One Way Functions with Non-Uniform Inputs

It is well-known that a One-Way Function is hard to invert on uniformly random inputs. As per Wikipedia,

for [sic] all randomized algorithms $$F$$, all positive integers $$c$$ and all sufficiently large $$n = \mathsf{length}(x)$$, $$\Pr[f(F(f(x))) = f(x)]\leq\mathsf{negl}(n)$$ where the probability is over the choice of $$x$$ from the uniform distribution on $$\{0,1\}^n$$ and the randomness of $$F$$.

Some basic facts about what happens in the case of non-uniform inputs can be easily ascertained: given, for instance, a pseudorandom distribution as input, the function is also hard to invert lest the existence of a distinguisher for the input distribution. It should be similarly easy to prove the hardness of inverting given an input distribution with a non-negligible amount of min-entropy.

What is the state-of-the-art on invertibility guarantees with non-uniform input distributions?

• "It should be similarly easy to prove the hardness of inverting given an input distribution with a non-negligible amount of min-entropy." that's not true. Let $f : \{0,1\}^n \to \{0,1\}^n$ be a OWF. Define $g : \{0,1\}^{2n} \to \{0,1\}^n$ as $g(x\Vert y) := \begin{cases}y&\text{if } x=0^n\\f(y)&\text{otherwise}\end{cases}$. This remains one-way, with a uniform input distribution, but is trivial to invert relative to the distribution with $x=0^n$ and $y$ uniform. Mar 27 at 9:54