RSA has been around a long time and is relatively easy to understand and implement. But for new deployments we’re supposed to use ECC instead. The comment earlier about P vs NP is unfortunately confused. Factoring is not known to be NP-hard and it probably isn’t.
We can actually make stronger claims about factoring. We know for certain do that it is not in NP hard, so you are correct there
And I’m not exactly a security expert, but moving away from RSA at this point makes sense. Early assumptions about the difficulty of factoring large semi-primes certainly hasn’t panned out (in particular in light of the growing risk of quantum computers.)
Factoring is not NP-complete because it is not a member of NP-hard.
You are correct that verifiability is in there, but you have to also be able to do the reduction.
There are plenty of problems out there where we lack polynomial time solutions, while the problem also lacks the expressiveness required to reduce back and forth.
Factoring is not in any category because we don’t have a proof for it. As it stands, the algorithms we have require exponential time in the worst case, but solutions can be verified in polynomial time, which is NP-complete. I said “at most” for a reason.
Also, every algorithm is verifiable. If there’s no better way to do it then re-running the algorithm and that algorithm is exponential, then it’s NP-Hard.
Factoring is in a category. Everything computable exists somewhere. We know factoring to be in NP.
What I was disagreeing with was the “at least” and “at most” characterization of NP completeness. It is a set, not a boundary. The actual diagram of the complexity zoo is much more complicated than concentric circles.
And for verifiability, I was not referring to it as an existential sort of thing. I was simply saying that I agreed with you in that particular facet, but it isn’t sufficient to describe NP completeness. You also need NP-hardness.
RSA has been around a long time and is relatively easy to understand and implement. But for new deployments we’re supposed to use ECC instead. The comment earlier about P vs NP is unfortunately confused. Factoring is not known to be NP-hard and it probably isn’t.
We can actually make stronger claims about factoring. We know for certain do that it is not in NP hard, so you are correct there
And I’m not exactly a security expert, but moving away from RSA at this point makes sense. Early assumptions about the difficulty of factoring large semi-primes certainly hasn’t panned out (in particular in light of the growing risk of quantum computers.)
Factoring is at most NP-complete because we have a polynomial time verification for it.
Factoring is not NP-complete because it is not a member of NP-hard.
You are correct that verifiability is in there, but you have to also be able to do the reduction.
There are plenty of problems out there where we lack polynomial time solutions, while the problem also lacks the expressiveness required to reduce back and forth.
Factoring is not in any category because we don’t have a proof for it. As it stands, the algorithms we have require exponential time in the worst case, but solutions can be verified in polynomial time, which is NP-complete. I said “at most” for a reason.
Also, every algorithm is verifiable. If there’s no better way to do it then re-running the algorithm and that algorithm is exponential, then it’s NP-Hard.
Factoring is in a category. Everything computable exists somewhere. We know factoring to be in NP.
What I was disagreeing with was the “at least” and “at most” characterization of NP completeness. It is a set, not a boundary. The actual diagram of the complexity zoo is much more complicated than concentric circles.
And for verifiability, I was not referring to it as an existential sort of thing. I was simply saying that I agreed with you in that particular facet, but it isn’t sufficient to describe NP completeness. You also need NP-hardness.