## Implications of An Infinitely Large Function if P ≠ NP

Posted by Michael Dickens on November 25, 2008

Prerequisite reading:

http://en.wikipedia.org/wiki/One-way_function

http://en.wikipedia.org/wiki/P_%3D_NP_problem

Let’s assume that one-way functions exist, meaning that there is something that’s unsolvable in reverse in polynomial time. So what if you’re using infinities? The polynomial of an infinity is still that infinity. But the exponential of an infinity is a greater infinity. So it may be possible to use a cipher that requires one to stay within the realm of one infinity. This would make it impossible to decrypt. Here’s an example.

P = plaintext

C = ciphertext

f(P) = C

where f is a function that takes aleph null steps. This works if there is some way to do aleph null steps, but it’s impossible to do aleph one steps. So the amount of time it takes to calculate is aleph null, but the number of steps it takes to solve is aleph one (x ^ aleph null = aleph one).

This all makes sense if you have infinite time. With infinite time you can do aleph null operations, but no more. Or if you can do infinite operations per second. But you can’t do aleph one operations in any amount of time, even infinite.

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