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11.4. THE KNAPSACK CIPHER SYSTEM 223
turning the easy knapsack problem into a hard one. Further, if we use the
Euclidean Algorithm to solve 39 × d ≡ 1 (mod 101) for d = 57, then we can
turn the hard problem back into the easy one, for then multiplying the U’s by
d will give us the W ’s back.
⋄
This example suggests somehow using the total weight to send messages. To
produce 80 we used W 2 , W 4 and W 5 , hinting at the pattern no-yes-no-yes-yes
or nynyy or 01011. What can 01011 mean
a = 00001 g = 00111 m = 01101 s = 10011 y = 11001
b = 00010 h = 01000 n = 01110 t = 10100 z = 11010
c = 00011 i = 01001 o = 01111 u = 10101
d = 00100 j = 01010 p = 10000 v = 10110
e = 00101 k = 01011 q = 10001 w = 10111
f = 00110 l = 01100 r = 10010 x = 11000
Figure 11.1: Binary Equivalents for the Alphabet
11.4 The Knapsack Cipher System
Many modern ciphers are built around a mathematical problem that is (hopefully)
impossible to solve unless you have some special information that makes
the problem very simple. One of the first was the suitably named Knapsack
Cipher System. It was invented by Ralph Merkle and Martin Hellman [MH]. 5
In this cipher the intractable problem is the general knapsack problem, and the
simple version is the super-increasing knapsack problem. The Knapsack Cipher
attempts to exploit this method for turning the simple problem into an
apparently impossible one.
The first four steps of the set-up we’ve already seen. The final two make
this into what was then a radically new type of cipher.
5 M. E. Hellman and R. C. Merkle, received U.S. Patent 4,218,582, filed October 6 1977,
issued August 19 1980 for “Public Key Cryptographic Apparatus and Method.” It expired in
1997.