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Quantum information allows us to build quantum money schemes, where a bank can issue banknotes in the form of authenticatable quantum states that cannot be cloned or counterfeited. Similar to paper banknotes, in existing quantum money schemes, a banknote consists of an unclonable quantum state and a classical serial number, signed by bank. Thus, they lack one of the most fundamental properties cryptographers look for in a currency scheme: privacy. In this work, we first further develop the formal definitions of privacy for quantum money schemes. Then, we construct the first public-key quantum money schemes that satisfy these security notions. Namely, - Assuming existence of indistinguishability obfuscation (iO) and hardness of Learning with Errors (LWE), we construct a public-key quantum money scheme with anonymity against users and traceability by authorities. Since it is a policy choice whether authorities should be able to track banknotes or not, we also construct an untraceable money scheme from the same cryptographic assumptions, where no one (not even the authorities) can track banknotes. Further, we show that the no-cloning principle, a result of quantum mechanics, allows us to construct schemes, with security guarantees that are classically impossible, for a seemingly unrelated application: voting! - Assuming iO and LWE, we construct a universally verifiable quantum voting scheme with classical votes. Finally, as a technical tool, we introduce the notion of publicly rerandomizable encryption with strong correctness, where no adversary is able to produce a malicious ciphertext and a malicious randomness such that the ciphertext before and after rerandomization decrypts to different values! We believe this might be of independent interest. - Assuming LWE, we construct a (post-quantum) classical publicly rerandomizable encryption scheme with strong correctness.

We initiate the study of untelegraphable encryption (UTE), founded on the no-telegraphing principle, which allows an encryptor to encrypt a message such that a binary string representation of the ciphertext cannot be decrypted by a user with the secret key, a task that is classically impossible. This is a natural relaxation of unclonable encryption, inspired by the recent work of Nehoran and Zhandry (ITCS 2024), who showed a computational separation between the no-cloning and no-telegraphing principles. In this work, we define and construct UTE information-theoretically in the plain model. Building off this, we give several applications of UTE and study the interplay of UTE with UE and well-studied tasks in quantum state learning, yielding the following contributions: - A construction of collusion-resistant UTE from standard secret-key encryption. We additionally show that hyper-efficient shadow tomography (HEST) is impossible assuming collusion-resistant UTE exists. By considering a relaxation of collusion-resistant UTE, we are able to show the impossibility of HEST assuming only pseudorandom state generators (which may not imply one-way functions). This almost completely answers an open inquiry of Aaronson (STOC 2019). - A construction of UTE from a one-shot message authentication code in the classical oracle model, such that there is an explicit attack that breaks UE security for an unbounded polynomial number of decryptors. - A construction of everlasting secure collusion-resistant UTE, where the decryptor adversary can run in unbounded time, in the quantum random oracle model (QROM), and formal evidence that a construction in the plain model is a challenging task. We additionally show that HEST with unbounded post-processing time is impossible in the QROM. - Constructions (and definitions) of untelegraphable secret sharing and untelegraphable functional encryption., Comment: 55 pages

We construct a quantum oracle relative to which BQP = QCMA but quantum-computation-classical-communication (QCCC) key exchange, QCCC commitments, and two-round quantum key distribution exist. We also construct an oracle relative to which BQP = QMA, but quantum lightning (a stronger variant of quantum money) exists. This extends previous work by Kretschmer [Kretschmer, TQC22], which showed that there is a quantum oracle relative to which BQP = QMA but pseudorandom state generators (a quantum variant of pseudorandom generators) exist. We also show that QCCC key exchange, QCCC commitments, and two-round quantum key distribution can all be used to build one-way puzzles. One-way puzzles are a version of "quantum samplable" one-wayness and are an intermediate primitive between pseudorandom state generators and EFI pairs, the minimal quantum primitive. In particular, one-way puzzles cannot exist if BQP = PP. Our results together imply that aside from pseudorandom state generators, there is a large class of quantum cryptographic primitives which can exist even if BQP = QCMA, but are broken if BQP = PP. Furthermore, one-way puzzles are a minimal primitive for this class. We denote this class "CountCrypt"., Comment: 50 pages, 1 figure

While in classical cryptography, one-way functions (OWFs) are widely regarded as the "minimal assumption," the situation in quantum cryptography is less clear. Recent works have put forward two concurrent candidates for the minimal assumption in quantum cryptography: One-way state generators (OWSGs), postulating the existence of a hard search problem with an efficient verification algorithm, and EFI pairs, postulating the existence of a hard distinguishing problem. Two recent papers [Khurana and Tomer STOC'24; Batra and Jain FOCS'24] showed that OWSGs imply EFI pairs, but the reverse direction remained open. In this work, we give strong evidence that the opposite direction does not hold: We show that there is a quantum unitary oracle relative to which EFI pairs exist, but OWSGs do not. In fact, we show a slightly stronger statement that holds also for EFI pairs that output classical bits (QEFID). As a consequence, we separate, via our oracle, QEFID, and one-way puzzles from OWSGs and several other Microcrypt primitives, including efficiently verifiable one-way puzzles and unclonable state generators. In particular, this solves a problem left open in [Chung, Goldin, and Gray Crypto'24]. Using similar techniques, we also establish a fully black-box separation (which is slightly weaker than an oracle separation) between private-key quantum money schemes and QEFID pairs. One conceptual implication of our work is that the existence of an efficient verification algorithm may lead to qualitatively stronger primitives in quantum cryptography., Comment: Minor corrections and typos were addressed in version 2

