# 4: Basing Cryptography on Limits of Computation

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John Nash was a mathematician who earned the 1994 Nobel Prize in Economics for his work in game theory. His life story was made into a successful movie, $$A$$ Beautiful Mind.

In 1955, Nash was in correspondence with the United States National Security Agency (NSA), $${ }^{1}$$ discussing new methods of encryption that he had devised. In these letters, he also proposes some general principles of cryptography (bold highlighting not in the original):

... in principle the enemy needs very little information to begin to break down the process. Essentially, as soon as $$\lambda$$ bits $$^{2}$$ of enciphered message have been transmitted the key is about determined. This is no security, for a practical key should not be too long. But this does not consider how easy or difficult it is for the enemy to make the computation determining the key. If this computation, although possible in principle, were sufficiently long at best then the process could still be secure in a practical sense.

Nash is saying something quite profound: it doesn’t really matter whether attacks are impossible, only whether attacks are computationally infeasible. If his letters hadn’t been kept classified until 2012, they might have accelerated the development of "modern" cryptography, in which security is based on intractable computations. As it stands, he was decades ahead of his time in identifying one of the most important concepts in modern cryptography.

1The original letters, handwritten by Nash, are available at: http://www.nsa.gov/public_info/press_room/ 2012/nash_exhibits.html.

2Nash is using r to denote the length of the key, in bits

This page titled 4: Basing Cryptography on Limits of Computation is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Mike Rosulek (Open Oregon State) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.