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226 CHAPTER 11. KNAPSACK CIPHERS trial-and-error.) But it is easy to use the Knapsack System as a polygraphic cipher. Examples: (1) Use the weights {3, 4, 7, 15, 30, 47, 93, 279, 466, 749}, and the values e = 211 and P = 1507 to encipher swim as pairs of letters. First, we compute the U’s: U 1 = (3 · 211)%1507 = 633 U 2 = (4 · 211)%1507 = 844 U 3 = (7 · 211)%1507 = 1477 U 4 = (15 · 211)%1507 = 151 U 5 = (30 · 211)%1507 = 302 U 6 = (47 · 211)%1507 = 875 U 7 = (93 · 211)%1507 = 32 U 8 = (279 · 211)%1507 = 96 U 9 = (466 · 211)%1507 = 371 U 10 = (749 · 211)%1507 = 1311 Then we encipher as before, just with more weights: sw = {10011}{10111} = 1001110111 → 1 · U 1 + 1 · U 4 + 1 · U 5 + 1 · U 6 + 1 · U 8 + 1 · U 9 + 1 · U 10 = 633 + 151 + 302 + 875 + 96 + 371 + 1311 = 3739 Similarly, im becomes 2585. (2) Decipher 2585. We first need the deciphering number. From the Euclidean algorithm d = 50 is the solution to (d × 211)%1507 = 1. Next, 2585 becomes (d × 2585)%1507 = 1155. Finally, write 1155 as a sum of the original weights: 1155 = 4 + 30 + 93 + 279 + 749 = U 2 + U 5 + U 7 + U 8 + U + 10 = 0100101101 = {01001}{01101} → im ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄
11.5. PUBLIC KEY CIPHER 227 Even with 10 weights our adversary needs to check at most 2 10 = 1024 possible combinations to determine each piece of plaintext, long by hand but quick by computer. Suppose, however, we used 100 weights. We could then encipher and decipher 20 = 100/5 letters at a time. Further, once we have computed the disguised weights, each block of 20 letters is enciphered with only 20 quick additions. Likewise, deciphering involves only one modular multiplication and then a quick greedy algorithm, per block of letters. So enciphering and deciphering are very quick. On the other hand, there are 2 100 = 1267650600228229401496703205376 ways for the 100 weights to be combined. Even with the use of a a supercomputer, our adversary will be unable to determine the meanings of the blocks of letters by brute force. 11.5 Public Key Cipher This is our first real example of a Public Key Cipher. 7 Modern companies and governments must strive to overcome not only the breakability of their ciphersystems but also the key-management problem: there are just too many people and to whom they’d like to send messages that doing all of (1) meeting with everyone who you’d like to send messages to and agreeing upon a ciphersystem and a private key, (2) keeping all of that information very secret (so no one who shouldn’t learns it), and (3) keeping all that information easily accessible (so you can send messages back and forth), is very difficult. “It’d sure be nice”, someone thought one day, “if there was a secret message system that worked just like the phone book: you just look up someone’s cipher information and send them a message.” 8 It seems contradictory, making public the very information used to send secret messages. But if it were possible, difficulties (1) and (3) would vanish. 9 The knapsack cipher method was the first example that seemed plausible to use. Once everyone made their U’s public, anyone could send anyone else a message and the only items needed to be kept secret were the individual’s W ’s, e, d and P . Unfortunately, what made this method into a viable method didn’t work. Multiplying by e is meant to destroy the super-increasing nature of the W ’s but doesn’t completely do it. Using some fancy Linear Algebra [Shamir] it is possible to recover the original weights even without knowing e, d or P . The knapsack ciphers seemed the wave of the future, but suddenly became merely an interesting relic from the past. 7 In Chapter 3 we discussed Koblitz’s Kid-RSA. It is only a toy system, never used. The Knapsack Ciphers, however, were (nearly) very important. 8 The someones were Martin Hellman, a Stanford professor of Electrical Engineering, and Whitfield Diffie, then (1976) an undergraduate. 9 As a very simple example (due to Saloma), think of a telephone book. To call someone on the phone all you need to know is their name: their telephone number you can always look up. Similarly, once you’ve gotten a telephone number anyone, including people you’ve never met, can call you up.
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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
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Chapter 7 Vigenère Ciphers It was
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7.2. TRITHEMIUS, THE FATHER OF BIBL
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7.3. BELASO, THE UNKNOWN AND PORTA,
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7.4. VIGENÈRE CIPHERS 115 to encip
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7.6. HOW TO BREAK VIGENÈRE CIPHERS
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7.6. HOW TO BREAK VIGENÈRE CIPHERS
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7.7. THE KASISKI TEST 121 Kasiski
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7.8. SUMMARY 123 Repetition Start P
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7.10. EXERCISES 125 2. Little Miss
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7.10. EXERCISES 127 to Gen. E. K. S
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7.10. EXERCISES 129 MEJMY JKXCD LCN
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7.10. EXERCISES 131 22. (a) Use Kas
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7.10. EXERCISES 133 (a) Construct a
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Chapter 8 Polyalphabetic Ciphers Th
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8.1. COINCIDENCES 137 (being born o
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8.2. THE MEASURE OF ROUGHNESS 139 I
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8.2. THE MEASURE OF ROUGHNESS 141 P
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8.3. THE FRIEDMAN TEST 143 The Frie
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8.4. MULTIPLE ENCIPHERINGS 145 Bein
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8.4. MULTIPLE ENCIPHERINGS 147 Just
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8.5. VIGENÈRE’S AUTO KEY CIPHER
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8.5. VIGENÈRE’S AUTO KEY CIPHER
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8.6. PERFECT SECRECY 153 invent now
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8.8. TERMS AND TOPICS 155 multiple
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8.9. EXERCISES 157 6. As is this on
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8.9. EXERCISES 159 13. It has been
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8.9. EXERCISES 161 This is Equation
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8.9. EXERCISES 163 A: AB CDE FGH I
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8.9. EXERCISES 165 26. Decipher SVQ
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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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Page 231 and 232: Chapter 12 RSA Take two large prime
Page 233 and 234: 12.1. FERMAT’S THEOREM 233 Exampl
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Page 239 and 240: 12.5. COMPLICATION IV: THE LAST ONE
Page 241 and 242: 12.6. PUTTING IT ALL TOGETHER 241 1
Page 243 and 244: 12.8. RSA 243 The RSA Algorithm Set
Page 245 and 246: 12.9. RSA AND PUBLIC KEYS 245 There
Page 247 and 248: 12.10. HOW TO BREAK RSA 247 had to
Page 249 and 250: 12.12. SUMMARY 249 cannot read the
Page 251 and 252: 12.14. EXERCISES 251 6. What is a
Page 253 and 254: 12.14. EXERCISES 253 Bob is going t
Page 255 and 256: Bibliography [Antonucci] Michael An
Page 257: BIBLIOGRAPHY 257 [Shamir] [SRA] A.
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