One-way Functions and Partial MCSP.

One-way functions (OWFs) are central objects of study in cryptography and computational complexity theory. In a seminal work, Liu and Pass (FOCS 2020) proved that the average-case hardness of computing time-bounded Kolmogorov complexity is equivalent to the existence of OWFs. It remained an open problem to establish such an equivalence for the average-case hardness of some NP -complete problem. In this paper, we make progress on this question by studying a polynomially-sparse variant of Partial Minimum Circuit Size Problem (Partial MCSP), which we call Sparse Partial MCSP , as follows. 1. First, we prove that if Sparse Partial MCSP is zero-error average-case hard on a polynomial fraction of its instances, then there exist OWFs. 2. Then, we observe that Sparse Partial MCSP is NP -complete under polynomial-time deterministic reductions. That is, there are NP -complete problems whose average-case hardness implies the existence of OWFs. 3. Finally, we prove that the existence of OWFs implies the nontrivial zero-error average-case hardness of Sparse Partial MCSP. Thus the existence of OWFs is inextricably linked to the average-case hardness of this NP -complete problem.