# 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 "unnatural" 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.