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arXiv:2510.04448

Quantum Cryptography and Hardness of Non-Collapsing Measurements

Tomoyuki Morimae, Shirakawa, Yuki, Takashi Yamakawa

Published Oct 6, 2025• 0 citations
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Abstract

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, $\mathbf{SampPDQP}$, which is a sampling version of the decision class $\mathbf{PDQP}$ introduced in [Aaronson, Bouland, Fitzsimons, and Lee, ITCS 2016]. We show that if $\mathbf{SampPDQP}$ is hard on average for quantum polynomial time, then OWPuzzs exist. $\mathbf{SampPDQP}$ 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 $\mathbf{SampPDQP}$. 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 $\mathbf{SampPDQP}$ (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.

DOI
10.48550/arxiv.2510.04448

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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 $UP\cap coUP$ 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 $UP\cap coUP$ 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 $UP\cap coUP$ implies $P^{\#P}\not\subseteq i.o.BQP/qpoly$, our result also implies that a non-relativizing proof technique is necessary to construct OWPuzzs solely from $P^{\#P}\not\subseteq i.o.BQP/qpoly$, which gives a partial negative answer to the open problem of [Khurana and Tomer 2025].

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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 $PP\neq BQP$, while the converse remains open. In classical cryptography, the analogous problem-whether OWFs can be constructed from $P \neq NP$-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 $P \neq NP$. 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 $PP \neq BQP$. If so, and a worst-case to average-case hardness reduction is achieved, it would imply OWPuzzs solely from $PP \neq BQP$. However, we show that this would be extremely difficult: if worst-case hardness is PP-hard (in a black-box reduction), then $SampBQP \neq SampBPP$ follows from the infiniteness of the polynomial hierarchy. Despite that, we show that $PP \neq BQP$ is equivalent to another standard notion of hardness of learning: agnostic. We prove that $PP \neq BQP$ 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 $PPT^{Σ_3^P}$ learners implies $SampBQP \neq SampBPP$.

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EFI pairs (Brakerski, Canetti, and Qian, ITCS 2023) and one-way puzzles (Khurana and Tomer, STOC 2024) are the leading candidates for the minimal assumption of quantum cryptography. The first are efficiently preparable quantum states, statistically far yet computationally indistinguishable; the second are classical puzzles, easy to sample and hard to solve. One-way puzzles imply EFI pairs, and whether the converse holds is open. We construct a single classical oracle relative to which one-way puzzles do not exist, even with an unbounded verifier, while an EFI pair survives every distinguisher that queries the oracle classically throughout and holds advice about it, making its one superposition query at the end. The oracle answers every question about the output probabilities of quantum samplers, which removes the puzzles, and hides a Haar-random half-dimensional subspace. To prove security we reduce it to communication complexity. An adversary whose knowledge of the subspace arrives as classical query answers can be simulated inside a two-party protocol against the party holding it, so it does no better than the best classical protocol for Vector-in-Subspace (Klartag and Regev, STOC 2011), whatever the oracle computes. That argument does not cover the superposition query, which we bound instead using tools from random matrix theory. The same attack gives a classical simulation of any quantum party in a classical-message protocol with no entanglement shared in advance, so relative to the oracle there is no proof of quantumness either. Quantum polynomial time therefore offers no advantage on any task with classical inputs and outputs, while the two quantum states stay indistinguishable. We state conjectures on removing the restriction on superposition queries.

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IACR Communications in Cryptology•Jan 13, 2025•0 citations

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One-way functions are central to classical cryptography. They are necessary for the existence of non-trivial classical cryptosystems, and also sufficient to realize meaningful primitives including commitments, pseudorandom generators and digital signatures. At the same time, a mounting body of evidence suggests that assumptions even weaker than one-way functions may suffice for many cryptographic tasks of interest in a quantum world, including bit commitments and secure multi-party computation. This work studies one-way state generators [Morimae-Yamakawa, CRYPTO 2022], a natural quantum relaxation of one-way functions. Given a secret key, a one-way state generator outputs a hard to invert quantum state. A fundamental question is whether this type of quantum one-wayness suffices to realize quantum cryptography. We obtain an affirmative answer to this question, by proving that one-way state generators with pure state outputs imply quantum bit commitments and secure multiparty computation. Along the way, we use efficient shadow tomography [Huang et. al., Nature Physics 2020] to build an intermediate primitive with classical outputs, which we call a (quantum) one-way puzzle. Our main technical contribution is a proof that one-way puzzles imply quantum bit commitments. This proof develops new techniques for pseudoentropy generation [Hastad et. al., SICOMP 1999] from arbitrary distributions, which may be of independent interest.

