Unconditionally secure quantum commitments with preprocessing

We demonstrate how to build computationally secure commitment schemes with the aid of quantum auxiliary inputs without unproven complexity assumptions. Furthermore, the quantum auxiliary input can be prepared either (1) efficiently through a trusted setup similar to the classical common random string model, or (2) strictly between the two involved parties in uniform exponential time. Classically this remains impossible without first proving $\mathsf{P} \neq \mathsf{NP}$.Comment: 16 page

Unconditionally secure quantum commitments with preprocessing

We demonstrate how to build computationally secure commitment schemes with the aid of quantum auxiliary inputs without unproven complexity assumptions. Furthermore, the quantum auxiliary input can be prepared either (1) efficiently through a trusted setup similar to the classical common random string model, or (2) strictly between the two involved parties in uniform exponential time. Classically this remains impossible without first proving $\mathsf{P} \neq \mathsf{NP}$

On the Computational Hardness Needed for Quantum Cryptography

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

Beating Classical Impossibility of Position Verification

Chandran et al. (SIAM J. Comput. \u2714) formally introduced the cryptographic task of position verification, where they also showed that it cannot be achieved by classical protocols. In this work, we initiate the study of position verification protocols with classical verifiers. We identify that proofs of quantumness (and thus computational assumptions) are necessary for such position verification protocols. For the other direction, we adapt the proof of quantumness protocol by Brakerski et al. (FOCS \u2718) to instantiate such a position verification protocol. As a result, we achieve classically verifiable position verification assuming the quantum hardness of Learning with Errors. Along the way, we develop the notion of 1-of-2 non-local soundness for a natural non-local game for 1-of-2 puzzles, first introduced by Radian and Sattath (AFT \u2719), which can be viewed as a computational unclonability property. We show that 1-of-2 non-local soundness follows from the standard 2-of-2 soundness (and therefore the adaptive hardcore bit property), which could be of independent interest

Lower Bounds for Function Inversion with Quantum Advice

Function inversion is the problem that given a random function $f: [M] \to [N]$, we want to find pre-image of any image $f^{-1}(y)$ in time $T$. In this work, we revisit this problem under the preprocessing model where we can compute some auxiliary information or advice of size $S$ that only depends on $f$ but not on $y$. It is a well-studied problem in the classical settings, however, it is not clear how quantum algorithms can solve this task any better besides invoking Grover's algorithm, which does not leverage the power of preprocessing. Nayebi et al. proved a lower bound $ST^2 \ge \tilde\Omega(N)$ for quantum algorithms inverting permutations, however, they only consider algorithms with classical advice. Hhan et al. subsequently extended this lower bound to fully quantum algorithms for inverting permutations. In this work, we give the same asymptotic lower bound to fully quantum algorithms for inverting functions for fully quantum algorithms under the regime where $M = O(N)$. In order to prove these bounds, we generalize the notion of quantum random access code, originally introduced by Ambainis et al., to the setting where we are given a list of (not necessarily independent) random variables, and we wish to compress them into a variable-length encoding such that we can retrieve a random element just using the encoding with high probability. As our main technical contribution, we give a nearly tight lower bound (for a wide parameter range) for this generalized notion of quantum random access codes, which may be of independent interest.Comment: ITC full versio

Pseudorandom (Function-Like) Quantum State Generators: New Definitions and Applications

Pseudorandom quantum states (PRS) are efficiently constructible states that are computationally indistinguishable from being Haar-random, and have recently found cryptographic applications. We explore new definitions, new properties and applications of pseudorandom states, and present the following contributions: 1. New Definitions: We study variants of pseudorandom function-like state (PRFS) generators, introduced by Ananth, Qian, and Yuen (CRYPTO'22), where the pseudorandomness property holds even when the generator can be queried adaptively or in superposition. We show feasibility of these variants assuming the existence of post-quantum one-way functions. 2. Classical Communication: We show that PRS generators with logarithmic output length imply commitment and encryption schemes with classical communication. Previous constructions of such schemes from PRS generators required quantum communication. 3. Simplified Proof: We give a simpler proof of the Brakerski--Shmueli (TCC'19) result that polynomially-many copies of uniform superposition states with random binary phases are indistinguishable from Haar-random states. 4. Necessity of Computational Assumptions: We also show that a secure PRS with output length logarithmic, or larger, in the key length necessarily requires computational assumptions

