Constructing One-Way Puzzles from Quantum-Average-Hardness of the GapK Problem

Core Concepts

Quantum-average-hardness of the GapK problem implies the existence of one-way puzzles, a fundamental primitive in quantum cryptography.

Abstract

The paper presents a construction of one-way puzzles (OWPuzzs) from the assumption of quantum-average-hardness of the GapK problem.

The key insights are:

1. Quantum-average-hardness of the GapK problem implies quantum-average-hardness of the probability estimation problem.
2. Quantum-average-hardness of the probability estimation problem implies the existence of OWPuzzs.

The GapK problem is a promise problem to decide whether a given bit string has a small Kolmogorov complexity or not. Its quantum-average-hardness means that an instance is sampled from a quantum-polynomial-time-samplable distribution, and no quantum-polynomial-time algorithm can solve the problem with high probability.

The authors show that this assumption, which is studied in the field of meta-complexity, can be used to construct OWPuzzs, a fundamental primitive in quantum cryptography. This is the first time a "Microcrypt" primitive is constructed based on concrete hardness assumptions that do not seem to imply one-way functions.

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Key Insights Distilled From

Quantum Cryptography from Meta-Complexity

by Taiga Hiroka... at arxiv.org 10-03-2024

https://arxiv.org/pdf/2410.01369.pdf

Deeper Inquiries

What other quantum cryptographic primitives could potentially be constructed from quantum-average-hardness assumptions beyond one-way puzzles?

Beyond one-way puzzles (OWPuzzs), several other quantum cryptographic primitives could potentially be constructed from quantum-average-hardness assumptions. These include:

Pseudorandom Unitaries (PRUs): PRUs are quantum analogues of classical pseudorandom generators. They can be constructed using quantum-average-hardness assumptions, as they require the ability to produce indistinguishable quantum states from truly random states. If a quantum-average-hardness assumption can ensure that distinguishing between these states is hard, PRUs can be realized.

Pseudorandom Function-like State Generators (PRFSGs): Similar to PRUs, PRFSGs can be developed from quantum-average-hardness assumptions. They generate quantum states that behave like pseudorandom functions, providing a foundation for secure quantum communication protocols.

Quantum Money Schemes: Quantum money schemes, which rely on the ability to create and verify quantum states that represent currency, could be constructed from quantum-average-hardness assumptions. If the hardness of distinguishing between valid and invalid quantum states can be guaranteed, secure quantum money systems can be established.

Quantum Digital Signatures: The construction of quantum digital signatures could also benefit from quantum-average-hardness assumptions. If it can be shown that certain quantum states are hard to forge or replicate, secure digital signatures can be implemented.

Commitment Schemes: Commitment schemes, which allow one party to commit to a value while keeping it hidden until a later time, could be derived from quantum-average-hardness assumptions. The security of these schemes can be enhanced if the underlying assumptions ensure that the committed value cannot be efficiently revealed or altered.

Multiparty Computation Protocols: Quantum-average-hardness assumptions could also facilitate the development of secure multiparty computation protocols, where multiple parties compute a function over their inputs while keeping those inputs private. The hardness of certain quantum problems can ensure that no party can gain an advantage in learning others' inputs.

These primitives leverage the unique properties of quantum mechanics and the computational challenges posed by quantum-average-hardness assumptions, providing a rich landscape for future research in quantum cryptography.

How might the quantum-average-hardness assumption on the GapK problem relate to or differ from other quantum complexity assumptions, such as those involving quantum advantage or quantum supremacy?

The quantum-average-hardness assumption on the GapK problem presents a unique perspective in the landscape of quantum complexity assumptions. Here’s how it relates to and differs from other assumptions:

Relation to Quantum Advantage: Quantum advantage refers to scenarios where quantum algorithms outperform classical algorithms for specific tasks. The quantum-average-hardness of the GapK problem suggests that certain instances are hard for quantum algorithms to solve, indicating a form of quantum advantage in the context of Kolmogorov complexity. This contrasts with general quantum advantage, which may not specifically address the hardness of problems related to complexity measures.

Comparison with Quantum Supremacy: Quantum supremacy is the demonstration that a quantum computer can perform a calculation that is infeasible for classical computers. While quantum supremacy focuses on specific computational tasks, the quantum-average-hardness of the GapK problem is more about the inherent difficulty of certain problems in quantum complexity theory. The GapK problem's hardness does not necessarily imply that a quantum computer can outperform classical computers in all tasks, but rather that it cannot efficiently solve specific instances of the problem.

Implications for Cryptographic Security: The quantum-average-hardness assumption on the GapK problem provides a foundation for constructing cryptographic primitives that do not rely on one-way functions. This is significant because many existing cryptographic systems are built on the assumption of one-way functions, which may not hold in a quantum context. In contrast, other quantum complexity assumptions may not directly lead to new cryptographic constructions or may still imply the existence of one-way functions.

Broader Context of Complexity Theory: The quantum-average-hardness of the GapK problem fits within a broader framework of complexity theory, where various hardness assumptions are explored. It provides a specific instance of a problem that can be studied in relation to other complexity classes, such as BQP (bounded-error quantum polynomial time) and QMA (quantum Merlin-Arthur). This specificity allows researchers to draw connections between different areas of quantum complexity and cryptography.

In summary, while the quantum-average-hardness assumption on the GapK problem shares some commonalities with other quantum complexity assumptions, it also presents distinct characteristics that make it particularly relevant for the development of new quantum cryptographic primitives.

Could the techniques used to derive one-way puzzles from quantum-average-hardness be extended to construct other cryptographic primitives that do not seem to imply one-way functions?

Yes, the techniques used to derive one-way puzzles (OWPuzzs) from quantum-average-hardness assumptions can potentially be extended to construct other cryptographic primitives that do not seem to imply one-way functions. Here are several ways in which this can be achieved:

Framework of Reductions: The construction of OWPuzzs from quantum-average-hardness involves a series of reductions that demonstrate how the hardness of one problem can imply the existence of another primitive. This framework can be adapted to show that other primitives, such as pseudorandom generators or quantum money schemes, can be constructed under similar hardness assumptions without relying on one-way functions.

Utilization of Quantum-Average-Hardness: The quantum-average-hardness of problems like GapK can be leveraged to create primitives that require a different set of security guarantees. For instance, if a primitive can be shown to be secure against adversaries that cannot efficiently solve instances of the GapK problem, it may be possible to construct secure protocols for commitments or digital signatures.

Exploration of Alternative Hardness Assumptions: The techniques used in the derivation of OWPuzzs can inspire the exploration of other hardness assumptions that do not imply one-way functions. By identifying problems that exhibit quantum-average-hardness characteristics, researchers can develop new primitives that are secure under these assumptions.

Generalization of Techniques: The methods employed to establish the relationship between quantum-average-hardness and OWPuzzs can be generalized to other cryptographic settings. For example, the construction of secure multiparty computation protocols could benefit from similar techniques, where the hardness of a problem ensures that no participant can gain an advantage in learning others' inputs.

Interplay with Other Cryptographic Concepts: The insights gained from the construction of OWPuzzs can be applied to other cryptographic concepts, such as zero-knowledge proofs or secure function evaluation. By understanding how quantum-average-hardness can be utilized to ensure security, researchers can create new protocols that do not rely on traditional assumptions like one-way functions.

In conclusion, the techniques used to derive one-way puzzles from quantum-average-hardness assumptions provide a valuable foundation for constructing a variety of other cryptographic primitives. This approach opens up new avenues for research in quantum cryptography, allowing for the development of secure systems that are resilient to the challenges posed by quantum computing.