Secure key leasing (a.k.a. key-revocable cryptography) enables us to lease a cryptographic key as a quantum state in such a way that the key can be later revoked in a verifiable manner. We propose a simple framework for constructing cryptographic primitives with secure key leasing via the certified deletion property of BB84 states. Based on our framework, we obtain the following schemes. - A public key encryption scheme with secure key leasing that has classical revocation based on any IND-CPA secure public key encryption scheme. Prior works rely on either quantum revocation or stronger assumptions such as the quantum hardness of the learning with errors (LWE) problem. - A pseudorandom function with secure key leasing that has classical revocation based on one-way functions. Prior works rely on stronger assumptions such as the quantum hardness of the LWE problem. - A digital signature scheme with secure key leasing that has classical revocation based on the quantum hardness of the short integer solution (SIS) problem. Our construction has static signing keys, i.e., the state of a signing key almost does not change before and after signing. Prior constructions either rely on non-static signing keys or indistinguishability obfuscation to achieve a stronger goal of copy-protection. In addition, all of our schemes remain secure even if a verification key for revocation is leaked after the adversary submits a valid certificate of deletion. To our knowledge, all prior constructions are totally broken in this setting. Moreover, in our view, our security proofs are much simpler than those for existing schemes., Comment: 55 pages

Quantum computational advantage refers to an existence of computational tasks that are easy for quantum computing but hard for classical one. Unconditionally showing quantum advantage is beyond our current understanding of complexity theory, and therefore some computational assumptions are needed. Which complexity assumption is necessary and sufficient for quantum advantage? In this paper, we show that inefficient-verifier proofs of quantumness (IV-PoQ) exist if and only if classically-secure one-way puzzles (OWPuzzs) exist. As far as we know, this is the first time that a complete cryptographic characterization of quantum advantage is obtained. IV-PoQ capture various types of quantum advantage previously studied, such as sampling-based quantum advantage and searching-based one. Previous work [Morimae and Yamakawa, Crypto 2024] showed that IV-PoQ can be constructed from OWFs, but a construction of IV-PoQ from weaker assumptions was left open. Our result solves the open problem. OWPuzzs are one of the most fundamental quantum cryptographic primitives implied by many quantum cryptographic primitives weaker than one-way functions (OWFs). The equivalence between IV-PoQ and classically-secure OWPuzzs therefore highlights that if there is no quantum advantage, then these fundamental primitives do not exist. The equivalence also means that quantum advantage is an example of the applications of OWPuzzs. Except for commitments, no application of OWPuzzs was known before. Our result shows that quantum advantage is another application of OWPuzzs, which solves the open question of [Chung, Goldin, and Gray, Crypto 2024]. Moreover, it is the first quantum-computation-classical-communication (QCCC) application of OWPuzzs., Comment: 58 pages, 2 figures

Unpredictable functions (UPFs) play essential roles in classical cryptography, including message authentication codes (MACs) and digital signatures. In this paper, we introduce a quantum analog of UPFs, which we call unpredictable state generators (UPSGs). UPSGs are implied by pseudorandom function-like states generators (PRFSs), which are a quantum analog of pseudorandom functions (PRFs), and therefore UPSGs could exist even if one-way functions do not exist, similar to other recently introduced primitives like pseudorandom state generators (PRSGs), one-way state generators (OWSGs), and EFIs. In classical cryptography, UPFs are equivalent to PRFs, but in the quantum case, the equivalence is not clear, and UPSGs could be weaker than PRFSs. Despite this, we demonstrate that all known applications of PRFSs are also achievable with UPSGs. They include IND-CPA-secure secret-key encryption and EUF-CMA-secure MACs with unclonable tags. Our findings suggest that, for many applications, quantum unpredictability, rather than quantum pseudorandomness, is sufficient., Comment: 38 pages, 1 figure

In classical cryptography, one-way functions are widely considered to be the minimal computational assumption. However, when taking quantum information into account, the situation is more nuanced. There are currently two major candidates for the minimal assumption: the search quantum generalization of one-way functions are one-way state generators (OWSG), whereas the decisional variant are EFI pairs. A well-known open problem in quantum cryptography is to understand how these two primitives are related. A recent breakthrough result of Khurana and Tomer (STOC'24) shows that OWSGs imply EFI pairs, for the restricted case of pure states. In this work, we make progress towards understanding the general case. To this end, we define the notion of inefficiently-verifiable one-way state generators (IV-OWSGs), where the verification algorithm is not required to be efficient, and show that these are precisely equivalent to EFI pairs, with an exponential loss in the reduction. Significantly, this equivalence holds also for mixed states. Thus our work establishes the following relations among these fundamental primitives of quantum cryptography: (mixed) OWSGs => (mixed) IV-OWSGs $\equiv_{\rm exp}$ EFI pairs, where $\equiv_{\rm exp}$ denotes equivalence up to exponential security of the primitives., Comment: 15 pages

We study the output length of one-way state generators (OWSGs), their weaker variants, and EFIs. - Standard OWSGs. Recently, Cavalar et al. (arXiv:2312.08363) give OWSGs with $m$-qubit outputs for any $m=\omega(\log \lambda)$, where $\lambda$ is the security parameter, and conjecture that there do not exist OWSGs with $O(\log \log \lambda)$-qubit outputs. We prove their conjecture in a stronger manner by showing that there do not exist OWSGs with $O(\log \lambda)$-qubit outputs. This means that their construction is optimal in terms of output length. - Inverse-polynomial-advantage OWSGs. Let $\epsilon$-OWSGs be a parameterized variant of OWSGs where a quantum polynomial-time adversary's advantage is at most $\epsilon$. For any constant $c\in \mathbb{N}$, we construct $\lambda^{-c}$-OWSGs with $((c+1)\log \lambda+O(1))$-qubit outputs assuming the existence of OWFs. We show that this is almost tight by proving that there do not exist $\lambda^{-c}$-OWSGs with at most $(c\log \lambda-2)$-qubit outputs. - Constant-advantage OWSGs. For any constant $\epsilon>0$, we construct $\epsilon$-OWSGs with $O(\log \log \lambda)$-qubit outputs assuming the existence of subexponentially secure OWFs. We show that this is almost tight by proving that there do not exist $O(1)$-OWSGs with $((\log \log \lambda)/2+O(1))$-qubit outputs. - Weak OWSGs. We refer to $(1-1/\mathsf{poly}(\lambda))$-OWSGs as weak OWSGs. We construct weak OWSGs with $m$-qubit outputs for any $m=\omega(1)$ assuming the existence of exponentially secure OWFs with linear expansion. We show that this is tight by proving that there do not exist weak OWSGs with $O(1)$-qubit outputs. - EFIs. We show that there do not exist $O(\log \lambda)$-qubit EFIs. We show that this is tight by proving that there exist $\omega(\log \lambda)$-qubit EFIs assuming the existence of exponentially secure PRGs., Comment: 28 pages

