> Because trust me, if it was a low exponent for x, we'd have found it already.
This seems not to understand the basic nature of the problem with finding a solution: It's not that there's a brute force method for finding a solution.
The difficulty here is that the solution will probably involve some trick or twist that nobody has thought of before.
It's a similar case to that of Fermat's original solution to FLT; it's not that he didn't have a solution (although most believe that he didn't); it's just that nobody ever figured out or found his proof. It's possible that the solution exists, and nobody has yet figured out what the mental gymnastics were to arrive at it.
And the solution of NP-complete problems would make number-theoretic cryptography significantly more difficult (if not impossible). It's not like there's another, better class of problems. The reason that the NP complete problems work for public key cryptography is that there is an easy solution in both directions given the right pieces of information. It doesn't rely upon a one-way function, but upon a reversible algorithm.
Quantum would be the way to go at that point.
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> Because trust me, if it was a low exponent for x, we'd have found it already.
This seems not to understand the basic nature of the problem with finding a solution: It's not that there's a brute force method for finding a solution.
The difficulty here is that the solution will probably involve some trick or twist that nobody has thought of before.
It's a similar case to that of Fermat's original solution to FLT; it's not that he didn't have a solution (although most believe that he didn't); it's just that nobody ever figured out or found his proof. It's possible that the solution exists, and nobody has yet figured out what the mental gymnastics were to arrive at it.
And the solution of NP-complete problems would make number-theoretic cryptography significantly more difficult (if not impossible). It's not like there's another, better class of problems. The reason that the NP complete problems work for public key cryptography is that there is an easy solution in both directions given the right pieces of information. It doesn't rely upon a one-way function, but upon a reversible algorithm. Quantum would be the way to go at that point.