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arXiv:1912.04769

Post-quantum Zero Knowledge in Constant Rounds

Nir Bitansky, Omri Shmueli

Published Dec 10, 2019• 0 citations
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Abstract

We construct a constant-round zero-knowledge classical argument for NP secure against quantum attacks. We assume the existence of Quantum Fully-Homomorphic Encryption and other standard primitives, known based on the Learning with Errors Assumption for quantum algorithms. As a corollary, we also obtain a constant-round zero-knowledge quantum argument for QMA. At the heart of our protocol is a new no-cloning non-black-box simulation technique.

DOI
10.48550/arxiv.1912.04769

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We construct a constant-round zero-knowledge classical argument for NP secure against quantum attacks. We assume the existence of Quantum Fully-Homomorphic Encryption and other standard primitives, known based on the Learning with Errors Assumption for quantum algorithms. As a corollary, we also obtain a constant-round zero-knowledge quantum argument for QMA. At the heart of our protocol is a new no-cloning non-black-box simulation technique.
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 εεε-zero-knowledge. Concretely, we construct the following protocols: - We construct a constant round interactive proof for NP that satisfies statistical soundness and black-box εεε-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 εεε-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 εεε-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.
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 ϵ\epsilonϵ-zero-knowledge. Concretely, we construct the following protocols: - We construct a constant round interactive proof for NP that satisfies statistical soundness and black-box ϵ\epsilonϵ-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 ϵ\epsilonϵ-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 ϵ\epsilonϵ-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.
We investigate the existence of constant-round post-quantum black-box zero-knowledge protocols for NP. As a main result, we show that there is no constant-round post-quantum black-box zero-knowledge argument for NP unlessNP⊆BQP\text{NP} \subseteq \text{BQP}NP⊆BQP. As constant-round black-box zero-knowledge arguments for 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 unlessNP⊆BQP\text{NP} \subseteq \text{BQP}NP⊆BQP, constant-round post-quantum zero-knowledge protocols for NP exist if and only if we use non-black-box techniques or relax certain security requirements such as relaxing standard zero-knowledge toϵ\epsilonϵ-zero-knowledge. Additionally, we also prove that three-round and public-coin constant-round post-quantum black-boxϵ\epsilonϵ-zero-knowledge arguments for NP do not exist unlessNP⊆BQP\text{NP} \subseteq \text{BQP}NP⊆BQP.
We investigate the existence of constant-round post-quantum black-box zero-knowledge protocols for NP\mathbf{NP}NP. As a main result, we show that there is no constant-round post-quantum black-box zero-knowledge argument for NP\mathbf{NP}NP unless NP⊆BQP\mathbf{NP}\subseteq \mathbf{BQP}NP⊆BQP. As constant-round black-box zero-knowledge arguments for NP\mathbf{NP}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 NP⊆BQP\mathbf{NP}\subseteq \mathbf{BQP}NP⊆BQP, constant-round post-quantum zero-knowledge protocols for NP\mathbf{NP}NP exist if and only if we use non-black-box techniques or relax certain security requirements such as relaxing standard zero-knowledge to εεε-zero-knowledge. Additionally, we also prove that three-round and public-coin constant-round post-quantum black-box εεε-zero-knowledge arguments for NP\mathbf{NP}NP do not exist unless NP⊆BQP\mathbf{NP}\subseteq \mathbf{BQP}NP⊆BQP.
We initiate the study of multi-party computation for classical functionalities (in the plain model) with security against malicious polynomial-time quantum adversaries. We observe that existing techniques readily give a polynomial-round protocol, but our main result is a construction of *constant-round* post-quantum multi-party computation. We assume mildly super-polynomial quantum hardness of learning with errors (LWE), and polynomial quantum hardness of an LWE-based circular security assumption. Along the way, we develop the following cryptographic primitives that may be of independent interest: 1. A spooky encryption scheme for relations computable by quantum circuits, from the quantum hardness of an LWE-based circular security assumption. This yields the first quantum multi-key fully-homomorphic encryption scheme with classical keys. 2. Constant-round zero-knowledge secure against multiple parallel quantum verifiers from spooky encryption for relations computable by quantum circuits. To enable this, we develop a new straight-line non-black-box simulation technique against *parallel* verifiers that does not clone the adversary's state. This forms the heart of our technical contribution and may also be relevant to the classical setting. 3. A constant-round post-quantum non-malleable commitment scheme, from the mildly super-polynomial quantum hardness of LWE.