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Notes on 10.62056/ayiv4fe-3 Zero-Knowledge Proofs of Quantumness

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Date:19 Tevet 5785

Phan, Wen, Yan, and Zheng, Zero-Knowledge Proofs of Quantumness. IACR Communications in Cryptology, 1:4, Jan 13 2025, doi: 10.62056/ayiv4fe-3.

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TL;DR:

This paper introduces a method to enhance the security of quantum computing verification by ensuring that classical verifiers cannot exploit the quantum systems they are testing, thereby protecting the intellectual property and capabilities of quantum devices.

For General Tech Enthusiasts:

This paper explores how to make sure that when we test if a quantum computer is truly "quantum," we don't accidentally give away 'leak' its secrets or let others misuse its power, thereby making quantum computing safer for everyone.

For Quantum Computing Researchers:

This paper formalizes zero-knowledge proofs of quantumness (ZKPoQ), demonstrating that existing PoQ schemes like Shor's factoring-based and LWE-based methods can be transformed into ZKPoQ, enhancing security against malicious classical verifiers without compromising verification efficiency.

For Cryptographers:

The authors propose a novel cryptographic primitive, ZKPoQ, that leverages extractable non-interactive zero-knowledge (NIZK) arguments to prevent classical verifiers from gaining any information beyond the validity of quantumness, thus extending zero-knowledge principles to the realm of quantum computation verification.

For Tech Companies Developing Quantum Computers:

This research provides a crucial security enhancement for quantum services, ensuring that classical clients cannot exploit interactions with quantum servers to solve problems beyond the intended verification, safeguarding valuable quantum resources.

Non-hype / Critical TL;DR:

While the paper introduces a valuable concept of zero-knowledge proofs of quantumness, its practical application relies heavily on the existence and efficiency of extractable NIZK arguments, which may introduce complexities and potential limitations in real-world deployment, especially concerning scalability and verifier-side overhead.

Application:

Problem: In current proofs of quantumness (PoQ) schemes, quantum provers (e.g., quantum computers) are vulnerable to exploitation by malicious classical verifiers. These verifiers might use the interaction to solve hard computational problems or gain unauthorized knowledge about the quantum device's capabilities. Formally, a malicious verifier V* could potentially use the interaction with a quantum prover P to gain an advantage in solving a problem that is assumed to be hard for classical computers.

Solution: The research introduces zero-knowledge proofs of quantumness (ZKPoQ). This ensures that the classical verifier gains no information from the interaction beyond the fact that the prover possesses quantum capabilities. It achieves this by requiring the verifier to provide an extractable non-interactive zero-knowledge (NIZK) argument, proving they are not acting maliciously. This prevents the verifier from exploiting the quantum prover's abilities while still allowing them to verify the quantumness. Formally, a ZKPoQ protocol ensures that for any verifier V, there exists a simulator SimV such that the view of V in the real interaction with P is computationally indistinguishable from the output of SimV.

Unexpected Findings:

  1. Classical Zero-Knowledge is Sufficient: Surprisingly, classical zero-knowledge proofs are sufficient to transform some existing PoQ schemes into ZKPoQ schemes. This means that existing cryptographic techniques can be adapted to enhance the security of quantum verification.
  2. Verifier-Side Zero-Knowledge: The paper finds that it is more general and effective to require zero-knowledge proofs on the verifier's side rather than the prover's side. This is counterintuitive, as traditionally, zero-knowledge proofs are used to protect the prover's secrets. However, in this context, it effectively regulates the verifier's behavior.
  3. Dual Role of Parties: Both the prover and verifier play a dual role in the proposed ZKPoQ schemes. They participate in both the PoQ and the classical zero-knowledge proof, adding a layer of complexity but enhancing security.
  4. Honest-but-Curious Verifier Assumption: The assumption that the verifier should be honest-but-curious rather than the prover was unexpected. This shifts the traditional focus of security measures in zero-knowledge protocols. Traditionally, ZKP aims to protect the prover's secrets from a potentially malicious verifier. Here, the focus is on preventing a potentially malicious verifier from exploiting the prover's quantum capabilities, even if the prover is assumed to be honest. This represents a significant paradigm shift in the application of zero-knowledge principles.

