inzicht - Quantum Computing - # Post-Quantum Cryptography

Quantum-Computable Trapdoor One-Way Functions Exist Even if P=NP (Relative to an Oracle)

Q: What are the practical implications of this research for the development of post-quantum cryptography, considering the reliance on an oracle in the current construction?

While groundbreaking from a theoretical standpoint, the reliance on a classical oracle in this research poses significant limitations to its practical implications for post-quantum cryptography. Here's why: Oracles are theoretical tools: Oracles are powerful theoretical constructs that provide black-box access to potentially intractable computations. However, they don't translate directly to real-world scenarios. We don't know of any concrete cryptographic primitives that can realistically instantiate the behavior of the oracle used in this research. Absence of concrete candidates: The paper itself acknowledges the lack of known non-oracular problems in BQP (Bounded-Error Quantum Polynomial Time) that are hard for PH (Polynomial Hierarchy). This absence makes it challenging to find real-world candidates for building practical cryptographic systems based on these principles. Bridging the gap between theory and practice: The paper hints at potential pathways, such as exploring quantum-computable classical cryptography instantiations based on assumptions like #P-hardness and quantum advantage conjectures. However, substantial further research is needed to bridge the gap between these theoretical possibilities and practical implementations. In essence, while this research provides exciting theoretical insights into the potential of quantum-computable one-way functions, it doesn't immediately translate to practical post-quantum cryptographic solutions. The focus now shifts towards identifying concrete, non-oracular problems that exhibit the necessary properties to instantiate these theoretical concepts in the real world.

Q: Could there be alternative explanations for the existence of quantum-computable one-way functions in a P=NP world that do not rely on the inherent computational differences between classical and quantum computers?

While the paper focuses on the inherent computational differences between classical and quantum computers as the primary explanation, exploring alternative explanations for the existence of quantum-computable one-way functions in a P=NP world is an intriguing avenue for future research. Some potential alternative hypotheses could involve: Exploiting structural properties of problems: Instead of solely relying on computational hardness, could there be specific structural properties of certain problems that make them amenable to efficient quantum evaluation but hard for classical inversion, even in a P=NP world? Leveraging quantum information theory: Could we devise quantum-computable one-way functions whose security relies on principles of quantum information theory, such as entanglement or no-cloning, in a way that circumvents the limitations imposed by P=NP? Hybrid approaches: Could we combine computational assumptions with information-theoretic principles to construct quantum-computable one-way functions that remain secure even if P=NP? It's important to note that these are speculative alternatives, and further investigation is needed to determine their plausibility and potential for yielding secure cryptographic constructions.

Q: How might the development of quantum algorithms for NP-complete problems influence the feasibility of building practical cryptographic systems based on the principles outlined in this paper?

The development of quantum algorithms for NP-complete problems would significantly impact the feasibility of building practical cryptographic systems based on the principles outlined in this paper. Here's how: Invalidation of the oracle separation: The core result of the paper relies on the existence of a classical oracle relative to which P=NP, but quantum-computable one-way functions exist. If a quantum algorithm for NP-complete problems were discovered, it would imply P=NP in the real world, invalidating the oracle separation. Undermining the foundation: The security of the proposed quantum-computable one-way functions hinges on the presumed hardness of inverting them for classical computers, even if P=NP. If NP-complete problems become efficiently solvable by quantum computers, this foundational assumption crumbles. Shifting the focus: The discovery of such algorithms would necessitate a fundamental shift in the search for post-quantum cryptographic primitives. Instead of relying on the separation between BQP and PH, we would need to explore alternative avenues that remain secure even in a world where quantum computers can efficiently solve NP-complete problems. In conclusion, the development of quantum algorithms for NP-complete problems would be a significant development with far-reaching consequences, rendering the specific constructions in this paper insecure. However, it would also open up new research directions in post-quantum cryptography, pushing us to explore alternative foundations and assumptions for building secure cryptographic systems in a world with significantly more powerful quantum computers.

Belangrijkste concepten

This paper demonstrates that even if P=NP, there exists a specially constructed oracle relative to which quantum computers could still have a cryptographic advantage over classical computers, enabling functionalities like public-key encryption and digital signatures.

