Abstract
One-way functions (OWFs) form the foundation of modern cryptography, yet their unconditional existence remains a major open question. In this work, we study this question by exploring its relation to lossy reductions, i.e., reductions R for which it holds that I(X;R(X)) ≪ n for all distributions X over inputs of size n. Our main result is that either OWFs exist or any lossy reduction for any promise problem Π runs in time 2^Ω(logτ_Π / loglog n), where τ_Π(n) is the infimum of the runtime of all (worst-case) solvers of Π on instances of size n. In fact, our result requires a milder condition, that R is lossy for sparse uniform distributions (which we call mild-lossiness). It also extends to f-reductions as long as f is a non-constant permutation-invariant Boolean function, which includes And-, Or-, Maj-, Parity-, Modulo_k, and Threshold_k-reductions. Additionally, we show that worst-case to average-case Karp reductions and randomized encodings are special cases of mildly-lossy reductions and improve the runtime above as 2^Ω(log τ_Π) when these mappings are considered. Restricting to weak fine-grained OWFs, this runtime can be further improved as Ω(τ_Π). Taking Π as kSAT, our results provide sufficient conditions under which (fine-grained) OWFs exist assuming the Exponential Time Hypothesis (ETH). Conversely, if (fine-grained) OWFs do not exist, we obtain impossibilities on instance compressions (Harnik and Naor, FOCS 2006) and instance randomizations of kSAT under the ETH. Finally, we partially extend these findings to the quantum setting; the existence of a pure quantum mildly-lossy reduction for Π within the runtime 2^o(logτ_Π / loglog n) implies the existence of one-way state generators.