> Modern cryptography relies on the hardness of integer factorization... P vs NP
This is not true for elliptic curve cryptography, or exactly true for RSA et al. ECC is based on the hardness of the elliptic discrete logarithm problem which is not exactly the same as solving integer factorization. However both can be tackled in polynomial time using Shor's algorithm [0], so both will be vulnerable once we have quantum computers. There are a handful of promising post quantum approaches, such as lattice cryptography. You don't need to solve P vs NP to break modern cryptography, you just need a computer that can run Shor.
This is not true for elliptic curve cryptography, or exactly true for RSA et al. ECC is based on the hardness of the elliptic discrete logarithm problem which is not exactly the same as solving integer factorization. However both can be tackled in polynomial time using Shor's algorithm [0], so both will be vulnerable once we have quantum computers. There are a handful of promising post quantum approaches, such as lattice cryptography. You don't need to solve P vs NP to break modern cryptography, you just need a computer that can run Shor.
[0] https://eprint.iacr.org/2017/598.pdf