Key Terms:

  • Proof of Quantumness (PoQ): A method to verify that a quantum device can perform tasks that are beyond the capabilities of classical computers.
  • Zero-Knowledge Proof (ZKP): A cryptographic method where one party (the prover) can prove to another party (the verifier) that a statement is true, without revealing any information beyond the truth of the statement.
  • Zero-Knowledge Proofs of Quantumness (ZKPoQ): An extension of PoQ where the verifier learns nothing beyond the fact that the prover possesses quantum capabilities.
  • Classical Verifier: A party that uses classical (non-quantum) computational resources to verify the quantumness of a prover.
  • Quantum Prover: A party that uses quantum computational resources to demonstrate its quantum capabilities.
  • Extractable Non-Interactive Zero-Knowledge (NIZK) Argument: A type of zero-knowledge proof where the verifier provides a proof that can be "extracted" by a simulator to gain the secret knowledge, ensuring that the verifier is not acting maliciously.
  • Shor's Factoring-Based Scheme: A PoQ scheme based on the ability of quantum computers to efficiently factor large numbers, a task believed to be hard for classical computers. Formally, given a composite number N = p * q, where p and q are large primes, a quantum prover can efficiently find p and q using Shor's algorithm.
  • Learning With Errors (LWE): A mathematical problem that is believed to be hard for both classical and quantum computers, used as a basis for cryptographic schemes. Formally, given a matrix A ∈ ℤqm×n and a vector s ∈ ℤqn, and a vector b = As + e, where e is a small error vector, the LWE problem is to find s.
  • Trapdoor Claw-Free (TCF) Hash Functions: Cryptographic functions used in the LWE-based PoQ scheme, designed to be hard to invert without a secret "trapdoor." Formally, a TCF family is a pair of injective functions {fk,0, fk,1}k that share the same image, and there exists a trapdoor td such that, given td, it is easy to find x0 and x1 such that fk,0(x0) = fk,1(x1) = y for a given y.
  • Computational Bell Test: A method to verify quantum entanglement, used in some PoQ schemes.
  • Post-Quantum Hardness: The assumption that certain computational problems remain hard even for quantum computers.
  • Claw-Free Property: A property of certain cryptographic functions where it is hard to find two different inputs that produce the same output. Formally, for a claw-free function family F = {fk,b : X → Y}k∈K, b∈{0,1}, it is computationally hard to find x0, x1 such that fk,0(x0) = fk,1(x1) without the trapdoor.
  • Preimage: An input to a function that produces a specific output.
  • Common Reference String (CRS): A shared piece of information used in some cryptographic protocols, including NIZK proofs.
  • Hellinger Distance: A measure of the difference between two probability distributions. Formally, for two probability distributions P and Q, the Hellinger distance is defined as H2(P, Q) = 1/2 * Σx (√P(x) - √Q(x))2.
  • Adaptive Hardcore Bit: A property of certain cryptographic functions related to the unpredictability of specific bits in the output.
  • Noisy Trapdoor Claw-Free (NTCF) Family: A specific type of TCF function used in the LWE-based PoQ scheme.
  • Quantum-Prover Interactive Proof (QPIP): An interactive protocol between a quantum prover and a classical verifier.
  • Probabilistic Polynomial Time (PPT): A class of algorithms that can be executed in polynomial time with a certain probability.
  • Quantum Polynomial Time (QPT): A class of algorithms that can be executed in polynomial time on a quantum computer.

Approach:

  1. Outline the Research Methodology:

    • The researchers formalize the concept of zero-knowledge proofs of quantumness (ZKPoQ). Formally, they define ZKPoQ as a QPIP system (P, V) that satisfies quantum completeness, classical soundness, and computational zero-knowledge.
    • They analyze two existing PoQ schemes: Shor's factoring-based scheme and the LWE-based scheme from [BCM+18].
    • They propose a method to transform these schemes into ZKPoQ by requiring the verifier to provide an extractable NIZK argument.
      • For the factoring-based scheme, the verifier provides an NIZK proof for the statement "N = p * q" where p and q are the prime factors of N.
      • For the LWE-based scheme, the verifier provides an NIZK proof for the statement "k = (A, As + e)" where s is the secret of the LWE instance.
    • They prove the security of the transformed schemes, demonstrating quantum completeness, classical soundness, and computational zero-knowledge.