We study digital signatures with revocation capabilities and show two results. First, we define and construct digital signatures with revocable signing keys from the LWE assumption. In this primitive, the signing key is a quantum state which enables a user to sign many messages and yet, the quantum key is also revocable, i.e., it can be collapsed into a classical certificate which can later be verified. Once the key is successfully revoked, we require that the initial recipient of the key loses the ability to sign. We construct digital signatures with revocable signing keys from a newly introduced primitive which we call two-tier one-shot signatures, which may be of independent interest. This is a variant of one-shot signatures, where the verification of a signature for the message ``0'' is done publicly, whereas the verification for the message ``1'' is done in private. We give a construction of two-tier one-shot signatures from the LWE assumption. As a complementary result, we also construct digital signatures with quantum revocation from group actions, where the quantum signing key is simply ``returned'' and then verified as part of revocation. Second, we define and construct digital signatures with revocable signatures from OWFs. In this primitive, the signer can produce quantum signatures which can later be revoked. Here, the security property requires that, once revocation is successful, the initial recipient of the signature loses the ability to find accepting inputs to the signature verification algorithm. We construct this primitive using a newly introduced two-tier variant of tokenized signatures. For the construction, we show a new lemma which we call the adaptive hardcore bit property for OWFs, which may enable further applications., Comment: 46 pages

We show the following unconditional results on quantum commitments in two related yet different models: 1. We revisit the notion of quantum auxiliary-input commitments introduced by Chailloux, Kerenidis, and Rosgen (Comput. Complex. 2016) where both the committer and receiver take the same quantum state, which is determined by the security parameter, as quantum auxiliary inputs. We show that computationally-hiding and statistically-binding quantum auxiliary-input commitments exist unconditionally, i.e., without relying on any unproven assumption, while Chailloux et al. assumed a complexity-theoretic assumption, ${\bf QIP}\not\subseteq{\bf QMA}$. On the other hand, we observe that achieving both statistical hiding and statistical binding at the same time is impossible even in the quantum auxiliary-input setting. To the best of our knowledge, this is the first example of unconditionally proving computational security of any form of (classical or quantum) commitments for which statistical security is impossible. As intermediate steps toward our construction, we introduce and unconditionally construct post-quantum sparse pseudorandom distributions and quantum auxiliary-input EFI pairs which may be of independent interest. 2. We introduce a new model which we call the common reference quantum state (CRQS) model where both the committer and receiver take the same quantum state that is randomly sampled by an efficient setup algorithm. We unconditionally prove that there exist statistically hiding and statistically binding commitments in the CRQS model, circumventing the impossibility in the plain model. We also discuss their applications to zero-knowledge proofs, oblivious transfers, and multi-party computations., Comment: 42 pages

A robust combiner combines many candidates for a cryptographic primitive and generates a new candidate for the same primitive. Its correctness and security hold as long as one of the original candidates satisfies correctness and security. A universal construction is a closely related notion to a robust combiner. A universal construction for a primitive is an explicit construction of the primitive that is correct and secure as long as the primitive exists. It is known that a universal construction for a primitive can be constructed from a robust combiner for the primitive in many cases. Although robust combiners and universal constructions for classical cryptography are widely studied, robust combiners and universal constructions for quantum cryptography have not been explored so far. In this work, we define robust combiners and universal constructions for several quantum cryptographic primitives including one-way state generators, public-key quantum money, quantum bit commitments, and unclonable encryption, and provide constructions of them. On a different note, it was an open problem how to expand the plaintext length of unclonable encryption. In one of our universal constructions for unclonable encryption, we can expand the plaintext length, which resolves the open problem., Comment: 55 pages

This paper studies the quantum computational complexity of the discrete logarithm (DL) and related group-theoretic problems in the context of generic algorithms -- that is, algorithms that do not exploit any properties of the group encoding. We establish a generic model of quantum computation for group-theoretic problems, which we call the quantum generic group model. Shor's algorithm for the DL problem and related algorithms can be described in this model. We show the quantum complexity lower bounds and almost matching algorithms of the DL and related problems in this model. More precisely, we prove the following results for a cyclic group $G$ of prime order. - Any generic quantum DL algorithm must make $\Omega(\log |G|)$ depth of group operations. This shows that Shor's algorithm is asymptotically optimal among the generic quantum algorithms, even considering parallel algorithms. - We observe that variations of Shor's algorithm can take advantage of classical computations to reduce the number of quantum group operations. We introduce a model for generic hybrid quantum-classical algorithms and show that these algorithms are almost optimal in this model. Any generic hybrid algorithm for the DL problem with a total number of group operations $Q$ must make $\Omega(\log |G|/\log Q)$ quantum group operations of depth $\Omega(\log\log |G| - \log\log Q)$. - When the quantum memory can only store $t$ group elements and use quantum random access memory of $r$ group elements, any generic hybrid algorithm must make either $\Omega(\sqrt{|G|})$ group operations in total or $\Omega(\log |G|/\log (tr))$ quantum group operations. As a side contribution, we show a multiple DL problem admits a better algorithm than solving each instance one by one, refuting a strong form of the quantum annoying property suggested in the context of password-authenticated key exchange protocol.

