Encryption and Decryption Part 4 Ken Dewey Rose State College Midwest City, OK 73110

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EncryptionandDecryption(Part 4)Ken
DeweyRose State CollegeMidwest City, OK 73110
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Merkle-Hellman Knapsack
Ralph C. Merkle
Martin Hellman
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How it works

• Pick
• Simple (Super Increasing) Knapsack
• Calculate
• Hard Knapsack
• then
• Encrypt Message
• Easy Enough?

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Merkle-Hellman Algorithm (contd)

• Sending an Encrypted Message
• Receiver picks a simple (super increasing)
  knapsack (S), multiplier (w) and modulus (n)
• S 1, 2, 6 w 11 n 13 (Note n is prime)
• Receiver computes hard knapsack (H h1, h2,
  h3)
• hi w si mod n
• h1 w s1 mod n 11 1 mod 13 11 mod 13
  11
• h2 w s2 mod n 11 2 mod 13 22 mod 13 9
• h3 w s3 mod n 11 6 mod 13 66 mod 13 1

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Merkle-Hellman Algorithm (contd)

• Sending an Encrypted Message (contd.)
• Receiver sends H 11, 9, 1 to sender
• Receiver keeps S, w and n secret
• Suppose sender wishes to transmit P 101 010
  011
• P 1 0 1 0 1 0 0 1 1
• 11 9 1 11 9 1 11 9 1
• C 12 9 10

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Modulus

• 33 mod 8 ?
• What is the remainder?
• 33 / 8 4.125
• 8 4 32 33-32 1
• 33 mod 8 1

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Merkle-Hellman Algorithm (contd)

• Decrypting an Encrypted Message
• H wS mod n (from previous slide)
• C HP wSP mod n (from previous slide)
• w-1C w-1HP w-1wSP mod n SP mod n
• w-1Ci SPi mod n (note S 1, 2, 6
• Remember w-1 6 C 12,9, 10
• w-1Ci mod n P
• 6 12 72 mod 13 7 101
• 6 9 54 mod 13 2 010
• 6 10 60 mod 13 8 011

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Example

• Consider the Merkle-Hellmen Cryptosystem.
• Assume that the
• simple knapsack (S) 1 3 6
• multiplier (w) 11
• modulus (n) 13
• Compute H (hard knapsack)
• Hi Si w mod n (General Formula)
• H1 1 11 mod 13 11 mod 13 11
• H2 3 11 mod 13 33 mod 13 7
• H3 6 11 mod 13 66 mod 13 1
• H 11 7 1

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Example (contd)

• Consider the Merkle-Hellmen Cryptosystem.
• Assume that the
• simple knapsack (S) 1 3 6
• multiplier (w) 11
• modulus (n) 13
• Encrypt 101 010 011
• H 11 7 1 11 7 1 11 7 1 (Previous Slide)
• P 1 0 1 0 1 0 0 1 1
• C 110112 0707 0718
• C 12 7 8

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Example (contd)

• Consider the Merkle-Hellmen Cryptosystem.
• Assume that the
• simple knapsack (S) 1 3 6
• multiplier (w) 11
• modulus (n) 13
• Compute w-1
• w w-1 mod n 1 11 ? mod n 1
• n13 w11 comparisons done
• 1 0 0
• 14 11 1
• 27 22 2
• 40 33 3
• 53 44 4
• 66 55 5
• 66 6
• w-1 6

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Example (contd)

• Consider the Merkle-Hellmen Cryptosystem.
• Assume that the
• simple knapsack (S) 1 3 6
• multiplier (w) 11
• modulus (n) 13
• Compute w-1
• w-1 ww mod n
• W-1 1111 mod 13 6
• w-1 6

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Example (contd)

• Consider the Merkle-Hellmen Cryptosystem.
• Assume that the
• simple knapsack (S) 1 3 6
• multiplier (w) 11
• modulus (n) 13
• Decrypt 19 18
• w-1 11-1 mod 13 6
• w-1 Ci mod n Intermediary Valuei
• w-1 C 6 19 mod 13 6 18 mod 13
• 10 4
• S 1 3 6 1 3 6
• P 1 1 1 1 1 0
• (11 31 61) (11 31 60)
• P 111 110

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Merkle-Hellman Algorithm (contd)

• Cryptanalysis
• Modulus n 200 bits long
• Si are chosen to be approx. 2200 apart!
• Knapsack has approx. 200 terms (m 200)
• Choose m random numbers between 0 and 2200
• Si 2200i-1 ri
• Each term is 200 to 400 bits long
• 1047 years to try 2200 choices for each Si
• Hard to break for large values of n m

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Merkle-Hellman Algorithm (contd)

• Weaknesses
• Shamir (1980) If modulus n is known, it is
  possible to determine simple knapsack S in
  polynomial time

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Test Question Example(What you WILL see)

• Consider the Merkle-Hellmen Cryptosystem.
• Assume that the
• simple knapsack (S) 1 3 6
• multiplier (w) 11
• modulus (n) 13.
• Compute H (hard knapsack)
• Encrypt 101 010 011
• Compute w-1
• Decrypt 19 18

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2nd Test Question

• This will be for Extra Credit
• You only have 30 seconds to solve it!
• Ready!

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Pick the One that Doesnt Belong