  2. Describe Problem-Solving Techniques:

    • Extractable NIZK Argument: The key technique is to require the verifier to provide an extractable NIZK proof that they know the secret (e.g., the factorization of a number or the solution to an LWE instance). This allows a simulator to extract the secret and simulate the interaction without needing quantum resources. Formally, the NIZK proof system Π = (Setup, P, V) must satisfy the extractability property, meaning there exists a polynomial-time extractor Ext such that for any adversary A, Pr[V(crs, x, π) = 1 ∧ (x, w) ∉ R] ≤ negl(λ), where (crs, td) ← Sim(1λ), (x, π) ← A(crs), and w ← Ext(crs, td, x, π).
    • One-Way Functions and Public Key Encryption: These are used to construct the extractable NIZK proof, ensuring its security and zero-knowledge properties.
    • Game-Based Security Proofs: The researchers use a sequence of games to prove the security properties of their ZKPoQ schemes, demonstrating that an adversary cannot distinguish between the real and simulated interactions. For example, in the proof of classical soundness, they define a sequence of games Gamei and show that the adversary's advantage in each game is negligible.

Results and Evaluation:

  1. Key Findings:

    • Existing PoQ schemes can be transformed into ZKPoQ schemes, enhancing their security against malicious verifiers.
    • Classical zero-knowledge proofs are sufficient for this transformation in the case of factoring-based and LWE-based schemes.
    • Requiring zero-knowledge proofs on the verifier's side is a more general and effective approach.

  2. Quantitative Results:

    • The paper does not provide specific quantitative results in terms of performance metrics. The focus is on the theoretical security guarantees of the proposed schemes.
    • The security of the schemes is based on the assumed hardness of factoring and LWE, as well as the security properties of the extractable NIZK argument.

  3. Notable Achievements:

    • Formalization of the concept of ZKPoQ, extending zero-knowledge principles to the quantum realm.
    • Demonstration that existing PoQ schemes can be made zero-knowledge, enhancing their security.
    • Introduction of a novel approach where the verifier provides the zero-knowledge proof, effectively regulating their behavior.

Practical Deployment and Usability:

Real-World Applicability: - The research has implications for the security of quantum computing services. It provides a mechanism to protect quantum resources from being exploited by malicious clients. - It can be applied in scenarios where quantum computers are used as a service, and clients need to verify their quantum capabilities without compromising their security. - It can also be used to certify randomness generated by quantum devices, ensuring that the randomness is truly quantum and not influenced by a malicious party.

Practicality and Ease of Use:

  • The paper focuses on the theoretical aspects, not evaluating the practical performance and scalability of the proposed schemes.

Examples: - A company offering quantum computing services can use ZKPoQ to allow clients to verify the quantum capabilities of their machines without revealing sensitive information about their hardware or algorithms. - A user can verify that a quantum device is generating truly random numbers without worrying that the device is being manipulated by a malicious party.

Limitations, Assumptions, and Caveats:

Limitations: - The proposed schemes rely on the existence and efficiency of extractable NIZK arguments. The practicality of these schemes depends on the performance of these arguments. - The paper focuses on factoring-based and LWE-based schemes. Further research is needed to extend these results to other PoQ schemes, such as sampling-based methods. - The analysis assumes that the verifier is classical. The case of a malicious quantum verifier is not considered.

Assumptions: - The security of the schemes is based on the assumed hardness of factoring and LWE. - It is assumed that the extractable NIZK argument is secure and zero-knowledge. - The paper assumes a common reference string (CRS) model for the NIZK proofs.

Caveats: - The practical implementation of the proposed schemes may face challenges related to the efficiency and scalability of the extractable NIZK arguments. - The verifier-side zero-knowledge requirement adds complexity to the protocol, which may impact its usability. - The paper focuses on the theoretical aspects, and further research is needed to evaluate the practical performance and security of the proposed schemes in real-world scenarios.

Promises and Horizons:

New Research Areas: - Exploring the use of other types of zero-knowledge proofs, such as interactive proofs, to achieve ZKPoQ. - Extending the ZKPoQ framework to other PoQ schemes, including sampling-based methods. - Investigating the security of ZKPoQ against malicious quantum verifiers. - Developing more efficient and scalable extractable NIZK arguments.

Evolution of the Research: - Future research may focus on improving the efficiency and practicality of ZKPoQ schemes. - There may be a shift towards developing ZKPoQ protocols that do not rely on a CRS model.

Conflict of Interest and Funding:

  • The research seems driven by scientific curiosity and the desire to advance the field of quantum cryptography. The authors do not declare any conflicts of interest.
  • The research is supported by the National Key Research and Development Program of China, the European Union's Horizon Europe research and innovation program, and the ANR Projects SecureCompute and HELO.