Chapter%203%20Encryption%20Algorithms%20

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Chapter 3Encryption Algorithms Systems (Part
B)
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Outline

• NP-completeness Encryption
• Symmetric (secret key) vs Asymmetric (public key)
  Encryptions
• Popular Encryption Algorithms
• Merkle-Hellman Knapsacks
• RSA Encryption
• El Gamal Algorithms
• DES
• Hashing Algorithms
• Key Escrow Clipper

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Merkle-Hellman Knapsacks

• A public key cryptosystem
• The public key is the set of integers of a
  knapsack problem (the general knapsack) the
  private key is a corresponding superincreasing
  knapsack (or simple knapsack).
• A sample general knapsack (17, 38, 73, 4, 11, 1)
• A sample superincreasing knapsack (1, 4, 11, 17,
  38, 73), where each item ak is greater than the
  sum of all the previous items.
• Merkle and Hellman provided an algorithm for the
  receiver to use a superincreasing knapsack (the
  private key) to decrypt the ciphertext.

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Merkle-Hellman Knapsacks

• Basic idea To encode a binary message as a
  solution to a knapsack problem, reducing the
  ciphertext to the target sum obtained by adding
  terms corresponding to the 1s in the plaintext.
• Example Fig. 3-5, p.84
• Two kinds of knapsacks
• Simple (superincreasing) knapsack
• Hard (general) knapsack

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Solving a simple knapsack problem

• Given a superincreasing knapsack S (S1, S2, ,
  Sn) and a target sum T, find combination of Si
  that equals to T.
• Hint No combination of terms less than a
  particular term can yield a sum as large as the
  term. That is, Si gt S1 Si-1.
• Example Given S (1, 4, 11, 17, 38, 73) and T
  96, determine which terms in S correspond to the
  1s of the plaintext.
• Solution 1, 4, 17, 73 (See Fig. 3-6, p.85).
  That is, the plaintext was 110101.
• Exercise S (1, 2, 5, 9, 20, 43), T 49.

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Deriving a Hard Knapsack from a Simple Knapsack

• An example on pages 87-89
• Choose the number (M) of items in a knapsack.
• Example M 4
• Create a simple knapsack (S) with M items.
• Example S (1, 2, 4, 9).
• Choose a multiplier w and a modulus n, where n gt
  S1 SM and w is relprime to n. Note
  Usually n is a prime, which is relprime to any
  number lt n.
• Example w 15 and n 17.
• Replace every item in the simple knapsack with
  the term hi w si mod n. Then H (h1, h2, ,
  hM) is the hard knapsack.
• Example H (15, 13, 9, 16).

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Encryption using Merkle-Hallman Knapsacks

• The plaintext is encrypted using the hard
  knapsack (the public key), while the simple
  knapsack, as well as w and n, are used as the
  private key.
• Example Encrypt the plaintext 0100 1011 1010
  0101 using the sample hard knapsack H (15, 13,
  9, 16).
• Ans
• C1 0100 H 13.
• C2 1011 H 15 9 16 40.
• C3 1010 H 15 9 24.
• C4 0101 H 13 16 29.
• So, ciphertext (13, 40, 24, 29)

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Decryption using Merkle-Hallman Knapsacks

• The receiver of the ciphertext uses w and n to
  calculate the multiplicative inverse of w, w-1.
• w w-1 mod n 1.
• Example w-1 15-1 mod 17 8 (use algorithm on
  p.81 or the inverse.java program)
• To decipher the ciphertext (C) Multiply each of
  the numbers in C by w-1.
• (w-1 C w-1 H P w-1 w S P S P
  )mod n
• Exercise Decrypt the ciphertext from the
  previous exercise, i.e., 13, 40, 24, 29, by using
  the simple knapsack S (1, 2, 4, 9).
• The answer next slide

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Decryption using Merkle-Hallman Knapsacks

• Given H (19, 28, 76, 171, 293, 46, 130, 150).
• C 13, 40, 24, 29
• S (1, 2, 4, 9)
• n 17, w 15, w -1 8.
• To get the target Ti Multiply each Ci by w-1
• T1 (w-1 C1) mod n 8 13 mod 17 2
• T2 (w-1 C1) mod n 8 40 mod 17 14
• T3 (w-1 C1) mod n 8 24 mod 17 5
• T4 (w-1 C1) mod n 8 29 mod 17 11
• Given the target sum Ti and the simple knapsack
  S, find the combination of items in S that
  produces T.
• e.g., The answer for T1 is 0100.

