And even then, we'd need to invent a cryptosystem whose hardness depends on an NP-complete problem, and be able to reliably find hard instance of that problem. There are interesting attempts at that which have failed e.g. the Knapsack cryptosystem:
https://en.wikipedia.org/wiki/Merkle%E2%80%93Hellman_knapsac...
What has cryptography got to do with p!=np ?
If P=NP, then you can invert in polynomial time (assuming this is practical) arbitrary boolean functions and thus solve any equations (that can be specified as a boolean circuit), find keys to any cryptosystem etc.
For example, you can assume part of the plaintext (like a header in transmission) and solve an equation (which can be restated as a SAT instance) that yields you a key assuming certain cryptographic algorithm was used. So pretty much all cryptography relies on P!=NP.
If p == np, then interger factorization can be reduced to k-sat, which is solvable in p time
https://cs.stackexchange.com/questions/6650/how-can-p-np-enh...
Re:My survey article on P vs. NP (2017)
by raverbashing on Tuesday March 23, 2021 @12:12PM
(#26556712)
> then integer factorization can be reduced to k-sat,
It can be reduced already, but then 3-sat can be solved in P time
Neat, thanks!
If there's an efficient algorithm for solving NP-complete problems then 99% of current cryptography would be broken. Only one-time-pads would still be secure.
But don't worry, if that happens the world would be so different there might no longer be a need for cryptography.
There exist crypto systems independent on P vs NP, both symmetric and asymnetric. The general approach exploits polynomial time complexity differences. However key-sizes increase accordingly: last I saw a key-pair was a few 100's of megabytes. (an enourmous reduction from originally GB's)
That's what we have, but it's not really comforting.
Now if you think what it'd mean to proove that e.g. a public key crypto system is secure, it almost implies that you'd need to proove p!=np before that. I.e. prooving p!=np would not proove that cryptography is secure, but it may be the first step and the understanding gained from prooving p!=np may mean that there's a much better understanding of complexity in general, which will help.