We present a general compiler to add the publicly verifiable deletion property for various cryptographic primitives including public key encryption, attribute-based encryption, and quantum fully homomorphic encryption. Our compiler only uses one-way functions, or more generally hard quantum planted problems for NP, which are implied by one-way functions. It relies on minimal assumptions and enables us to add the publicly verifiable deletion property with no additional assumption for the above primitives. Previously, such a compiler needs additional assumptions such as injective trapdoor one-way functions or pseudorandom group actions [Bartusek-Khurana-Poremba, ePrint:2023/370]. Technically, we upgrade an existing compiler for privately verifiable deletion [Bartusek-Khurana, ePrint:2022/1178] to achieve publicly verifiable deletion by using digital signatures., Comment: 15 pages

We construct quantum public-key encryption from one-way functions. In our construction, public keys are quantum, but ciphertexts are classical. Quantum public-key encryption from one-way functions (or weaker primitives such as pseudorandom function-like states) are also proposed in some recent works [Morimae-Yamakawa, eprint:2022/1336; Coladangelo, eprint:2023/282; Barooti-Grilo-Malavolta-Sattath-Vu-Walter, eprint:2023/877]. However, they have a huge drawback: they are secure only when quantum public keys can be transmitted to the sender (who runs the encryption algorithm) without being tampered with by the adversary, which seems to require unsatisfactory physical setup assumptions such as secure quantum channels. Our construction is free from such a drawback: it guarantees the secrecy of the encrypted messages even if we assume only unauthenticated quantum channels. Thus, the encryption is done with adversarially tampered quantum public keys. Our construction is the first quantum public-key encryption that achieves the goal of classical public-key encryption, namely, to establish secure communication over insecure channels, based only on one-way functions. Moreover, we show a generic compiler to upgrade security against chosen plaintext attacks (CPA security) into security against chosen ciphertext attacks (CCA security) only using one-way functions. As a result, we obtain CCA secure quantum public-key encryption based only on one-way functions., Comment: 47pages

It is a long-standing open question to construct a classical oracle relative to which BQP/qpoly $\neq$ BQP/poly or QMA $\neq$ QCMA. In this paper, we construct classically-accessible classical oracles relative to which BQP/qpoly $\neq$ BQP/poly and QMA $\neq$ QCMA. Here, classically-accessible classical oracles are oracles that can be accessed only classically even for quantum algorithms. Based on a similar technique, we also show an alternative proof for the separation of QMA and QCMA relative to a distributional quantumly-accessible classical oracle, which was recently shown by Natarajan and Nirkhe., Comment: 31 pages. Added classically-accessible classical oracle separation of QMA and QCMA and updated the abstract. v4: Fixed an issue with the proof of Claim 5.2

We introduce the notion of public key encryption with secure key leasing (PKE-SKL). Our notion supports the leasing of decryption keys so that a leased key achieves the decryption functionality but comes with the guarantee that if the quantum decryption key returned by a user passes a validity test, then the user has lost the ability to decrypt. Our notion is similar in spirit to the notion of secure software leasing (SSL) introduced by Ananth and La Placa (Eurocrypt 2021) but captures significantly more general adversarial strategies. In more detail, our adversary is not restricted to use an honest evaluation algorithm to run pirated software. Our results can be summarized as follows: 1. Definitions: We introduce the definition of PKE with secure key leasing and formalize security notions. 2. Constructing PKE with Secure Key Leasing: We provide a construction of PKE-SKL by leveraging a PKE scheme that satisfies a new security notion that we call consistent or inconsistent security against key leasing attacks (CoIC-KLA security). We then construct a CoIC-KLA secure PKE scheme using 1-key Ciphertext-Policy Functional Encryption (CPFE) that in turn can be based on any IND-CPA secure PKE scheme. 3. Identity Based Encryption, Attribute Based Encryption and Functional Encryption with Secure Key Leasing: We provide definitions of secure key leasing in the context of advanced encryption schemes such as identity based encryption (IBE), attribute-based encryption (ABE) and functional encryption (FE). Then we provide constructions by combining the above PKE-SKL with standard IBE, ABE and FE schemes., Comment: 68 pages, 4 figures. added related works and a comparison with a concurrent work (2023-04-07)

We show how to obfuscate pseudo-deterministic quantum circuits in the classical oracle model, assuming the quantum hardness of learning with errors. Given the classical description of a quantum circuit $Q$, our obfuscator outputs a quantum state $\ket{\widetilde{Q}}$ that can be used to evaluate $Q$ repeatedly on arbitrary inputs. Instantiating the classical oracle using any candidate post-quantum indistinguishability obfuscator gives us the first candidate construction of indistinguishability obfuscation for all polynomial-size pseudo-deterministic quantum circuits. In particular, our scheme is the first candidate obfuscator for a class of circuits that is powerful enough to implement Shor's algorithm (SICOMP 1997). Our approach follows Bartusek and Malavolta (ITCS 2022), who obfuscate \emph{null} quantum circuits by obfuscating the verifier of an appropriate classical verification of quantum computation (CVQC) scheme. We go beyond null circuits by constructing a publicly-verifiable CVQC scheme for quantum \emph{partitioning} circuits, which can be used to verify the evaluation procedure of Mahadev's quantum fully-homomorphic encryption scheme (FOCS 2018). We achieve this by upgrading the one-time secure scheme of Bartusek (TCC 2021) to a fully reusable scheme, via a publicly-decodable \emph{Pauli functional commitment}, which we formally define and construct in this work. This commitment scheme, which satisfies a notion of binding against committers that can access the receiver's standard and Hadamard basis decoding functionalities, is constructed by building on techniques of Amos, Georgiou, Kiayias, and Zhandry (STOC 2020) introduced in the context of equivocal but collision-resistant hash functions.

