A fundamental pursuit in complexity theory concerns reducing worst-case problems to average-case problems. There exist complexity classes such as PSPACE that admit worst-case to average-case reductions. However, for many other classes such as NP, the evidence so far is typically negative, in the sense that the existence of such reductions would cause collapses of the polynomial hierarchy(PH). Basing cryptographic primitives, e.g., the average-case hardness of inverting one-way permutations, on NP-completeness is a particularly intriguing instance. As there is evidence showing that classical reductions from NP-hard problems to breaking these primitives result in PH collapses, it seems unlikely to base cryptographic primitives on NP-hard problems. Nevertheless, these results do not rule out the possibilities of the existence of quantum reductions. In this work, we initiate a study of the quantum analogues of these questions. Aside from formalizing basic notions of quantum reductions and demonstrating powers of quantum reductions by examples of separations, our main result shows that if NP-complete problems reduce to inverting one-way permutations using certain types of quantum reductions, then coNP ⊆ QIP( 2 ) .
A fundamental pursuit in complexity theory concerns reducing worst-case problems to average-case problems. There exist complexity classes such as PSPACE that admit worst-case to average-case reductions. However, for many other classes such as NP, the evidence so far is typically negative, in the sense that the existence of such reductions would cause collapses of the polynomial hierarchy(PH). Basing cryptographic primitives, e.g., the average-case hardness of inverting one-way permutations, on NP-completeness is a particularly intriguing instance. As there is evidence showing that classical reductions from NP-hard problems to breaking these primitives result in PH collapses, it seems unlikely to base cryptographic primitives on NP-hard problems. Nevertheless, these results do not rule out the possibilities of the existence of quantum reductions. In this work, we initiate a study of the quantum analogues of these questions. Aside from formalizing basic notions of quantum reductions and demonstrating powers of quantum reductions by examples of separations, our main result shows that if NP-complete problems reduce to inverting one-way permutations using certain types of quantum reductions, then coNP ⊆ QIP( 2 ) .
The existence of one-way functions (OWFs) forms the minimal assumption in classical cryptography. However, this is not necessarily the case in quantum cryptography. One-way puzzles (OWPuzzs), introduced by Khurana and Tomer, provide a natural quantum analogue of OWFs. The existence of OWPuzzs implies PP=BQP, while the converse remains open. In classical cryptography, the analogous problem-whether OWFs can be constructed from P=NP-has long been studied from the viewpoint of hardness of learning. Hardness of learning in various frameworks (including PAC learning) has been connected to OWFs or to P=NP. In contrast, no such characterization previously existed for OWPuzzs. In this paper, we establish the first complete characterization of OWPuzzs based on the hardness of a well-studied learning model: distribution learning. Specifically, we prove that OWPuzzs exist if and only if proper quantum distribution learning is hard on average. A natural question that follows is whether the worst-case hardness of proper quantum distribution learning can be derived from PP=BQP. If so, and a worst-case to average-case hardness reduction is achieved, it would imply OWPuzzs solely from PP=BQP. However, we show that this would be extremely difficult: if worst-case hardness is PP-hard (in a black-box reduction), then SampBQP=SampBPP follows from the infiniteness of the polynomial hierarchy. Despite that, we show that PP=BQP is equivalent to another standard notion of hardness of learning: agnostic. We prove that PP=BQP if and only if agnostic quantum distribution learning with respect to KL divergence is hard. As a byproduct, we show that hardness of agnostic quantum distribution learning with respect to statistical distance against PPTΣ3P learners implies SampBQP=SampBPP.
One-way puzzles (OWPuzzs) introduced by Khurana and Tomer [STOC 2024] are a natural quantum analogue of one-way functions (OWFs), and one of the most fundamental primitives in ''Microcrypt'' where OWFs do not exist but quantum cryptography is possible. OWPuzzs are implied by almost all quantum cryptographic primitives, and imply several important applications such as non-interactive commitments and multi-party computations. A significant goal in the field of quantum cryptography is to base OWPuzzs on plausible assumptions that will not imply OWFs. In this paper, we base OWPuzzs on hardness of non-collapsing measurements. To that end, we introduce a new complexity class, SampPDQP, which is a sampling version of the decision class PDQP introduced in [Aaronson, Bouland, Fitzsimons, and Lee, ITCS 2016]. We show that if SampPDQP is hard on average for quantum polynomial time, then OWPuzzs exist. SampPDQP is the class of sampling problems that can be solved by a classical polynomial-time algorithm that can make a single query to a non-collapsing measurement oracle, which is a ''magical'' oracle that can sample measurement results on quantum states without collapsing the states. Such non-collapsing measurements are highly unphysical operations that should be hard to realize in quantum polynomial-time. We also study upperbounds of the hardness of SampPDQP. We introduce a new primitive, distributional collision-resistant puzzles (dCRPuzzs), which are a natural quantum analogue of distributional collision-resistant hashing [Dubrov and Ishai, STOC 2006]. We show that dCRPuzzs imply average-case hardness of SampPDQP (and therefore OWPuzzs as well). We also show that two-message honest-statistically-hiding commitments with classical communication and one-shot signatures [Amos, Georgiou, Kiayias, Zhandry, STOC 2020] imply dCRPuzzs.