Samenvatting

Bibliographic Information: Kretschmer, W., Qian, L., & Tal, A. (2024). Quantum-Computable One-Way Functions without One-Way Functions. arXiv preprint arXiv:2411.02554.
Research Objective: This paper investigates the possibility of achieving quantum cryptography with classical communication in a world where P=NP by constructing quantum-computable trapdoor one-way functions relative to a classical oracle.
Methodology: The authors develop a novel oracle construction based on a specific encoding of a random oracle using the Forrelation problem. This encoding allows quantum algorithms to efficiently evaluate the oracle while remaining indistinguishable from random to classical adversaries, even with access to an NP oracle. The core technical innovation is a new distributional block-insensitivity lemma for AC0 circuits, which forms the basis for proving the security of their construction.
Key Findings: The paper presents a construction of a quantum-computable trapdoor one-way function relative to a classical oracle where P=NP. This implies the existence of other cryptographic primitives like public-key encryption, digital signatures, and oblivious transfer in the same oracle world. Additionally, the construction yields many-copy-secure pseudorandom states and pseudorandom unitaries, strengthening previous oracle separation results.
Main Conclusions: The existence of quantum-computable trapdoor one-way functions in a P=NP world suggests that quantum computers could retain some cryptographic advantage even if classical cryptography collapses. This highlights a potential path towards building cryptographic systems based on the computational limitations of classical computers even in the face of significant algorithmic breakthroughs.
Significance: This work significantly advances our understanding of the relationship between classical and quantum cryptography, particularly in worlds where P=NP. It provides a theoretical foundation for exploring new cryptographic assumptions and constructions that leverage the unique capabilities of quantum computation.
Limitations and Future Research: The current construction relies on an oracle and does not directly translate to concrete cryptographic assumptions in the real world. Future research could explore instantiating these primitives under concrete hardness assumptions or investigating the possibility of stronger separations, such as separating quantum-computable collision-resistant hash functions from P=NP.

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Quantum-Computable One-Way Functions without One-Way Functions

by William Kret... om arxiv.org 11-06-2024

https://arxiv.org/pdf/2411.02554.pdf

Quantum-Computable One-Way Functions without One-Way Functions

Diepere vragen

What are the practical implications of this research for the development of post-quantum cryptography, considering the reliance on an oracle in the current construction?

While groundbreaking from a theoretical standpoint, the reliance on a classical oracle in this research poses significant limitations to its practical implications for post-quantum cryptography. Here's why:

Oracles are theoretical tools: Oracles are powerful theoretical constructs that provide black-box access to potentially intractable computations. However, they don't translate directly to real-world scenarios. We don't know of any concrete cryptographic primitives that can realistically instantiate the behavior of the oracle used in this research.
Absence of concrete candidates: The paper itself acknowledges the lack of known non-oracular problems in BQP (Bounded-Error Quantum Polynomial Time) that are hard for PH (Polynomial Hierarchy). This absence makes it challenging to find real-world candidates for building practical cryptographic systems based on these principles.
Bridging the gap between theory and practice:  The paper hints at potential pathways, such as exploring quantum-computable classical cryptography instantiations based on assumptions like #P-hardness and quantum advantage conjectures. However, substantial further research is needed to bridge the gap between these theoretical possibilities and practical implementations.
In essence, while this research provides exciting theoretical insights into the potential of quantum-computable one-way functions, it doesn't immediately translate to practical post-quantum cryptographic solutions. The focus now shifts towards identifying concrete, non-oracular problems that exhibit the necessary properties to instantiate these theoretical concepts in the real world.

Could there be alternative explanations for the existence of quantum-computable one-way functions in a P=NP world that do not rely on the inherent computational differences between classical and quantum computers?

While the paper focuses on the inherent computational differences between classical and quantum computers as the primary explanation, exploring alternative explanations for the existence of quantum-computable one-way functions in a P=NP world is an intriguing avenue for future research. Some potential alternative hypotheses could involve:

Exploiting structural properties of problems:  Instead of solely relying on computational hardness, could there be specific structural properties of certain problems that make them amenable to efficient quantum evaluation but hard for classical inversion, even in a P=NP world?
Leveraging quantum information theory:  Could we devise quantum-computable one-way functions whose security relies on principles of quantum information theory, such as entanglement or no-cloning, in a way that circumvents the limitations imposed by P=NP?
Hybrid approaches: Could we combine computational assumptions with information-theoretic principles to construct quantum-computable one-way functions that remain secure even if P=NP?
It's important to note that these are speculative alternatives, and further investigation is needed to determine their plausibility and potential for yielding secure cryptographic constructions.

How might the development of quantum algorithms for NP-complete problems influence the feasibility of building practical cryptographic systems based on the principles outlined in this paper?

The development of quantum algorithms for NP-complete problems would significantly impact the feasibility of building practical cryptographic systems based on the principles outlined in this paper. Here's how:

Invalidation of the oracle separation: The core result of the paper relies on the existence of a classical oracle relative to which P=NP, but quantum-computable one-way functions exist. If a quantum algorithm for NP-complete problems were discovered, it would imply P=NP in the real world, invalidating the oracle separation.
Undermining the foundation: The security of the proposed quantum-computable one-way functions hinges on the presumed hardness of inverting them for classical computers, even if P=NP. If NP-complete problems become efficiently solvable by quantum computers, this foundational assumption crumbles.
Shifting the focus: The discovery of such algorithms would necessitate a fundamental shift in the search for post-quantum cryptographic primitives.  Instead of relying on the separation between BQP and PH, we would need to explore alternative avenues that remain secure even in a world where quantum computers can efficiently solve NP-complete problems.
In conclusion, the development of quantum algorithms for NP-complete problems would be a significant development with far-reaching consequences, rendering the specific constructions in this paper insecure. However, it would also open up new research directions in post-quantum cryptography, pushing us to explore alternative foundations and assumptions for building secure cryptographic systems in a world with significantly more powerful quantum computers.