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228 CHAPTER 11. KNAPSACK CIPHERS
11.6 Summary
In the past 25 years it has become fashionable to take “intractable” mathematical
problems and try to turn them into cryptosystems. One of the first examples
of a problem put to such use was the Knapsack Problem. In its full generality
(from a large set of items choose some subset that maximizes value while
keeping the total weight under some given bound) the Knapsack Problem, also
known as the Subset Sum Problem, is known to be such an intractable problem.
In 1977 Merkle and Hellman announced a cipher system based on a version
of the Knapsack problem. It had the advantage of being a public key system,
plus was relatively fast. This system starts with a set of super-increasing weights
(each successive weight is larger than the sum of all the previously chosen ones).
After a modular multiplication (and a permutation of their order) the weights
are made public. To send a binary message send the sum of the weights corresponding
to the 1’s in the message. The creator of the system can undo the
modular multiplication, and so to read the message needs only to solve the
sum-subset problem for a super-increasing set, which is easy.
It was quickly suspected that hiding the super-increasing nature of the
weights by a modular multiplication and permutation was not really sufficient,
and this proved to be the case. By 1984 most forms of the Knapsack Ciphers
had been shown to be insecure. Nonetheless, they provide a lovely case study
of a modern cryptosystem.
11.7 Topics and Techniques
1. What is the general Knapsack Problem What is the meaning of “knapsack”
in the name
2. What is super-increasing
3. Why is a Knapsack Problem with super-increasing weights easy to solve
How to solve it
4. How can a set of super-increasing weights be modified into a set of not
super-increasing weights
5. Explain the setup of a Knapsack Cipher.
6. How does a Knapsack Cipher encipher What steps are involved
11.8 Exercises
1. Given the weights 41, 28, 39, 57, 49 and 31, and being allowed to use them
possibly multiple times, can the weight 782 be achieved, and how