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224 CHAPTER 11. KNAPSACK CIPHERS
Knapsack Cipher Setup:
1. Pick a set of super-increasing numbers: W 1 , W 2 , W 3 , W 4 , W 5 and P .
2. Pick e < P with gcd(P, e) = 1. Use the Euclidean Algorithm to find
d such that e × d ≡ 1 (mod P ).
3. Compute the U’s: U i = e × W i %P .
4. Make the U’s public. Keep the W ’s, e, d and P secret.
To Encipher:
1. Turn each letter of the message into its binary equivalent
(a 1 , a 2 , a 3 , a 4 , a 5 ).
2. Use the U’s to find the enciphered number M, of each letter:
To Decipher:
M = a 1 · U 1 + a 2 · U 2 + a 3 · U 3 + a 4 · U 4 + a 5 · U 5 .
1. For each received cipher number M, compute N = (d · M)%P .
2. Solve N = x 1 · W 1 + x 2 · W 2 + x 3 · W 3 + x 4 · W 4 + x 5 · W 5 for the x’s.
3. Convert (x 1 , x 2 , x 3 , x 4 , x 5 ) into its letter equivalent.
Examples:
(1) Use the weights U 1 = 16, U 2 = 72, U 3 = 64, U 4 = 27 and U 5 = 93 to
encipher swim.
s = 10011 → 1 · U 1 + 0 · U 2 + 0 · U 3 + 1 · U 4 + 1 · U 5
= 16 + 27 + 93 = 136.
w = 10111 → 1 · U 1 + 0 · U 2 + 1 · U 3 + 1 · U 4 + 1 · U 5
= 16 + 64 + 27 + 93 = 200.
i = 01001 → 0 · U 1 + 1 · U 2 + 0 · U 3 + 0 · U 4 + 1 · U 5
= 72 + 93 = 165.
m = 01101 → 0 · U 1 + 1 · U 2 + 1 · U 3 + 0 · U 4 + 1 · U 5
= 72 + 64 + 93 = 229.
The ciphertext message is 136, 200, 165, 229.
(2) Decipher 136, 200, 165, 229.
Of course, we know the answer. But how could we find it if we didn’t
already know it First, if we didn’t know e and P (the private key) but