We study certified everlasting secure functional encryption (FE) and many other cryptographic primitives in this work. Certified everlasting security roughly means the following. A receiver possessing a quantum cryptographic object can issue a certificate showing that the receiver has deleted the cryptographic object and information included in the object was lost. If the certificate is valid, the security is guaranteed even if the receiver becomes computationally unbounded after the deletion. Many cryptographic primitives are known to be impossible (or unlikely) to have information-theoretical security even in the quantum world. Hence, certified everlasting security is a nice compromise (intrinsic to quantum). In this work, we define certified everlasting secure versions of FE, compute-and-compare obfuscation, predicate encryption (PE), secret-key encryption (SKE), public-key encryption (PKE), receiver non-committing encryption (RNCE), and garbled circuits. We also present the following constructions: - Adaptively certified everlasting secure collusion-resistant public-key FE for all polynomial-size circuits from indistinguishability obfuscation and one-way functions. - Adaptively certified everlasting secure bounded collusion-resistant public-key FE for NC1 circuits from standard PKE. - Certified everlasting secure compute-and-compare obfuscation from standard fully homomorphic encryption and standard compute-and-compare obfuscation - Adaptively (resp., selectively) certified everlasting secure PE from standard adaptively (resp., selectively) secure attribute-based encryption and certified everlasting secure compute-and-compare obfuscation. - Certified everlasting secure SKE and PKE from standard SKE and PKE, respectively. - Certified everlasting secure RNCE from standard PKE. - Certified everlasting secure garbled circuits from standard SKE., Comment: This is a major update version of arXiv:2207.13878 with many new results

We demonstrate quantum advantage with several basic assumptions, specifically based on only the existence of OWFs. We introduce inefficient-verifier proofs of quantumness (IV-PoQ), and construct it from classical bit commitments. IV-PoQ is an interactive protocol between a verifier and a quantum prover consisting of two phases. In the first phase, the verifier is probabilistic polynomial-time, and it interacts with the prover. In the second phase, the verifier becomes inefficient, and makes its decision based on the transcript of the first phase. If the prover is honest, the inefficient verifier accepts with high probability, but any classical malicious prover only has a small probability of being accepted by the inefficient verifier. Our construction demonstrates the following results: (1)If one-way functions exist, then IV-PoQ exist. (2)If distributional collision-resistant hash functions exist (which exist if hard-on-average problems in $\mathbf{SZK}$ exist), then constant-round IV-PoQ exist. We also demonstrate quantum advantage based on worst-case-hard assumptions. We define auxiliary-input IV-PoQ (AI-IV-PoQ) that only require that for any malicious prover, there exist infinitely many auxiliary inputs under which the prover cannot cheat. We construct AI-IV-PoQ from an auxiliary-input version of commitments in a similar way, showing that (1)If auxiliary-input one-way functions exist (which exist if $\mathbf{CZK}\not\subseteq\mathbf{BPP}$), then AI-IV-PoQ exist. (2)If auxiliary-input collision-resistant hash functions exist (which is equivalent to $\mathbf{PWPP}\nsubseteq \mathbf{FBPP}$) or $\mathbf{SZK}\nsubseteq \mathbf{BPP}$, then constant-round AI-IV-PoQ exist., Comment: 52pages

Recently, Aaronson et al. (arXiv:2009.07450) showed that detecting interference between two orthogonal states is as hard as swapping these states. While their original motivation was from quantum gravity, we show its applications in quantum cryptography. 1. We construct the first public key encryption scheme from cryptographic \emph{non-abelian} group actions. Interestingly, the ciphertexts of our scheme are quantum even if messages are classical. This resolves an open question posed by Ji et al. (TCC '19). We construct the scheme through a new abstraction called swap-trapdoor function pairs, which may be of independent interest. 2. We give a simple and efficient compiler that converts the flavor of quantum bit commitments. More precisely, for any prefix X,Y $\in$ {computationally,statistically,perfectly}, if the base scheme is X-hiding and Y-binding, then the resulting scheme is Y-hiding and X-binding. Our compiler calls the base scheme only once. Previously, all known compilers call the base schemes polynomially many times (Cr\'epeau et al., Eurocrypt '01 and Yan, Asiacrypt '22). For the security proof of the conversion, we generalize the result of Aaronson et al. by considering quantum auxiliary inputs., Comment: 51 pages

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., Comment: 50 pages, 1 figure

