What is the relation between the existence of a cryptographic hash function and the existence of a PRG? Does the existence of a PRF/PRG/one way function imply the existence of a hash function? I did not find a proof of the existence of a cryptographic hash function, so I want to know whether it is based the same axiom (the PRG axiom)?
There is a black-box separation between one-way functions and collision resistant hash functions. This was proven at Eurocrypt 1998 by Dan Simon, in the paper entitled Finding collisions on a one-way street: Can secure hash functions be based on general assumptions?. Of course, this doesn't mean that it's not possible using non-black-box reductions, but no one knows how to do this.