Can irreversibility of trapdoor functions generally not be proved?

This seems to be an open research problem. If you look at https://en.wikipedia.org/wiki/One-way_function it says that the existence of one-way functions is currently unproven. So for cryptography applications there are two pieces missing for a rigorous proof. First, one-way functions do exist and second, the particular function used in the cryptographic algorithm is such a one-way function.

Edit: I would say that the statement using 'always' on the German wikipedia is wrong. With current knowledge it is based on unproven assumptions. If at some later time it is proven that one-way functions exist one could prove the security of public key cryptography. If on the other hand it is shown that no one-way functions exists, this proves that all public key cryptography is a priori unsecure. It might just be the case that we don't know how to invert a particular function even though we know it can't be a one-way function.

This is due to the fact that the existence of One-Way Functions (OWF) implies that $P \neq NP$. In other words, with contrapositive, if $P = NP$ then OWF doesn't exit. So if we have one you would know it in the news. Therefore, the security of the cryptographic systems is based on unproven results.

There are candidates for OWF functions like;

• Multiplication and Factoring
• The Rabin function used in Rabin Cryptosystem
• Discrete logarithms used in ECC, ElGamal encryption, DHKE
• Cryptographically secure hash functions

After extensive research, these are neither proven as an OWF or the reverse.