Assume that Alice can do only classical probabilistic polynomial-time computing while Bob can do quantum polynomial-time computing. Alice and Bob communicate over only classical channels, and finally Bob gets a state $|x_0\rangle+|x_1\rangle$ with some bit strings $x_0$ and $x_1$. Is it possible that Alice can know $\{x_0,x_1\}$ but Bob cannot? Such a task, called {\it remote state preparations}, is indeed possible under some complexity assumptions, and is bases of many quantum cryptographic primitives such as proofs of quantumness, (classical-client) blind quantum computing, (classical) verifications of quantum computing, and quantum money. A typical technique to realize remote state preparations is to use 2-to-1 trapdoor collision resistant hash functions: Alice sends a 2-to-1 trapdoor collision resistant hash function $f$ to Bob, and Bob evaluates it on superposition and measures the image. Bob's post-measurement state is $|x_0\rangle+|x_1\rangle$, where $f(x_0)=f(x_1)=y$. With the trapdoor, Alice can learn $\{x_0,x_1\}$, but due to the collision resistance, Bob cannot. This Alice's advantage can be leveraged to realize the quantum cryptographic primitives listed above. It seems that the collision resistance is essential here. In this paper, surprisingly, we show that the collision resistance is not necessary for a restricted case: we show that (non-verifiable) remote state preparations of $|x_0\rangle+|x_1\rangle$ secure against {\it classical} probabilistic polynomial-time Bob can be constructed from classically-secure (full-domain) trapdoor permutations. Trapdoor permutations are not likely to imply the collision resistance, because black-box reductions from collision-resistant hash functions to trapdoor permutations are known to be impossible. As an application of our result, we construct proofs of quantumness from classically-secure (full-domain) trapdoor permutations., Comment: 20 pages

Computational security in cryptography has a risk that computational assumptions underlying the security are broken in the future. One solution is to construct information-theoretically-secure protocols, but many cryptographic primitives are known to be impossible (or unlikely) to have information-theoretical security even in the quantum world. A nice compromise (intrinsic to quantum) is certified everlasting security, which roughly means the following. A receiver with possession of quantum encrypted data can issue a certificate that shows that the receiver has deleted the encrypted data. If the certificate is valid, the security is guaranteed even if the receiver becomes computationally unbounded. Although several cryptographic primitives, such as commitments and zero-knowledge, have been made certified everlasting secure, there are many other important primitives that are not known to be certified everlasting secure. In this paper, we introduce certified everlasting FE. In this primitive, the receiver with the ciphertext of a message m and the functional decryption key of a function f can obtain f(m) and nothing else. The security holds even if the adversary becomes computationally unbounded after issuing a valid certificate. We, first, construct certified everlasting FE for P/poly circuits where only a single key query is allowed for the adversary. We, then, extend it to q-bounded one for NC1 circuits where q-bounded means that q key queries are allowed for the adversary with an a priori bounded polynomial q. For the construction of certified everlasting FE, we introduce and construct certified everlasting versions of secret-key encryption, public-key encryption, receiver non-committing encryption, and a garbling scheme, which are of independent interest., Comment: 57 pages

We provide the first $\mathit{constant}$-$\mathit{round}$ construction of post-quantum non-malleable commitments under the minimal assumption that $\mathit{post}$-$\mathit{quantum}$ $\mathit{one}$-$\mathit{way}$ $\mathit{functions}$ exist. We achieve the standard notion of non-malleability with respect to commitments. Prior constructions required $\Omega(\log^*\lambda)$ rounds under the same assumption. We achieve our results through a new technique for constant-round non-malleable commitments which is easier to use in the post-quantum setting. The technique also yields an almost elementary proof of security for constant-round non-malleable commitments in the classical setting, which may be of independent interest. When combined with existing work, our results yield the first constant-round quantum-secure multiparty computation for both classical and quantum functionalities $\mathit{in}$ $\mathit{the}$ $\mathit{plain}$ $\mathit{model}$, under the $\mathit{polynomial}$ hardness of quantum fully-homomorphic encryption and quantum learning with errors.

We show the following hold, unconditionally unless otherwise stated, relative to a random oracle: - There are NP search problems solvable by quantum polynomial-time machines but not classical probabilistic polynomial-time machines. - There exist functions that are one-way, and even collision resistant, against classical adversaries but are easily inverted quantumly. Similar separations hold for digital signatures and CPA-secure public key encryption (the latter requiring the assumption of a classically CPA-secure encryption scheme). Interestingly, the separation does not necessarily extend to the case of other cryptographic objects such as PRGs. - There are unconditional publicly verifiable proofs of quantumness with the minimal rounds of interaction: for uniform adversaries, the proofs are non-interactive, whereas for non-uniform adversaries the proofs are two message public coin. - Our results do not appear to contradict the Aaronson-Ambanis conjecture. Assuming this conjecture, there exist publicly verifiable certifiable randomness, again with the minimal rounds of interaction. By replacing the random oracle with a concrete cryptographic hash function such as SHA2, we obtain plausible Minicrypt instantiations of the above results. Previous analogous results all required substantial structure, either in terms of highly structured oracles and/or algebraic assumptions in Cryptomania and beyond., Comment: 56 pages, fixed the proof of Theorem 3.11 etc

In the classical world, the existence of commitments is equivalent to the existence of one-way functions. In the quantum setting, on the other hand, commitments are not known to imply one-way functions, but all known constructions of quantum commitments use at least one-way functions. Are one-way functions really necessary for commitments in the quantum world? In this work, we show that non-interactive quantum commitments (for classical messages) with computational hiding and statistical binding exist if pseudorandom quantum states exist. Pseudorandom quantum states are sets of quantum states that are efficiently generated but their polynomially many copies are computationally indistinguishable from the same number of copies of Haar random states [Ji, Liu, and Song, CRYPTO 2018]. It is known that pseudorandom quantum states exist even if $\BQP=\QMA$ (relative to a quantum oracle) [Kretschmer, TQC 2021], which means that pseudorandom quantum states can exist even if no quantum-secure classical cryptographic primitive exists. Our result therefore shows that quantum commitments can exist even if no quantum-secure classical cryptographic primitive exists. In particular, quantum commitments can exist even if no quantum-secure one-way function exists. In this work, we also consider digital signatures, which are other fundamental primitives in cryptography. We show that one-time secure digital signatures with quantum public keys exist if pseudorandom quantum states exist. In the classical setting, the existence of digital signatures is equivalent to the existence of one-way functions. Our result, on the other hand, shows that quantum signatures can exist even if no quantum-secure classical cryptographic primitive (including quantum-secure one-way functions) exists., Comment: 26 pages

