The Merkle-Hellman Knapsack Cryptosystem is a public key cryptosystem based on the subset sum problem. It involves the creation of a 'hard' knapsack problem (the public key) from an 'easy' knapsack problem (the private key).

Here's how to generate two pairs of Merkle-Hellman Knapsack keys:

Pair 1 (size 5):

Step 1: Choose a superincreasing sequence (private key). A sequence is superincreasing if every element is greater than the sum of all previous elements. Let's choose (2, 3, 7, 14, 30).

Step 2: Choose a number n greater than the sum of the sequence, and a number m that is coprime with n. Let's choose n = 60 and m = 37.

Step 3: Calculate the public key by multiplying each element of the private key by m mod n. This gives us (74, 111, 57, 54, 30).

Pair 2 (size 6):

Step 1: Choose a superincreasing sequence (private key). Let's choose (2, 3, 7, 14, 30, 62).

Step 2: Choose a number n greater than the sum of the sequence, and a number m that is coprime with n. Let's choose n = 120 and m = 73.

Step 3: Calculate the public key by multiplying each element of the private key by m mod n. This gives us (146, 219, 121, 102, 90, 26).

Justification of selected numbers:

The private keys were chosen to be superincreasing sequences, as required by the Merkle-Hellman system. The numbers n were chosen to be greater than the sum of the private keys, and the numbers m were chosen to be coprime with n, also as required by the system. The public keys were then calculated according to the Merkle-Hellman system.

Question

The Merkle-Hellman Knapsack Cryptosystem is a public key cryptosystem based on the subset sum problem. It involves the creation of a 'hard' knapsack problem (the public key) from an 'easy' knapsack problem (the private key).

Here's how to generate two pairs of Merkle-Hellman Knapsack keys:

Pair 1 (size 5):

Step 1: Choose a superincreasing sequence (private key). A sequence is superincreasing if every element is greater than the sum of all previous elements. Let's choose (2, 3, 7, 14, 30).

Step 2: Choose a number n greater than the sum of the sequence, and a number m that is coprime with n. Let's choose n = 60 and m = 37.

Step 3: Calculate the public key by multiplying each element of the private key by m mod n. This gives us (74, 111, 57, 54, 30).

Pair 2 (size 6):

Step 1: Choose a superincreasing sequence (private key). Let's choose (2, 3, 7, 14, 30, 62).

Step 2: Choose a number n greater than the sum of the sequence, and a number m that is coprime with n. Let's choose n = 120 and m = 73.

Step 3: Calculate the public key by multiplying each element of the private key by m mod n. This gives us (146, 219, 121, 102, 90, 26).

Justification of selected numbers:

The private keys were chosen to be superincreasing sequences, as required by the Merkle-Hellman system. The numbers n were chosen to be greater than the sum of the private keys, and the numbers m were chosen to be coprime with n, also as required by the system. The public keys were then calculated according to the Merkle-Hellman system.

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