So, in cryptography, it is sometimes the case that in order to set up a cryptosystem, one needs some output $f(x)$ for randomly generated $x$ where $f$ is a one-way function but where the input $x$ can be used to break the cryptosystem (i.e. the input $x$ is toxic waste that needs to be destroyed or the output $f(x)$ needs to be computed using some sort of secure multi-party computation).
Does there exist an efficient (by efficient I mean polynomial time) deterministic algorithm that given a one-way function $f$, returns a circuit $C_{f}$ such that for all inputs $x$, there is some $c$ where $C_{f}(x)=f(c)$ but where there is no efficient algorithm that given $x$ returns some $d$ with $f(d)=C_{f}(x)$ (preferably where the function $C_{f}$ is collision resistant)? Or does there exist some impossibility result for such a cryptosystem? If there is no impossibility result, then what is the weakest cryptosystem that ensures an algorithm $f\mapsto C_{f}$ exists?