From the minimal assumption of post-quantum semi-honest oblivious transfers, we build the first $\epsilon$-simulatable two-party computation (2PC) against quantum polynomial-time (QPT) adversaries that is both constant-round and black-box (for both the construction and security reduction). A recent work by Chia, Chung, Liu, and Yamakawa (FOCS'21) shows that post-quantum 2PC with standard simulation-based security is impossible in constant rounds, unless either $\mathbf{NP} \subseteq \mathbf{BQP}$ or relying on non-black-box simulation. The $\epsilon$-simulatability we target is a relaxation of the standard simulation-based security that allows for an arbitrarily small noticeable simulation error $\epsilon$. Moreover, when quantum communication is allowed, we can further weaken the assumption to post-quantum secure one-way functions (PQ-OWFs), while maintaining the constant-round and black-box property. Our techniques also yield the following set of constant-round and black-box two-party protocols secure against QPT adversaries, only assuming black-box access to PQ-OWFs: - extractable commitments for which the extractor is also an $\epsilon$-simulator; - $\epsilon$-zero-knowledge commit-and-prove whose commit stage is extractable with $\epsilon$-simulation; - $\epsilon$-simulatable coin-flipping; - $\epsilon$-zero-knowledge arguments of knowledge for $\mathbf{NP}$ for which the knowledge extractor is also an $\epsilon$-simulator; - $\epsilon$-zero-knowledge arguments for $\mathbf{QMA}$. At the heart of the above results is a black-box extraction lemma showing how to efficiently extract secrets from QPT adversaries while disturbing their quantum state in a controllable manner, i.e., achieving $\epsilon$-simulatability of the post-extraction state of the adversary.

In known constructions of classical zero-knowledge protocols for NP, either of zero-knowledge or soundness holds only against computationally bounded adversaries. Indeed, achieving both statistical zero-knowledge and statistical soundness at the same time with classical verifier is impossible for NP unless the polynomial-time hierarchy collapses, and it is also believed to be impossible even with a quantum verifier. In this work, we introduce a novel compromise, which we call the certified everlasting zero-knowledge proof for QMA. It is a computational zero-knowledge proof for QMA, but the verifier issues a classical certificate that shows that the verifier has deleted its quantum information. If the certificate is valid, even unbounded malicious verifier can no longer learn anything beyond the validity of the statement. We construct a certified everlasting zero-knowledge proof for QMA. For the construction, we introduce a new quantum cryptographic primitive, which we call commitment with statistical binding and certified everlasting hiding, where the hiding property becomes statistical once the receiver has issued a valid certificate that shows that the receiver has deleted the committed information. We construct commitment with statistical binding and certified everlasting hiding from quantum encryption with certified deletion by Broadbent and Islam [TCC 2020] (in a black box way), and then combine it with the quantum sigma-protocol for QMA by Broadbent and Grilo [FOCS 2020] to construct the certified everlasting zero-knowledge proof for QMA. Our constructions are secure in the quantum random oracle model. Commitment with statistical binding and certified everlasting hiding itself is of independent interest, and there will be many other useful applications beyond zero-knowledge., Comment: 33 pages

Broadbent and Islam (TCC '20) proposed a quantum cryptographic primitive called quantum encryption with certified deletion. In this primitive, a receiver in possession of a quantum ciphertext can generate a classical certificate that the encrypted message is deleted. Although their construction is information-theoretically secure, it is limited to the setting of one-time symmetric key encryption (SKE), where a sender and receiver have to share a common key in advance and the key can be used only once. Moreover, the sender has to generate a quantum state and send it to the receiver over a quantum channel in their construction. Although deletion certificates are privately verifiable, which means a verification key for a certificate has to be kept secret, in the definition by Broadbent and Islam, we can also consider public verifiability. In this work, we present various constructions of encryption with certified deletion. - Quantum communication case: We achieve (reusable-key) public key encryption (PKE) and attribute-based encryption (ABE) with certified deletion. Our PKE scheme with certified deletion is constructed assuming the existence of IND-CPA secure PKE, and our ABE scheme with certified deletion is constructed assuming the existence of indistinguishability obfuscation and one-way function. These two schemes are privately verifiable. - Classical communication case: We also achieve PKE with certified deletion that uses only classical communication. We give two schemes, a privately verifiable one and a publicly verifiable one. The former is constructed assuming the LWE assumption in the quantum random oracle model. The latter is constructed assuming the existence of one-shot signatures and extractable witness encryption., Comment: 51 pages

We investigate the existence of constant-round post-quantum black-box zero-knowledge protocols for $\mathbf{NP}$. As a main result, we show that there is no constant-round post-quantum black-box zero-knowledge argument for $\mathbf{NP}$ unless $\mathbf{NP}\subseteq \mathbf{BQP}$. As constant-round black-box zero-knowledge arguments for $\mathbf{NP}$ exist in the classical setting, our main result points out a fundamental difference between post-quantum and classical zero-knowledge protocols. Combining previous results, we conclude that unless $\mathbf{NP}\subseteq \mathbf{BQP}$, constant-round post-quantum zero-knowledge protocols for $\mathbf{NP}$ exist if and only if we use non-black-box techniques or relax certain security requirements such as relaxing standard zero-knowledge to $\epsilon$-zero-knowledge. Additionally, we also prove that three-round and public-coin constant-round post-quantum black-box $\epsilon$-zero-knowledge arguments for $\mathbf{NP}$ do not exist unless $\mathbf{NP}\subseteq \mathbf{BQP}$., Comment: 46 pages

