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    "text": "Cryptography from Lossy Reductions: Towards OWFs from ETH, and Beyond (Pouria Fallahpour, Alex B. Grilo, Garazi Muguruza, Mahshid Riahinia) ia.cr/2025/778",
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          "alt": "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)/loglogn)), where τ_(Π)(n) is the infimum of the runtime of all (worst-case) solvers of Π on instances of size n. More precisely, by having a reduction with a better runtime, for an arbitrary promise problem Π, 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 And-, Or-, Maj-, Parity-, Mod_(k)-, and Threshold_(k)-reductions.\n\nAdditionally, 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^(Ω(logτ_(Π))) when these mappings are considered. Restricting to weak fine-grained OWFs, this runtime can be further improved as Ω(τ_(Π)). Intuitively, the latter asserts that if weak fine-grained OWFs do not exist then any instance randomization of any Π has the same runtime (up to a constant factor) as the best worst-case solver of Π.\n\nTaking 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. Moreover, the analysis can be adapted to studying such properties of any NP-complete problem.\n\nFinally, 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)/loglogn)) implies the existence of one-way state generators, where τ_(Π) is defined with respect to quantum solvers.\n",
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    "createdAt": "2025-05-03T20:15:51.008237Z"
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