Well for one, the safety of encryption rests on certain problems being intractable. (In a theoretical sense; there are always implementation bugs that destroy security).
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.
Is proving P = NP equivalent to knowing how any intractable problem can be solved? Is it possible for P=NP and yet a class of intractable problems to remain unsolved?
It would mean that a large class of problems that have solutions that can be verified quickly can be solved quickly. Which cuts both ways.
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.
'quickly' is doing a lot of work in that sentence.
"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.
There's also the chance that while we may be able to come up with a polynomial algorithm for integer factorization, it's not actually practical to run still. Remember computational complexity discards the constants on that polynomial. Practically speaking x^2 + x is a lot different from 2^64x^2 + 2^32x + 2^16 :)
Among other things, mostly encryption. Most of our current methods depends on P != NP. So no need for 0 days if you can just read at encrypted data as if it wasn't.
