Unformatted text preview: will show that both 2 and 3 are in the knapsack and the total weight is brought to 0, which indicates that a solution has been found. Were this a MerkleHellman knapsack encryption block, the plaintext that resulted from a ciphertext value of 70 would be 110101. Nonsuperincreasing, or normal, knapsacks are hard problems; they have no known quick algorithm. The only known way to determine which items are in the knapsack is to methodically test possible solutions until you stumble on the correct one. The fastest algorithms, taking into account the various heuristics, grow exponentially with the number of possible weights in the knapsack. Add one item to the sequence of weights, and it takes twice as long to find the solution. This is much more difficult than a superincreasing knapsack where, if you add one more weight to the sequence, it simply takes another operation to find the solution. Figure 19.1 Encryption with knapsacks. The MerkleHellman algorithm is based on this property. The private key is a sequence of weights for a superincreasing knapsack problem. The public key is a sequence of weights for a normal knapsack problem with the same solution. Merkle and Hellman developed a technique for converting a superincreasing knapsack problem into a normal knapsack problem. They did this using modular arithmetic. Creating the Public Key from the Private Key
Without going into the number theory, this is how the algorithm works: To get a normal knapsack sequence, take a superincreasing knapsack sequence, for example {2, 3, 6, 13, 27, 52}, and multiply all of the values by a number n, mod m. The modulus should be a number greater than the sum of all the numbers in the sequence: for example, 105. The multiplier should have no factors in common with the modulus: for example, 31. The normal knapsack sequence would then be 2 * 31 mod 105 = 62 3 * 31 mod 105 = 93 6 * 31 mod 105 = 81 13 * 31 mod 105 = 88 27 * 31 mod 105 = 102 52 * 31 mod 105 = 37 The knapsack would then be {62, 93, 81...
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This note was uploaded on 10/18/2010 for the course MATH CS 301 taught by Professor Aliulger during the Fall '10 term at Koç University.
 Fall '10
 ALIULGER
 Cryptography

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