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Notes 2024-1847 limitations of quantum cryptography in constructing secure protocols

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Status:In Progress Date: 21 Mar-Cheshvan 5785

Pappu & Lu (2024) "Notions of Quantum Reductions and Impossibility of Statistical NIZK" paper

Layman TL;DR:

Certain cryptographic constructs may remain inherently insecure in the quantum realm.

ZKP SME TL;DR:

Extends the classical impossibility results of S-NIZKs to the quantum realm, highlighting the challenges due to quantum rewinding and the lack of a direct quantum analogue for programming classical random tapes. They introduce a new framework, the Q-CAP system, for categorizing quantum reductions, which is an extension of the classical CAP notation.

Cryptography TL;DR:

Shows possibly fundamental inherent limitations of quantum cryptography. The standard approach of basing security on computationally hard problems falls short.

Non-Hype TL;DR:

Lu and Pappu carve out a very specific niche of NIZKs that are provably impossible to construct under their stated conditions: Adaptively sound, statistically zero-knowledge, with black-box reductions to falsifiable assumptions. The vast majority of practical NIZKs fall outside this niche. NIZKs based on e.g. algebraic structures because they might use computational zero-knowledge, static soundness, non-black-box reductions, or rely on assumptions stronger than the ones considered in the impossibility result. The impact on a particular NIZK depends entirely on its construction and its security proof.

Unexpected Findings

  1. Impossibility of Adaptive Soundness in Quantum Settings It is impossible to construct adaptive sound S-NIZK protocols based on any falsifiable assumption in a qauntum setting. This finding is significant because it challenges the prevalent belief that quantum techniques could help overcome classical cryptographic limitations. The authors demonstrate that even with the power of quantum computations, the fundamental security properties required for adaptive soundness cannot be achieved, indicating that certain cryptographic constructs may remain inherently insecure.

  2. Quantum Black-Box Reductions which allow classical queries to a quantum adversary, do not provide the expected security guarantees. This result is surprising because it suggests that the integration of quantum techniques does not necessarily lead to stronger security results in cryptography. The authors argue that the traditional methods of proving security through black-box reductions fail in the quantum context, as they do not account for the unique properties of quantum information and computation, such as entanglement and superposition.

Key Terms

  • Non-Interactive Zero-Knowledge (NIZK): A cryptographic protocol enabling a prover to convince a verifier of a statement's truth without revealing extra information, using a single message. This single message from the prover to the verifier constitutes the entire interaction. The protocol must satisfy completeness (a true statement is accepted), soundness (a false statement is rejected with high probability), and zero-knowledge (the verifier learns nothing beyond the statement's validity).

  • Statistical NIZK (S-NIZK): A variant of NIZK where the zero-knowledge property holds information-theoretically. This means the zero-knowledge guarantee is unconditional, not relying on computational hardness assumptions. Even a computationally unbounded verifier gains no information beyond the statement's validity. The soundness property, however, typically relies on computational assumptions.

  • Adaptive Soundness: A strong security property for NIZKs (and other cryptographic protocols). It ensures that a malicious prover cannot create a convincing proof for a false statement, even if it has seen the common reference string (CRS) and can choose the statement to be proven after observing the CRS. This contrasts with static soundness, where the statement must be chosen before seeing the CRS.

  • Quantum Black-Box Reduction: A security reduction technique used in the quantum setting. The reduction treats the adversary as a black box, meaning its internal workings are unknown. The reduction can make classical queries to the adversary (e.g., providing it with challenges and receiving responses), but the adversary itself operates using quantum computations. The restriction to classical queries is crucial; the paper explores the challenges of extending this to quantum queries.

  • Falsifiable Assumption: A cryptographic assumption that can be disproven through experimentation. If the assumption is false, there exists an efficient algorithm (either classical or quantum, depending on the context) that can break the underlying hard problem. The existence of such an algorithm can be demonstrated through practical experimentation, unlike non-falsifiable assumptions which may be inherently unfalsifiable. In the context of this paper, the focus is on quantum falsifiable assumptions, where the efficient algorithm may be a quantum algorithm.

  • Quantum Falsifiable Assumptions: A cryptographic assumption where the existence of an efficient quantum algorithm to break the assumption can be demonstrated experimentally. The assumption is defined through an interactive game between a quantum polynomial-time (QPT) challenger and an adversary. The challenger may interact quantumly with the adversary and output a bit indicating if the adversary has broken the assumption. The assumption is considered true if no QPT adversary can cause the challenger to output 1 with a probability significantly higher than a specified threshold.

