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.
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.
The existence of one-way functions is one of the most fundamental assumptions in classical cryptography. In the quantum world, on the other hand, there are evidences that some cryptographic primitives can exist even if one-way functions do not exist. We therefore have the following important open problem in quantum cryptography: What is the most fundamental element in quantum cryptography? In this direction, Brakerski, Canetti, and Qian recently defined a notion called EFI pairs, which are pairs of efficiently generatable states that are statistically distinguishable but computationally indistinguishable, and showed its equivalence with some cryptographic primitives including commitments, oblivious transfer, and general multi-party computations. However, their work focuses on decision-type primitives and does not cover search-type primitives like quantum money and digital signatures. In this paper, we study properties of one-way state generators (OWSGs), which are a quantum analogue of one-way functions. We first revisit the definition of OWSGs and generalize it by allowing mixed output states. Then we show the following results. (1) We define a weaker version of OWSGs, weak OWSGs, and show that they are equivalent to OWSGs. (2) Quantum digital signatures are equivalent to OWSGs. (3) Private-key quantum money schemes (with pure money states) imply OWSGs. (4) Quantum pseudo one-time pad schemes imply both OWSGs and EFI pairs. (5) We introduce an incomparable variant of OWSGs, which we call secretly-verifiable and statistically-invertible OWSGs, and show that they are equivalent to EFI pairs.
There is a large body of work studying what forms of computational hardness are needed to realize classical cryptography. In particular, one-way functions and pseudorandom generators can be built from each other, and thus require equivalent computational assumptions to be realized. Furthermore, the existence of either of these primitives implies that P ≠ N P , which gives a lower bound on the necessary hardness.One can also define versions of each of these primitives with quantum output: respectively one-way state generators and pseudorandom state generators. Unlike in the classical setting, it is not known whether either primitive can be built from the other. Although it has been shown that pseudorandom state generators for certain parameter regimes can be used to build one-way state generators, the implication has not been previously known in full generality. Furthermore, to the best of our knowledge, the existence of one-way state generators has no known implications in complexity theory.We show that pseudorandom states compressing n bits to log n + 1 qubits can be used to build one-way state generators and pseudorandom states compressing n bits to ω ( log n ) qubits are one-way state generators. This is a nearly optimal result since pseudorandom states with fewer than c log n -qubit output can be shown to exist unconditionally. We also show that any one-way state generator can be broken by a quantum algorithm with classical access to a P P oracle.An interesting implication of our results is that a t ( n ) -copy one-way state generator exists unconditionally, for every t ( n ) = o ( n / log n ) . This contrasts nicely with the previously known fact that O ( n ) -copy one-way state generators require computational hardness. We also outline a new route towards a black-box separation between one-way state generators and quantum bit commitments.
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.
Quantum cryptographic primitives beyond key distribution remain a less well understood area of research. In classical cryptography, one-way functions underpin nearly all standard cryptographic protocols, motivating the search for meaningful quantum analogues and for a clear understanding of the physical and computational mechanisms that could enforce one-wayness. In this article, we review quantum one-way functions and a range of closely related quantum-state primitives, including one-way state generators, pseudorandom quantum states, and efficiently indistinguishable pairs of states. We discuss both computational and information-theoretic notions of quantum one-wayness, emphasizing the different adversarial models and security assumptions that underlie these constructions. We compare and contrast the various proposed primitives, and clarify their conceptual relationships. Particular emphasis is placed on questions of physical realizability, experimental feasibility, and robustness to noise. Finally, we outline open problems and future directions toward the development of practical quantum cryptographic primitives beyond key distribution, and the emergence of a broader quantum-cryptographic ecosystem.
Classically, it is well-known that several fundamental cryptographic primitives, including one-way functions, pseudorandom number generators, commitments, and signatures, characterize the same cryptographic world, which is known as "Minicrypt". In this paper, we investigate to what extent this picture persists in the quantum-computation classical-communication (QCCC) setting, where parties may perform quantum local computation, but communicate through only classical messages.
We demonstrate that the classical Minicrypt landscape breaks down in the QCCC model by oracle separations. We construct two oracle worlds where efficiently verifiable one-way puzzles (EV-OWPuzz) exist. By a known equivalence, QCCC one-time signatures also exist in both worlds. In the first, one-way functions do not exist, even if we allow quantum pseudodeterministic evaluation. In the second, QCCC bit commitments do not exist. Thus, the classical Minicrypt reductions from signatures to one-way functions and bit commitments have no fully black-box counterparts in the QCCC setting. Our proofs develop techniques for analyzing the random phase states, including a concentration theorem and an LOCC decoupling theorem.
As a separate result, we establish quantum sum-binding for the NOVY protocol, settling the question of its post-quantum security. We also generalize it to a fully black-box construction of a QCCC commitment from EV-OWPuzz with nearly uniform puzzle distribution, complementing our oracle separation results.
A key challenge in quantum cryptography is to build quantum one-wayness and pseudorandomness without the use of (quantum computable) one-way functions. So far, this has turned out to be a difficult task, with only a few proposed candidates that are not directly built from one-way functions. Even within these few proposed candidates, we have lacked the techniques to analyze when proposed constructions may inadvertently yield one-way functions.
In this work, we introduce efficient NP-aided shadow tomography of quantum states. We prove that collections of {\em computable} pure states can be learned via efficient NP-aided shadow tomography, where we say that a state is computable if the amplitude and phase on any computational basis term can be classically efficiently computed given the description of an efficient preparation circuit for the state. We also give new algorithms for NP-aided learning of unitaries given polynomially many queries to the unitary.
By building on this, we show that many existing architectures for PRS and PRU, including some that were explicitly introduced for the purposes of avoiding one-way functions (e.g., Hamiltonian Phase States, Bostanci et. al., TQC 2025), actually do imply the existence of one-way functions or imply \(\NP\) hardness. We hope that these no-go results will inform future investigations into building PRS and PRUs from assumptions that are plausibly outside the complexity class NP.