Re: Other forms of strong cryptography

[smb@research.att.com]

	 Why is it that the idea of taking a difficult problem, such
	 as a knapsack problem, and using it to encode ciphers,
	 was abandoned? Too many trapdoors? These NP-complete
	 type problems seem ideal since they can be verified
	 in polynomial time, but are practically impossible to
	 solve for any significant input. Verification of a solution
	 could be decryption, where the solution is the key,
	 and the problem could be used to encode the text somehow.

	 I understand that Shamir broke the knapsack problem. So,
	 is that enough reason to completely abandon this approach?
	 Nobody seems to talk about it anymore.

The approach hasn't been abandoned; it's just a lot harder than it
looks, for a number of reasons.

First is that complexity theory says nothing about the average difficulty
of solving a problem, as opposed to the worst case.  A cryptosystem that
only hides 1% of the messages isn't very useful.  Second, finding
a suitable problem -- one that has a keyed back door isn't that easy.
Third -- and this is what sunk the knapsack problem -- you need a
cryptosystem that exploits the full NP-complete problem, as opposed
to just a simple case.  (The knapsack problem was solvable by someone
who knew the key because it wasn't a general knapsack, but a super-
increasing sequence -- each number in it was greater than the sum
of all of its predecessors.  (This was the simplest version; there
were, I believe, some others.))