We propose three constructions of classically verifiable non-interactive zero-knowledge proofs and arguments (CV-NIZK) for QMA in various preprocessing models. - We construct a CV-NIZK for QMA in the quantum secret parameter model where a trusted setup sends a quantum proving key to the prover and a classical verification key to the verifier. It is information theoretically sound and zero-knowledge. - Assuming the quantum hardness of the learning with errors problem, we construct a CV-NIZK for QMA in a model where a trusted party generates a CRS and the verifier sends an instance-independent quantum message to the prover as preprocessing. This model is the same as one considered in the recent work by Coladangelo, Vidick, and Zhang (CRYPTO '20). Our construction has the so-called dual-mode property, which means that there are two computationally indistinguishable modes of generating CRS, and we have information theoretical soundness in one mode and information theoretical zero-knowledge property in the other. This answers an open problem left by Coladangelo et al, which is to achieve either of soundness or zero-knowledge information theoretically. To the best of our knowledge, ours is the first dual-mode NIZK for QMA in any kind of model. - We construct a CV-NIZK for QMA with quantum preprocessing in the quantum random oracle model. This quantum preprocessing is the one where the verifier sends a random Pauli-basis states to the prover. Our construction uses the Fiat-Shamir transformation. The quantum preprocessing can be replaced with the setup that distributes Bell pairs among the prover and the verifier, and therefore we solve the open problem by Broadbent and Grilo (FOCS '20) about the possibility of NIZK for QMA in the shared Bell pair model via the Fiat-Shamir transformation., Comment: 46 pages This is a major update version of arXiv:2003.10712. A new result, NIZK via Fiat-Shamir, is added. (Sec.5)

In a recent seminal work, Bitansky and Shmueli (STOC '20) gave the first construction of a constant round zero-knowledge argument for NP secure against quantum attacks. However, their construction has several drawbacks compared to the classical counterparts. Specifically, their construction only achieves computational soundness, requires strong assumptions of quantum hardness of learning with errors (QLWE assumption) and the existence of quantum fully homomorphic encryption (QFHE), and relies on non-black-box simulation. In this paper, we resolve these issues at the cost of weakening the notion of zero-knowledge to what is called $\epsilon$-zero-knowledge. Concretely, we construct the following protocols: - We construct a constant round interactive proof for NP that satisfies statistical soundness and black-box $\epsilon$-zero-knowledge against quantum attacks assuming the existence of collapsing hash functions, which is a quantum counterpart of collision-resistant hash functions. Interestingly, this construction is just an adapted version of the classical protocol by Goldreich and Kahan (JoC '96) though the proof of $\epsilon$-zero-knowledge property against quantum adversaries requires novel ideas. - We construct a constant round interactive argument for NP that satisfies computational soundness and black-box $\epsilon$-zero-knowledge against quantum attacks only assuming the existence of post-quantum one-way functions. At the heart of our results is a new quantum rewinding technique that enables a simulator to extract a committed message of a malicious verifier while simulating verifier's internal state in an appropriate sense., Comment: Fixed a minor technical issue (see Footnote 17 in page 21) and improved the proof of Claim 4.5. (10/30/2023)

Secure software leasing (SSL) is a quantum cryptographic primitive that enables users to execute software only during the software is leased. It prevents users from executing leased software after they return the leased software to its owner. SSL can make software distribution more flexible and controllable. Although SSL is an attractive cryptographic primitive, the existing SSL scheme is based on public key quantum money, which is not instantiated with standard cryptographic assumptions so far. Moreover, the existing SSL scheme only supports a subclass of evasive functions. In this work, we present SSL schemes based on the learning with errors assumption (LWE). Specifically, our contributions consist of the following. - We construct an SSL scheme for pseudorandom functions from the LWE assumption against quantum adversaries. - We construct an SSL scheme for a subclass of evasive functions from the LWE assumption against sub-exponential quantum adversaries. - We construct SSL schemes for the functionalities above with classical communication from the LWE assumption against (sub-exponential) quantum adversaries. SSL with classical communication means that entities exchange only classical information though they run quantum computation locally. Our crucial tool is two-tier quantum lightning, which is introduced in this work and a relaxed version of quantum lighting. In two-tier quantum lightning schemes, we have a public verification algorithm called semi-verification and a private verification algorithm called full-verification. An adversary cannot generate possibly entangled two quantum states whose serial numbers are the same such that one passes the semi-verification, and the other also passes the full-verification. We show that we can construct a two-tier quantum lightning scheme from the LWE assumption., Comment: 40 pages. Fixed minor issues in Sec 1.5 and 3.2

In this paper, we extend the protocol of classical verification of quantum computations (CVQC) recently proposed by Mahadev to make the verification efficient. Our result is obtained in the following three steps: $\bullet$ We show that parallel repetition of Mahadev's protocol has negligible soundness error. This gives the first constant round CVQC protocol with negligible soundness error. In this part, we only assume the quantum hardness of the learning with error (LWE) problem similar to the Mahadev's work. $\bullet$ We construct a two-round CVQC protocol in the quantum random oracle model (QROM) where a cryptographic hash function is idealized to be a random function. This is obtained by applying the Fiat-Shamir transform to the parallel repetition version of the Mahadev's protocol. $\bullet$ We construct a two-round CVQC protocol with the efficient verifier in the CRS+QRO model where both prover and verifier can access to a (classical) common reference string generated by a trusted third party in addition to quantum access to QRO. Specifically, the verifier can verify a $QTIME(T)$ computation in time $poly(n,log T)$ where $n$ is the security parameter. For proving soundness, we assume that a standard model instantiation of our two-round protocol with a concrete hash function (say, SHA-3) is sound and the existence of post-quantum indistinguishability obfuscation and post-quantum fully homomorphic encryption in addition to the quantum hardness of the LWE problem.