If OWF were to exist, do we know for sure that one of the candidate OWF would indeed be a OWF? We have several candidates for OWF, like multiplication/factoring and discrete exponencial/logarithm. What I am asking is:
Does the existence of one way functions imply that our candidate functions are indeed one way functions?
Is there a candidate one way function whose one wayness is directly tied to the existance of OWF?
Yes, you are looking for the notion of a universal one-way function.
Rafael Pass/abhi shelat's notes contain a construction on page 49. The construction is &quot;unnatural&quot; in the sense that it involves parsing the input to the OWF $y$ as a pair $\langle M\rangle || x$, where $\langle M\rangle$ is interpreted as the description of a Turing machine. Then you simulate $M(x)$'s executation for a bounded number of steps.
This is fairly typical of &quot;universal&quot; constructions (others exist, namely Levin's universal search). It is far from being feasible in practice though.
The notes mention that a &quot;natural&quot; universal OWF construction is a very open problem (circa 2008-ish).