Asymmetric cryptographic algorithms

Is it possible for any of them to survive to P=NP, assuming that NP-complete O(2^N) problems collapse to at best O(N²)?

>>12 huh?
>>6
This is a stupid question: no.
Technically, any algorithm that uses factorization can be broken with a quantum computer.
As I said, not every asymmetric cryptography algorithm uses integer factorisation or discrete inverse logarithm that are solvable in polynomial time on Quantum Computers (theoretically).

Then, first, the precise relation between BQP (Bounded Quantum Polynomial) and NP is not known.

Second, BQP does not seem to allow you to "check all possible private keys to see if any matches the public key (and signature) as fast as you can decrypt/verify with a single key". It might be more powerful than NP in some cases (though not in integer factorization and such), but it seems to be less powerful for those other asymmetric cryptography algorithms.