656 + 1 N 2 (k − 1) 2k 26 k = .0656(N − k) + N(k − 1) 1 26 k(N − 1) .0656(N − k) + .0385N(k − 1) = k(N − 1) (a) If our work has been correct, substituting k = 1 into 8.3 should produce the value of Φ in a monoalphabetic cipher, 0.0656. Does it (b) If k = N, then the keylength is the same as the length of the text. This should produce a very flat frequency count, resulting in a value of .0385 for Φ. Does it (c) It is sometimes of value to have a formula for k, the keylength. Multiply both sides of 8.3 by k(N − 1) and then solve for k to find one. 20. (a) In section 8.3 we saw that Φ = .0656 when the language is English. Referring to Figure 1.5, compute Φ for French, German and Spanish. (b) Jeremiah 25:26 and 51:41 (taken from BibleGateway.org) are 25:26: and all the kings of the north, near and far, one after the other-all the kingdoms on the face of the earth. And after all of them, the king of Sheshach will drink it too. 51:41: How Sheshach will be captured, the boast of the whole earth seized! What a horror Babylon will be among the nations! (Sheshach is Babel (= Babylon) written with a reversed alphabet, in Hebrew, of course.) Translating the combined verses into German and French, and then enciphering with a monoalphabetic cipher produces Text 1: FINPC DYCVN TCSPC YGIYU IVTNU GVNKL YCNPY DNTBU YCFPJ PUCYI FPJFP IVYCY IFINP CDYCV NQFPM YCSPM NUSYO PTCNU ICPVD FRFKY SYDFI YVVYY IDYVN TSYCK LYCKL FKANT VFFGV YCYPJ YLOPN TCKLY CKLFK YCIGV TCYKY DDYSN UIDFB DNTVY VYMGD TCCFT IINPI YDFIY VVYYC IKNUO PTCYY LOPNT AFAQD NUYYC ISYIV PTIYF PMTDT YPSYC UFITN UC 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 3 2 27 14 0 19 5 0 21 3 10 7 4 20 4 20 2 1 8 17 11 17 0 0 42 0 257 and Text 2: PUSGD DYRMU TCYSY AUMVS YUAST YUGKY UPUSS TYEYV UYUYT UYUUG JKSYL GUSYV UPUSG DDYRM UTCVY TJKYS YVYVS YSTYG PESYV EDGJK YSYAY VSBMS YUAAT USPUS SYVRM UTCWM UAJKY AJKGJ KAMDD UGJKT KUYUH VTURY UNTYT AHAJK YAJKG JKYTU CYUML LYUPU SSYVV PKLSY VCGUX YUYVS YYVMB YVHNT YTAHB GBYDX PLYUH AYHXY UCYNM VSYUP UHYVS YUUGH TMUYU 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 13 4 6 8 3 0 13 8 0 10 13 5 11 3 0 9 0 4 25 17 40 18 1 3 51 0 265
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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
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6.3. SUKHOTIN’S METHOD FOR FINDIN
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6.4. FINAL MONOALPHABETIC TRICKS 99
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6.5. SUMMARY 101 our ciphertext muc
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6.7. EXERCISES 103 Frequency count:
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6.7. EXERCISES 105 53++!305))6*;482
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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 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: 8.9. EXERCISES 159 13. It has been
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
Page 195 and 196: 10.5. TRANSPOSITION VS. SUBSTITUTIO
Page 197 and 198: 10.6. LETTER CONNECTIONS 197 F 2 G
Page 199 and 200: 10.7. BREAKING THE COLUMNAR TRANSPO
Page 201 and 202: 10.8. DOUBLE TRANSPOSITION 201 Sinc
Page 203 and 204: 10.9. TRANSPOSITION DURING THE CIVI
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Page 207 and 208: 10.10. THE BATTLE OF THE CIVIL WAR
Page 209 and 210: 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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