Paper 2025/778

Cryptography from Lossy Reductions: Towards OWFs from ETH, and Beyond

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 $I(X;R(X)) \ll n$ 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 $\Pi$ runs in time $2^{\Omega(\log\tau_\Pi / \log\log n)}$, where $\tau_\Pi(n)$ is the infimum of the runtime of all (worst-case) solvers of $\Pi$ on instances of size $n$. More precisely, by having a reduction with a better runtime, for an arbitrary promise problem $\Pi$, and by using a non-uniform advice, we construct (a family of) OWFs. 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 $\text{And-, Or-, Maj-, Parity-}$, $\text{Mod}_k$-, and $\text{Threshold}_k$-reductions. Additionally, we show that worst-case to average-case Karp reductions and randomized encodings are special cases of mild-lossy reductions and improve the runtime above as $2^{\Omega(\log \tau_\Pi)}$ when these mappings are considered. Restricting to weak fine-grained OWFs, this runtime can be further improved as $\Omega(\tau_\Pi)$. Intuitively, the latter asserts that if weak fine-grained OWFs do not exist then any instance randomization of any $\Pi$ has the same runtime (up to a constant factor) as the best worst-case solver of $\Pi$. Taking $\Pi$ as $k\text{Sat}$, 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 $k\text{Sat}$ under the ETH. Moreover, the analysis can be adapted to studying such properties of any $\text{NP}$-complete problem. Finally, we partially extend these findings to the quantum setting; the existence of a pure quantum mildly-lossy reduction for $\Pi$ within the runtime $2^{o(\log\tau_\Pi / \log\log n)}$ implies the existence of one-way state generators, where $\tau_\Pi$ is defined with respect to quantum solvers.