Forgive my ignorance, but what would they be - the complete implosion of all forms of known security, or something else?
This is a bit beyond my ken :)
This is a bit beyond my ken :)
If P=NP, then those previously thought to be intractable problems, are actually tractable. And the foundation of a lot of security-related engineering collapses.
While that means most protocols used for cryptography would need to be replaced (hashing, digital signatures, etc) it also means other combinatorics algorithms (traveling salesman, protein structure prediction) would become solvable which may been boon for logistics and/or computational science.
(I think this is correct) If P=NP there will still be intractable problems; they would be ones where the solutions can't be verified in polynomial time... along the lines of verifying the solution is correct is as complicated as brute forcing the solution.
Note: it's been a while since my computation theory class. ;) I am reading over https://en.wikipedia.org/wiki/P_versus_NP_problem and relearning the fine house of cards theoreticians have divided this problem into. There is a "consequences of P=NP" towards the bottom that sums it up better than I can.
"A proof that P = NP could have stunning practical consequences if the proof leads to efficient methods for solving some of the important problems in NP."
Note the "if". It is extremely important to the meaning. It's very possible for P to equal NP, but for that "if" to be false.
Not really. They depend on guessing being slow. P, just P, can do slow. P can be bigger than the universe even with low values of n.