# Examples of inverse of trapdoor functions non-learnable with black-box query access?

The (informal) definition of trapdoor function is a one-way function with a trapdoor $$t$$ such that

1. given $$f$$, it is easy to compute $$f(x)$$ for any $$x$$, but hard to compute $$f^{-1}(y)$$ efficiently
2. given a trapdoor $$t$$, it will be easy and efficient to compute $$f^{-1}(y)$$ for any $$y$$.

By non-learnable with blackbox, I mean: Give a probabilistic polynomial-time adversary the description $$f$$, and an oracle that computes $$f^{-1}(y)$$ on any input $$y$$, but we do not give the adversary the trapdoor $$t$$. After only polynomial queries, adversary cannot efficiently and correctly compute $$f^{-1}(y)$$ for some randomly chosen $$y$$( not queried before) with high probability.

What are some constructions for trapdoor functions where their inverses are non-learnable through black-box access?