arXiv:2310.11526•ACM Symposium on Theory of Computing (STOC)•Jun 10, 2024•23 citations
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\mathbf{SampPDQP}SampPDQP, which is a sampling version of the decision class PDQP\mathbf{PDQP}PDQP introduced in [Aaronson, Bouland, Fitzsimons, and Lee, ITCS 2016]. We show that if SampPDQP\mathbf{SampPDQP}SampPDQP is hard on average for quantum polynomial time, then OWPuzzs exist. SampPDQP\mathbf{SampPDQP}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\mathbf{SampPDQP}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\mathbf{SampPDQP}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∩coUPUP\cap coUPUP∩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∩coUPUP\cap coUPUP∩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∩coUPUP\cap coUPUP∩coUP implies P#P⊈i.o.BQP/qpolyP^{\#P}\not\subseteq i.o.BQP/qpolyP#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/qpolyP^{\#P}\not\subseteq i.o.BQP/qpolyP#P⊆i.o.BQP/qpoly, which gives a partial negative answer to the open problem of [Khurana and Tomer 2025].
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≠BQPPP\neq BQPPP=BQP, while the converse remains open. In classical cryptography, the analogous problem-whether OWFs can be constructed from P≠NPP \neq NPP=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≠NPP \neq NPP=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≠BQPPP \neq BQPPP=BQP. If so, and a worst-case to average-case hardness reduction is achieved, it would imply OWPuzzs solely from PP≠BQPPP \neq BQPPP=BQP. However, we show that this would be extremely difficult: if worst-case hardness is PP-hard (in a black-box reduction), then SampBQP≠SampBPPSampBQP \neq SampBPPSampBQP=SampBPP follows from the infiniteness of the polynomial hierarchy. Despite that, we show that PP≠BQPPP \neq BQPPP=BQP is equivalent to another standard notion of hardness of learning: agnostic. We prove that PP≠BQPPP \neq BQPPP=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 PPTΣ3PPPT^{Σ_3^P}PPTΣ3P​ learners implies SampBQP≠SampBPPSampBQP \neq SampBPPSampBQP=SampBPP.
Founding Quantum Cryptography on Quantum Advantage, or, Towards Cryptography from #P\mathsf{\#P}#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\mathsf{P^{\#P}}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\mathsf{P^{\#P}} \not\subseteq \mathsf{(io)BQP/qpoly}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.
EFI pairs (Brakerski, Canetti, and Qian, ITCS 2023) and one-way puzzles (Khurana and Tomer, STOC 2024) are the leading candidates for the minimal assumption of quantum cryptography. The first are efficiently preparable quantum states, statistically far yet computationally indistinguishable; the second are classical puzzles, easy to sample and hard to solve. One-way puzzles imply EFI pairs, and whether the converse holds is open. We construct a single classical oracle relative to which one-way puzzles do not exist, even with an unbounded verifier, while an EFI pair survives every distinguisher that queries the oracle classically throughout and holds advice about it, making its one superposition query at the end. The oracle answers every question about the output probabilities of quantum samplers, which removes the puzzles, and hides a Haar-random half-dimensional subspace. To prove security we reduce it to communication complexity. An adversary whose knowledge of the subspace arrives as classical query answers can be simulated inside a two-party protocol against the party holding it, so it does no better than the best classical protocol for Vector-in-Subspace (Klartag and Regev, STOC 2011), whatever the oracle computes. That argument does not cover the superposition query, which we bound instead using tools from random matrix theory. The same attack gives a classical simulation of any quantum party in a classical-message protocol with no entanglement shared in advance, so relative to the oracle there is no proof of quantumness either. Quantum polynomial time therefore offers no advantage on any task with classical inputs and outputs, while the two quantum states stay indistinguishable. We state conjectures on removing the restriction on superposition queries.
In classical cryptography, one-way functions (OWFs) play a central role as the minimal primitive that (almost) all primitives imply. The situation is more complicated in quantum cryptography, in which honest parties and adversaries can use quantum computation and communication, and it is known that analogues of OWFs in the quantum setting might not be minimal. In this work we ask whether OWFs are minimal for the intermediate setting of post-quantum cryptography, in which the protocols are classical while they shall resist quantum adversaries. We show that for a wide range of natural settings, if a primitive Q implies OWFs, then so does its (uniformly or non-uniformly secure) post-quantum analogue. In particular, we show that if a primitive Q implies any other primitive P that has a 2-message security game (e.g., OWFs) through a black-box classical security reduction R, then one can always (efficiently) turn any polynomial-size quantum adversary breaking P into a polynomial-size quantum adversary breaking Q. Note that this result holds even if the implementation of P using that of Q is arbitrarily non-black-box. We also prove extensions of this result for when the reduction R anticipates its oracle adversary to be deterministic, whenever either of the following conditions hold: (1) the adversary needs to win the security game of Q only with non-negligible probability (e.g., Q is collision-resistant hashing) or (2) that either of P and Q have “falsifiable” security games (this is the case when P is OWFs). Our work leaves open answering our main question when Q implies OWFs through a non-black-box security reduction, or when P uses a more complicated security game than a two-message one.
One-way functions are central to classical cryptography. They are necessary for the existence of non-trivial classical cryptosystems, and also sufficient to realize meaningful primitives including commitments, pseudorandom generators and digital signatures. At the same time, a mounting body of evidence suggests that assumptions even weaker than one-way functions may suffice for many cryptographic tasks of interest in a quantum world, including bit commitments and secure multi-party computation. This work studies one-way state generators [Morimae-Yamakawa, CRYPTO 2022], a natural quantum relaxation of one-way functions. Given a secret key, a one-way state generator outputs a hard to invert quantum state. A fundamental question is whether this type of quantum one-wayness suffices to realize quantum cryptography. We obtain an affirmative answer to this question, by proving that one-way state generators with pure state outputs imply quantum bit commitments and secure multiparty computation. Along the way, we use efficient shadow tomography [Huang et. al., Nature Physics 2020] to build an intermediate primitive with classical outputs, which we call a (quantum) one-way puzzle. Our main technical contribution is a proof that one-way puzzles imply quantum bit commitments. This proof develops new techniques for pseudoentropy generation [Hastad et. al., SICOMP 1999] from arbitrary distributions, which may be of independent interest.