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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?

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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.

This is fairly typical of "universal" constructions (others exist, namely Levin's universal search). It is far from being feasible in practice though. The notes mention that a "natural" universal OWF construction is a very open problem (circa 2008-ish).

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