  • Hard Languages: Languages in NP (Nondeterministic Polynomial time) that are computationally hard to decide. Specifically, for a hard language L, there exist efficiently sampleable distributions of statements in L and statements not in L that are computationally indistinguishable by quantum polynomial-time (QPT) algorithms, even with quantum advice. The existence of such languages is linked to the existence of post-quantum one-way functions.

  • Quantum Statistical Non-Interactive Zero-Knowledge Arguments (Quantum S-NIZKs): A quantum version of S-NIZKs. The setup, prover, and verifier algorithms are all quantum polynomial-time (QPT) algorithms. The protocol operates with quantum states and communications. It maintains the properties of completeness, adaptive soundness (against QPT provers), and statistical adaptive zero-knowledge (with a quantum simulator).

  • Quantum Oracle Algorithms: Algorithms that interact with an oracle (a subroutine with unknown internal workings) using quantum operations. The algorithm can make queries to the oracle, receive responses, and incorporate these responses into its quantum computations. The paper focuses on algorithms that make classical queries to the oracle, even though the algorithm itself is quantum. This means the algorithm doesn't query the oracle in superposition.

  • CAP (Cryptographic Accumulator with Proofs): A cryptographic primitive that allows one to accumulate a set of values into a single value, called an accumulator. The accumulator can be used to efficiently prove membership or non-membership of elements in the set. In the context of this paper, CAPs are discussed in terms of their quantum-resistant properties, where proofs must be verifiable even against quantum adversaries.

  • Q-CAP (Quantum Cryptographic Accumulator with Proofs): An extension of CAP to the quantum setting. Here, the accumulator and the proofs are designed to withstand quantum attacks. This includes ensuring that the proofs remain valid and efficient in a world where quantum computers could potentially solve classical cryptographic problems. Q-CAPs might involve quantum-resistant hash functions, quantum state commitments, or other quantum cryptographic techniques to maintain security properties like collision resistance and the ability to generate and verify proofs in a quantum environment.

Approach

  1. Outline of Research Methodology: The researchers extend the classical impossibility results of Pass by adapting the meta-reduction paradigm to the quantum setting. They analyze the implications of quantum computations and communication on the security of S-NIZK protocols. The methodology involves theoretical proofs and the establishment of impossibility results based on the properties of quantum information.

  2. Problem-Solving Techniques: The authors demonstrate that for any S-NIZK protocol for an NP-complete language, its adaptive soundness cannot be reduced to a falsifiable assumption using a quantum black-box reduction. They employ a combination of theoretical analysis and proofs to establish this result, utilizing the meta-reduction paradigm to show that existing techniques are insufficient in the quantum context.

Results and Evaluation

Theorem 1 (Informal)

Let Π be a non-interactive quantum protocol for an NP-complete language L, satisfying the statistical zero-knowledge property.

Let R be a quantum black-box reduction that has classical access to an attacker A, such that for every attacker A that breaks the adaptive soundness of Π, R_A breaks some falsifiable assumption C.

Then, under the assumption of the existence of post-quantum one-way functions, assumption C is false.

  1. Key Findings: The main result is that no quantum black-box reduction can prove the security of S-NIZK protocols with adaptive soundness based on any falsifiable assumption. The authors provide a clear theoretical framework that illustrates these limitations.

  2. Quantitative Results: The paper does not present numerical data in the traditional sense, as it focuses on theoretical proofs: It establishes the impossibility of constructing adaptive sound S-NIZK protocols from falsifiable assumptions, asserting that for any such protocol Π, if there exists a quantum black-box reduction R that breaks some assumption C, then C must be false.

  3. Notable Achievements: The research extends classical results to the quantum domain, providing a clearer understanding of the limitations of quantum cryptography in constructing secure protocols. The Q-CAP notation system for categorizing quantum reductions is IMHO the most significant contribution, offering a structured way to analyze the relationships between various cryptographic primitives in the quantum context.

Limitations, Assumptions, and Caveats: The research assumes the existence of post-quantum one-way functions and focuses on NP-complete languages. The findings may not generalize to other classes of problems or cryptographic primitives.

Conflict of Interest: The authors disclose no apparent conflicts of interest that could affect the integrity of the research. Their affiliations with Portland State University and funding from the US National Science Foundation indicate a focus on advancing knowledge in the field rather than commercial interests.

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