Existence of one-way functions is a widely accepted conjecture in complexity theory. A function is one-way if it is computable in polynomial-time but not invertible in polynomial-time (this is different from the notion used in cryptography where average-case hardness is required). It seems we don not have any proof techniques that proves one-wayness.

Let us relax the requirement such that one-wayness means the function $ f(x)$ is computable in $ O(n^{c})$ but $ f^{-1}$ is not computable in $ O(n^{c-t})$ time for some integer $ t \gt 0$ .

Is there any known current techniques for proving this relaxed notion of one-wayness? Is there a natural function $ f$ that was proven to be one-way in this relaxed setting?

I am interested in honest functions where $ |x|< p(|f(x)|)$ for some polynomial $ p$ .