In the classical model of computation, it is well established that one-way functions (OWF) are minimal for computational cryptography: They are essential for almost any cryptographic application that cannot be realized with respect to computationally unbounded adversaries. In the quantum setting, however, OWFs appear not to be essential (Kretschmer 2021; Ananth et al., Morimae and Yamakawa 2022), and the question of whether such a minimal primitive exists remains open. We consider EFI pairs - efficiently samplable, statistically far but computationally indistinguishable pairs of (mixed) quantum states. Building on the work of Yan (2022), which shows equivalence between EFI pairs and statistical commitment schemes, we show that EFI pairs are necessary for a large class of quantum-cryptographic applications. Specifically, we construct EFI pairs from minimalistic versions of commitments schemes, oblivious transfer, and general secure multiparty computation, as well as from QCZK proofs from essentially any non-trivial language. We also construct quantum computational zero knowledge (QCZK) proofs for all of QIP from any EFI pair. This suggests that, for much of quantum cryptography, EFI pairs play a similar role to that played by OWFs in the classical setting: they are simple to describe, essential, and also serve as a linchpin for demonstrating equivalence between primitives.
In the classical model of computation, it is well established that one-way functions (OWF) are minimal for computational cryptography: They are essential for almost any cryptographic application that cannot be realized with respect to computationally unbounded adversaries. In the quantum setting, however, OWFs appear not to be essential (Kretschmer 2021; Ananth et al., Morimae and Yamakawa 2022), and the question of whether such a minimal primitive exists remains open. We consider EFI pairs - efficiently samplable, statistically far but computationally indistinguishable pairs of (mixed) quantum states. Building on the work of Yan (2022), which shows equivalence between EFI pairs and statistical commitment schemes, we show that EFI pairs are necessary for a large class of quantum-cryptographic applications. Specifically, we construct EFI pairs from minimalistic versions of commitments schemes, oblivious transfer, and general secure multiparty computation, as well as from QCZK proofs from essentially any non-trivial language. We also construct quantum computational zero knowledge (QCZK) proofs for all of QIP from any EFI pair. This suggests that, for much of quantum cryptography, EFI pairs play a similar role to that played by OWFs in the classical setting: they are simple to describe, essential, and also serve as a linchpin for demonstrating equivalence between primitives.
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.
While one-way functions (OWFs) serve as the minimal assumption for computational cryptography in the classical setting, in quantum cryptography, we have even weaker cryptographic assumptions such as pseudo-random states, and EFI pairs, among others. Moreover, the minimal assumption for computational quantum cryptography remains an open question. Recently, it has been shown that pseudoentanglement is necessary for the existence of quantum cryptography (Goulão and Elkouss 2024), but no cryptographic construction has been built from it. In this work, we study the cryptographic usefulness of quantum pseudoresources—a pair of families of quantum states that exhibit a gap in their resource content yet remain computationally indistinguishable. We show that quantum pseudoresources imply a variant of EFI pairs, which we call EPFI pairs, and that these are equivalent to quantum commitments and thus EFI pairs. Our results suggest that, just as randomness is fundamental to classical cryptography, quantum resources may play a similarly crucial role in the quantum setting. Finally, we focus on the specific case of entanglement, analyzing different definitions of pseudoentanglement and their implications for constructing EPFI pairs. Moreover, we propose a new cryptographic functionality that is intrinsically dependent on entanglement as a resource.
EFI⇔1PRS
A series of recent works has established EFI pairs as the "minimal" assumption for quantum cryptography. Recently, Cavalar et al. [CCC+26] showed that EFI pairs are equivalent to non-uniform expanding single-copy pseudorandom state generators (1PRS), and obtained a non-uniform meta-complexity characterization of EFI pairs. We remove the non-uniformity in their construction and prove that EFI pairs are equivalent to expanding 1PRS. As a corollary, we obtain the first uniform meta-complexity characterization of EFI pairs. Together with the equivalence of statistically verifiable one-way state generators and commitments proved by Batra and Jain [BJ24], this gives a quantum analogue of the classical equivalence between one-way functions, commitments, and pseudorandom generators.
Regarding minimal assumptions, most of classical cryptography is known to depend on the existence of One-Way Functions (OWFs). However, recent evidence has shown that this is not the case when considering quantum resources. Besides the well known unconditional security of Quantum Key Distribution, it is now known that computational cryptography may be built on weaker primitives than OWFs, e.g., pseudo-random states [JLS18], one-way state generators [MY23], or EFI pairs of states [BCQ23]. We consider a new quantum resource, pseudo-entanglement, and show that the existence of EFI pairs, one of the current main candidates for the weakest computational assumption for cryptography (necessary for commitments, oblivious transfer, secure multi-party computation, computational zero-knowledge proofs), implies the existence of pseudo-entanglement, as defined by [ABF+24, ABV23] under some reasonable adaptations. We prove this by constructing a new family of pseudo-entangled quantum states given only EFI pairs. Our result has important implications for the field of computational cryptography. It shows that if pseudo-entanglement does not exist, then most of cryptography cannot exist either. Moreover, it establishes pseudo-entanglement as a new minimal assumption for most of computational cryptography, which may pave the way for the unification of other assumptions into a single primitive. Finally, pseudo-entanglement connects physical phenomena and efficient computation, thus, our result strengthens the connection between cryptography and the physical world.
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].
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.