[Someone]

> I'm not sure what security properties are really "proven" about existing cryptographic hash functions

AFAIK, we don’t even know whether trapdoor functions exist.

https://en.wikipedia.org/wiki/Trapdoor_function:

“As of 2004, the best known trapdoor function (family) candidates are the RSA and Rabin families of functions”

Also note that the ‘examples’ section starts with:

“In the following two examples, we always assume it is difficult to factorize a large composite number (see Integer factorization).”

[aidenn0]

IIRC, if P=NP then trapdoor functions do not exist, so proving that one existed would be a huge deal even outside of cryptography.

[Dylan16807]

Only if your definitions of "easy" and "hard" are based entirely on complexity classes.

If you show me a setup where "easy" is n^3 and "hard" is n^15 I will happily call that a trapdoor function.