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Another Example

• Step 1 Derive a Hard Knapsack from a Simple
  Knapsack
• Choose the number (M) of items in a knapsack.
• Example M can be 8 when the plaintext is in
  ascii.
• Create a simple knapsack (S) with M items.
• Example S (1, 2, 4, 9, 17, 34, 70, 150).
• Choose a multiplier w and a modulus n, where n gt
  S1 SM and w is relprime to n. Note
  Usually n is a prime, which is relprime to any
  number lt n.
• Example w 19 and n 303.
• Replace every item in the simple knapsack with
  the term hi w si mod n. Then H (h1, h2, ,
  hM) is the hard knapsack.
• Example H (19, 38, 76, 171, 20, 40, 118, 123).

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Another Example

• Step 2 Encrypt a plaintext using the hard
  knapsack
• Encrypt the plaintext PEACE using the sample
  hard knapsack H, H (19, 38, 76, 171, 20, 40,
  118, 123).
• P 50h E 45h A 41h C 43h
• Ans
• C1 50h H 0101 0000 H 38 171 209.
• C2 45h H 0100 0101 H 38 40 123
  201.
• C3 41h H 0100 0001 H 38 123 161.
• C4 43h H 0100 0011 H 38 118 123
  279.
• C5 C2 228.
• So, ciphertext (PEACE) 209 201 161 279 228

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Another Example

• Step 3 Decrypt using C, w, n and S
• The receiver of the ciphertext uses w and n to
  calculate the multiplicative inverse of w, w-1.
• w w-1 mod n 1.
• Example w-1 19-1 mod 303 16 (use algorithm
  on p.81 or the inverse.java program)
• To decipher the ciphertext (C) Multiply each of
  the numbers in C by w-1.
• (w-1 C w-1 H P w-1 w S P S P
  )mod n
• Exercise Decrypt the ciphertext from the
  previous exercise, i.e., 209 201 161 279 228, by
  using the simple knapsack S (1, 2, 4, 9, 17,
  34, 70, 150).
• The answer next slide

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Another Example

• Given H (19, 28, 76, 171, 293, 46, 130, 150).
• C 209 201 161 279 228, n 303, w 19, w-1
  16.
• S (1, 2, 4, 9, 17, 34, 70, 150).
• To get the target Ti Multiply each Ci by w-1
• T1 (w-1 C1) mod n 16 209 mod 303 11
• T2 (w-1 C1) mod n 16 201 mod 303 186
• T3 (w-1 C1) mod n 16 161 mod 303 152
• T4 (w-1 C1) mod n 16 279 mod 303 222
• Given the target sum Ti and the simple knapsack
  S, find the combination of items in S that
  produces T.
• The answer for T1 is 0101 0000, which is 11.
• S (1, 2, 4, 9, 17, 34, 70, 150)
• Ans. for T1 (0 1 0 1 0 0 0
  0) ? 50h

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Weakness of the Merkle-Hellman Algorithm

• p.90
• The modulus n can be guessed. S, the secret key,
  and the multiplier, w, are then exposed.
• 1980 Shamir found that if the value of the
  modulus n is known, it may be possible to
  determine the simple knapsack S.
• 1982 Shamir came up with an approach to deduce w
  and n from H, the hard knapsack, alone.

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Summary

• Merkle-Hellman is a public-key cryptosystem based
  on the knapsack problem.
• The encryption can be broken, mainly due to its
  use of simple knapsacks as the secret keys.
• Next
• RSA Encryption
• El Gamal Algorithms
• DES
• Hashing Algorithms
• Key Escrow Clipper