[bhrgunatha]

To be clear I'm not talking about how useful asymptotic analysis and big O notation are in general, I'm talking specifically about the case if we eventually do prove P = NP.

Internet lore and popular media then assume that immediately for example all encryption will be trivially broken. Mathematically it may be true because then all NP problems would be known to be polynomial, but there's still the issue of the practical steps involved factorising a huge number which may have enormous constants or very large exponents. Or if it turns out we can reduce it to some other known polynomial problem, there's still the actual transformation which itself may be polynomial with large constants or exponents.

[smitherfield]

If I'm not mistaken, isn't it also the case that factorization hasn't been proven to be in NP?

[teraflop]

That's not quite correct. Factorization is definitely in NP, because verifying whether a number has a particular factor is easy.

Maybe you meant that factorization hasn't been proven to be NP-complete, which is true. In fact it would be very surprising if it was NP-complete, because that would imply other surprising things (e.g. NP = co-NP).

[Beltiras]

If P=NP is proven than that makes factorization an easy problem (and included in the proof) which would make many algorithms for encryption "trivially broken".

[lomnakkus]

Right, but it's worth mentioning that factorization itself is not known to be NP-hard (and it's suspected not to be, by most, I think).

Also, as others have pointed out, the constants involved in any "generic" algorithm for solving any NP problem in Polynomial type may be astronomical, so even if it turns out that P=NP, then it may not ever be feasible to actually such any algorithm for anything practical.