Homological characterization of one‑way functions

Prove that a function f is one‑way if and only if the inversion problem L_f = { (y, x) : f(x) = y } has positive homological complexity, i.e., h(L_f) > 0, meaning the computational homology of inverting f has nontrivial higher homology groups.

Background

Seeking a topological foundation for cryptographic hardness, the authors conjecture an equivalence between one‑wayness and the presence of essential cycles in the computational complex of inversion. This would turn cryptographic security into a homological property of the underlying inversion problem.

They connect the claim to their homological lower‑bound principle, arguing nontrivial homology should obstruct efficient inversion algorithms.