Pessiland is a world where NP is hard on average but one-way functions (OWFs) do not exist [Impagliazzo 1995]. Because almost all classical cryptographic primitives imply OWFs [Impagliazzo and Luby 1989], there is almost no classical cryptography in Pessiland. On the other hand, quantum cryptography can exist even when OWFs do not [Kretschmer 2021; Morimae and Yamakawa 2022; Ananth, Qian and Yuen 2022]. Is there a quantum analogue of Pessiland where NP is hard on average but even quantum cryptography does not exist? In this paper, we show that such a miserable world, Quantum Pessiland, exists: there is a quantum oracle relative to which UP∩coUP is hard on average against quantum polynomial-time algorithms with quantum advice, yet auxiliary-input EFI pairs do not exist. We also show that there is a classical oracle relative to which UP∩coUP is hard on average against quantum polynomial-time algorithms with quantum advice, yet classically-secure auxiliary-input one-way puzzles (OWPuzzs) do not exist. Almost all quantum cryptographic primitives imply EFI pairs or OWPuzzs, and therefore these results mean that there is almost no quantum cryptography relative to these oracles. We further show that relative to the classical oracle, SampBQP = SampBPP, and therefore there is no sampling-based quantum advantage in Quantum Pessiland. Finally, because our average-case hardness of UP∩coUP implies P#P⊆i.o.BQP/qpoly, our result also implies that a non-relativizing proof technique is necessary to construct OWPuzzs solely from P#P⊆i.o.BQP/qpoly, which gives a partial negative answer to the open problem of [Khurana and Tomer 2025].
Founding Quantum Cryptography on Quantum Advantage, or, Towards Cryptography from #P-Hardness
Recent oracle separations [Kretschmer, TQC'21, Kretschmer et. al., STOC'23] have raised the tantalizing possibility of building quantum cryptography from sources of hardness that persist even if the polynomial hierarchy collapses. We realize this possibility by building quantum bit commitments and secure computation from unrelativized, well-studied mathematical problems that are conjectured to be hard for P#P -- such as approximating the permanents of complex Gaussian matrices, or approximating the output probabilities of random quantum circuits. Indeed, we show that as long as any one of the conjectures underlying sampling-based quantum advantage (e.g., BosonSampling, Random Circuit Sampling, IQP, etc.) is true, quantum cryptography can be based on the extremely mild assumption that P#P⊆(io)BQP/qpoly. We prove that the following hardness assumptions are equivalent. (1) The hardness of approximating the probability assigned to a randomly chosen string in the support of certain efficiently sampleable distributions (upto inverse polynomial multiplicative error).(2) The existence of one-way puzzles, where a quantum sampler outputs a pair of classical strings -- a puzzle and its key -- and where the hardness lies in finding the key corresponding to a random puzzle. These are known to imply quantum bit commitments [Khurana and Tomer, STOC'24]. (3) The existence of state puzzles, or one-way state synthesis, where it is hard to synthesize a secret quantum state given a public classical identifier. These capture the hardness of search problems with quantum inputs (secrets) and classical outputs (challenges). These are the first constructions of quantum cryptographic primitives (one-way puzzles, quantum bit commitments, state puzzles) from concrete, well-founded mathematical assumptions that do not imply the existence of classical cryptography.
From Worst-Case Hardness of NP to Quantum Cryptography via Quantum Indistinguishability Obfuscation
Indistinguishability obfuscation (iO) has emerged as a powerful cryptographic primitive with many implications. While classical iO, combined with the infinitely-often worst-case hardness of NP, is known to imply one-way functions (OWFs) and a range of advanced cryptographic primitives, the cryptographic implications of quantum iO remain poorly understood. In this work, we initiate a study of the power of quantum iO. We define several natural variants of quantum iO, distinguished by whether the obfuscation algorithm, evaluation algorithm, and description of obfuscated program are classical or quantum. For each variant, we identify quantum cryptographic primitives that can be constructed under the assumption of quantum iO and the infinitely-often quantum worst-case hardness of NP (i.e., NP⊆i.o.BQP). In particular, we construct pseudorandom unitaries, QCCC quantum public-key encryption and (QCCC) quantum symmetric-key encryption, and several primitives implied by them such as one-way state generators, (efficiently-verifiable) one-way puzzles, and EFI pairs, etc. While our main focus is on quantum iO, even in the classical setting, our techniques yield a new and arguably simpler construction of OWFs from classical (imperfect) iO and the infinitely-often worst-case hardness of NP.
Inspired by connections to two dimensional quantum theory, we define several models of computation based on permuting distinguishable particles (which we call balls), and characterize their computational complexity. In the quantum setting, we find that the computational power of this model depends on the initial input states. More precisely, with a standard basis input state, we show how to approximate the amplitudes of this model within additive error using the model DQC1 (the class of problems solvable with one clean qubit), providing evidence that the model in this case is weaker than universal quantum computing. However, for specific choices of input states, the model is shown to be universal for BQP in an encoded sense. We use representation theory of the symmetric group to partially classify the computational complexity of this model for arbitrary input states. Interestingly, we find some input states which yield a model intermediate between DQC1 and BQP. Furthermore, we consider a restricted version of this model based on an integrable scattering problem in 1+1 dimensions. We show it is universal under postselection, if we allow intermediate destructive measurements and specific input states. Therefore, the existence of any classical procedure to sample from the output distribution of this model within multiplicative error implies collapse of polynomial hierarchy to its third level. Finally, we define a classical version of this model in which one can probabilistically permute balls. We find this yields a complexity class which is intermediate between L and BPP. Moreover, we find a nondeterministic version of this model is NP-complete.