Cryptography from Pseudorandom Quantum States

Pseudorandom states, introduced by Ji, Liu and Song (Crypto\u2718), are efficiently-computable quantum states that are computationally indistinguishable from Haar-random states. One-way functions imply the existence of pseudorandom states, but Kretschmer (TQC\u2720) recently constructed an oracle relative to which there are no one-way functions but pseudorandom states still exist. Motivated by this, we study the intriguing possibility of basing interesting cryptographic tasks on pseudorandom states. We construct, assuming the existence of pseudorandom state generators that map a $\lambda$-bit seed to a $\omega(\log\lambda)$-qubit state, (a) statistically binding and computationally hiding commitments and (b) pseudo one-time encryption schemes. A consequence of (a) is that pseudorandom states are sufficient to construct maliciously secure multiparty computation protocols in the dishonest majority setting. Our constructions are derived via a new notion called {\em pseudorandom function-like states} (PRFS), a generalization of pseudorandom states that parallels the classical notion of pseudorandom functions. Beyond the above two applications, we believe our notion can effectively replace pseudorandom functions in many other cryptographic applications

An efficient quantum parallel repetition theorem and applications

We prove a tight parallel repetition theorem for $3$-message computationally-secure quantum interactive protocols between an efficient challenger and an efficient adversary. We also prove under plausible assumptions that the security of $4$-message computationally secure protocols does not generally decrease under parallel repetition. These mirror the classical results of Bellare, Impagliazzo, and Naor [BIN97]. Finally, we prove that all quantum argument systems can be generically compiled to an equivalent $3$-message argument system, mirroring the transformation for quantum proof systems [KW00, KKMV07]. As immediate applications, we show how to derive hardness amplification theorems for quantum bit commitment schemes (answering a question of Yan [Yan22]), EFI pairs (answering a question of Brakerski, Canetti, and Qian [BCQ23]), public-key quantum money schemes (answering a question of Aaronson and Christiano [AC13]), and quantum zero-knowledge argument systems. We also derive an XOR lemma [Yao82] for quantum predicates as a corollary

Collusion Resistant Copy-Protection for Watermarkable Functionalities

Copy-protection is the task of encoding a program into a quantum state to prevent illegal duplications. A line of recent works studied copy-protection schemes under ``1 -> 2 attacks\u27\u27: the adversary receiving one program copy can not produce two valid copies. However, under most circumstances, vendors need to sell more than one copy of a program and still ensure that no duplicates can be generated. In this work, we initiate the study of collusion resistant copy-protection in the plain model. Our results are twofold: (*) The feasibility of copy-protecting all watermarkable functionalities is an open question raised by Aaronson et al. (CRYPTO\u27 21). In the literature, watermarking decryption, digital signature schemes and PRFs have been extensively studied. For the first time, we show that digital signature schemes can be copy-protected. Together with the previous work on copy-protection of decryption and PRFs by Coladangelo et al. (CRYPTO\u27 21), it suggests that many watermarkable functionalities can be copy-protected, partially answering the above open question by Aaronson et al. (*) We make all the above schemes (copy-protection of decryption, digital signatures, and PRFs) k bounded collusion resistant for any polynomial k, giving the first bounded collusion resistant copy-protection for various functionalities in the plain model

Unitary Complexity and the Uhlmann Transformation Problem

State transformation problems such as compressing quantum information or breaking quantum commitments are fundamental quantum tasks. However, their computational difficulty cannot easily be characterized using traditional complexity theory, which focuses on tasks with classical inputs and outputs. To study the complexity of such state transformation tasks, we introduce a framework for unitary synthesis problems, including notions of reductions and unitary complexity classes. We use this framework to study the complexity of transforming one entangled state into another via local operations. We formalize this as the Uhlmann Transformation Problem, an algorithmic version of Uhlmann's theorem. Then, we prove structural results relating the complexity of the Uhlmann Transformation Problem, polynomial space quantum computation, and zero knowledge protocols. The Uhlmann Transformation Problem allows us to characterize the complexity of a variety of tasks in quantum information processing, including decoding noisy quantum channels, breaking falsifiable quantum cryptographic assumptions, implementing optimal prover strategies in quantum interactive proofs, and decoding the Hawking radiation of black holes. Our framework for unitary complexity thus provides new avenues for studying the computational complexity of many natural quantum information processing tasks.Comment: 126 pages, comments welcom