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John Nash's Declassified 1955 Letter To the NSA
An anonymous reader writes "In 1955, John Nash sent an amazing letter (PDF) to the NSA in order to support an encryption design that he suggested. In it, he anticipates computational complexity theory as well as modern cryptography. He also proposes that the security of encryption can be based on computational hardness and makes the distinction between polynomial time and exponential time: 'So a logical way to classify enciphering processes is by the way in which the computation length for the computation of the key increases with increasing length of the key. This is at best exponential and at worst probably at most a relatively small power of r, ar^2 or ar^3, as in substitution ciphers.'"
What Nash seems to be describing is a feedback shift register. This has potential as a cryptosystem, but isn't a very good one. As the NSA pointed out, it "affords only limited security".
When Nash wrote this, Friedman had already developed the theory that allowed general cryptanalysis of rotor-type machines. But that was still highly classified. Friedman, of course, was responsible for breaking the Japanese "Purple" cypher, plus many others. Before Friedman, cryptanalysis was about guessing. After Friedman, it was about number crunching.
Friedman was the head cryptanalyst at NSA at the time. Within NSA, it would have been known that a linear feedback shift register was a weak key generator. So this idea was, properly, rejected. At least NSA looked at it. Friedman's hard line on that subject was "No new encryption system is worth looking at unless it comes from someone who has already broken a very hard one."
The fact that a problem is NP-hard isn't enough to make it a good key generator. The Merkle-Hellman knapsack cryptosystem, the first public-key cryptosystem published, is based on an NP-hard problem. But, like many NP-hard problems, it's only NP-hard in the worst case. The average case is only P-hard. (Linear programming problems, and problems which can be converted to a linear programming problem, are like that.) So that public-key system was cracked.
We still don't have cryptosystems which are provably NP-hard for all cases. Factoring and elliptic curves are as good as it gets, and there's still the possibility that a breakthrough could make factoring easy.