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Tight Parallel Repetition Theorem for Efficient Quantum Interactive Protocols


Core Concepts
The authors prove a tight parallel repetition theorem for 3-message computationally-secure quantum interactive protocols between an efficient challenger and an efficient adversary. They also show that the security of 4-message computationally secure protocols does not generally decrease under parallel repetition.
Abstract
The paper presents several key results: Tight Parallel Repetition Theorem for 3-Message Quantum Protocols: The authors prove that if a 3-message quantum protocol has computational soundness 𝛾, then its 𝑘-fold parallel repetition has computational soundness 𝛾^𝑘 + negl(𝜆), where 𝜆 is the security parameter. This is tight, as the 𝛾^𝑘 term is inherent. The proof combines techniques from quantum rewinding, the quantum singular value transform, and other new ideas to handle the challenges of the quantum setting. Barriers to Parallel Repetition Beyond 3 Messages: The authors show that their 3-message parallel repetition theorem cannot extend to 4-message protocols under reasonable cryptographic assumptions, mirroring a classical result. They construct a 4-message post-quantum protocol whose 𝑘-fold computational security does not decrease, assuming the existence of post-quantum non-malleable commitments. Round Compression for Quantum Arguments: The authors show that all quantum argument systems can be generically compiled into an equivalent 3-message argument system, preserving the communication and verifier complexity. This allows them to obtain a general round-preserving soundness amplification theorem for quantum arguments. Applications: The authors derive hardness amplification theorems for various quantum cryptographic primitives, including quantum bit commitments, EFI pairs, public-key quantum money, and quantum zero-knowledge arguments. They also obtain a quantum analogue of Yao's XOR lemma as a corollary. Overall, the paper makes significant advances in the theory of hardness amplification for quantum cryptographic primitives, with applications to a wide range of quantum constructions.
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Key Insights Distilled From

by John Bostanc... at arxiv.org 04-18-2024

https://arxiv.org/pdf/2311.10681.pdf
An efficient quantum parallel repetition theorem and applications

Deeper Inquiries

How can the techniques developed in this work be extended to handle quantum protocols with more than 4 messages

To extend the techniques developed in this work to handle quantum protocols with more than 4 messages, we can build upon the insights gained from the non-uniform and uniform reductions for 2-fold parallel repetition. For protocols with more than 4 messages, we would need to consider a more intricate structure of the quantum states and operations involved in the protocol. This would involve analyzing the entanglement and interactions between the various registers in the protocol, as well as devising efficient methods for post-selection and singular value threshold measurements for higher-dimensional systems. One approach could be to generalize the alternating projectors technique to handle a larger number of messages by iteratively applying projective measurements and unitaries to simulate the protocol's execution. This iterative process would need to carefully manage the entanglement and coherence of the quantum states to ensure the success of the reduction. Additionally, leveraging advanced quantum algorithmic tools such as the quantum singular value transformation (QSVT) could help in efficiently transforming and manipulating the quantum states to achieve the necessary post-selection conditions for higher-fold repetitions. By adapting and optimizing these techniques for protocols with more messages, we can potentially extend the parallel repetition results to a broader class of quantum protocols.

Are there other quantum cryptographic primitives beyond the ones considered in this work that could benefit from the hardness amplification results

Beyond the quantum cryptographic primitives considered in this work, such as quantum commitments, quantum money schemes, and quantum arguments, there are several other primitives that could benefit from the hardness amplification results obtained in this study. Some of these primitives include: Quantum Key Distribution (QKD): Hardness amplification techniques could enhance the security guarantees of QKD protocols by strengthening the resistance against quantum attacks and eavesdropping attempts. By amplifying the security parameters of QKD schemes, it may be possible to achieve higher levels of quantum key secrecy and reliability. Quantum Oblivious Transfer (QOT): Hardness amplification could be applied to quantum oblivious transfer protocols to enhance the privacy and security of information transfer between parties. By amplifying the security of QOT schemes, it may be possible to achieve stronger guarantees of information confidentiality and integrity. Quantum Secure Multi-Party Computation (QSMPC): Hardness amplification techniques could be utilized to improve the security of quantum multi-party computation protocols. By amplifying the security of QSMPC schemes, it may be possible to enhance the privacy and correctness of computations performed by multiple parties without revealing sensitive information. By applying the parallel repetition and round compression results to these and other quantum cryptographic primitives, it is possible to strengthen the security and resilience of quantum communication and computation protocols in various application scenarios.

What are the implications of the quantum parallel repetition and round compression results for the complexity-theoretic foundations of quantum cryptography

The quantum parallel repetition and round compression results presented in this work have significant implications for the complexity-theoretic foundations of quantum cryptography. Complexity of Quantum Protocols: The results provide insights into the complexity of quantum interactive protocols and the amplification of security guarantees in quantum cryptographic primitives. By demonstrating the feasibility of parallel repetition and round compression for quantum protocols, the study sheds light on the computational hardness and resilience of quantum cryptographic schemes. Quantum Security Amplification: The results offer a systematic approach to enhancing the security of quantum cryptographic primitives through hardness amplification. By showing that the security of quantum protocols can be exponentially amplified through repetition and compression techniques, the study contributes to the development of robust and secure quantum communication protocols. Foundations of Quantum Cryptography: The findings contribute to the theoretical foundations of quantum cryptography by establishing new techniques for strengthening the security of quantum protocols. The results pave the way for further research in quantum complexity theory and cryptographic protocol design, offering new avenues for enhancing the security of quantum communication and computation systems.
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