Well, most modern cryptography is based on assumptions that can not be proven, so having different standards based on different assumptions is probably the only way to safeguard against if one of the assumptions would be proven false in the future.
To nitpick, afaik, its not that they cannot be proven, its that they have not been, and look very hard to prove, which is slightly different (not my area of expertise, but i assume this would be tied to p vs np)
I it not tied to P vs NP as far as I’m aware. But it is the same sort of situation: number theory assumptions that are completely unproven despite many attempts.
I was thinking, if you could definitively prove these assumptions are hard, that would prove P != NP, because if P=NP that would imply there would be an algorithm to solve these types of problems, since they are the type of thing that can be solved in polynomial time with the key, but cannot without a key. (I'm a bit out of my depth here)
Hash functions too. If P=NP then you can reverse a hash in polynomial time.
NP is the set of all functions that you can verify a solution to in polyomial time, and the solution of the inverse-hash function is just a plaintext that hashes to the right value, and obviously you can check if a plaintext is right in polynomial time by just hashing it and comparing the hashes. Thus reversing a hash function is in NP, so if P=NP it's in P.
There's some subtlety here in that "reversing" a hash function really just means coming up with a plaintext that generates the right hash, not the original one, but you can put any polynomial-time set of constraints on the plaintext and finding a plaintext that satisfies those constraints (and hashes to the right value) is still in NP, so the subtlety really doesn't save you much.
Edit:
Side point, but since we're talking quantum, we should really be saying BQP=NP not P=NP, BQP being the problems solvable in polynomial time on a quantum computer, it's a superset of P and a subset of NP, but we don't know if it's equal to either or both. I.e. P=NP implies BQP=NP, BQP != NP implies P != NP, BQP != P implies P != NP, but the reverse of all of those statements isn't known to be true.
Maybe it safeguards them from looking like they've screwed this up, but in terms of providing a concrete recommendation to system implementers, how does this safeguard anything? How does offering multiple algorithms in the PQC category help me make systems safer? What am I actually supposed to do here (how do I reflect this hedge in a system design)?
They didn't feel the need to provide multiple recommendations during the AES, or the SHA-3 process, even though Rijndael and Keccak used different constructions relative to RC6/TwoFish and SHA-2/Blake2. Why now?
The recommendations look clear to me: you should use CRYSTALS-Dilithium (unless you need smaller signatures, in which case use FALCON), but you should also be prepared to switch to SPHINCS+ on short notice if someone breaks CRYSTALS-Dilithium (or structured lattices in general).
So best practice would seem to be to implement both CRYSTALS-Dilithium and SPHINCS+, set CRYSTALS-Dilithium as the default, and provide a switch (config setting, whatever) to switch to SPHINCS+. If you have long-term keys, you should have both forms set up & ready to use.
> They didn't feel the need to provide multiple recommendations during the AES, or the SHA-3 process, even though Rijndael and Keccak used different constructions relative to RC6/TwoFish and SHA-2/Blake2. Why now?
SHA-3 was explicitly alternative reccomendation. The entire point was to come up with something that was not based on sha-2, because they were worried that the attacks on md5/sha1 could be extended to sha2 (which didn't really happen the way people were worried about). Even to this day, general advice is not to use sha3.
Less clear cut for aes, but at time of standardization (and even now afaik), triple des was considered secure, so its not like there wasn't a secure alternative.
These standards arent meant as implementation guides. You still need cryptography knowledge to securely use them.
