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Post-quantum \(\kappa \)-to-1 trapdoor claw-free functions from extrapolated dihedral cosets

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Abstract

Noisy trapdoor claw-free function (NTCF) is a powerful post-quantum cryptographic tool that can efficiently constrain actions of untrusted quantum devices within a classical–quantum interactive cryptographic model. Although NTCF is powerful, its essence remains a 2-to-1 one-way function (NTCF\(^1_2\)), which is inefficient in some cryptographic tasks. This raises an intriguing question: Can NTCF be extended to higher dimensions based on standard cryptographic hardness assumptions? Inspired by the extrapolated dihedral cosets, this work focuses on developing many-to-one trapdoor claw-free functions with polynomially bounded preimage sizes. The main results can be summarized as follows: Firstly, we introduce the definition of \(\kappa \)-to-1 NTCF\(^1_{\kappa }\) where \(\kappa \) is a polynomial integer, and present an efficient construction of NTCF\(^1_{\kappa }\) assuming quantum hardness of the learning with errors (LWE) problem. Secondly, we illustrate a key application of NTCFs in establishing a reduction from the LWE problem to the dihedral coset problems (DCPs). Specifically, our approach, leveraging NTCF\(^1_2\) (resp. NTCF\(^1_{\kappa }\)), reveals a new quantum reduction pathway from the LWE problem to the DCP (resp. an extrapolated version of DCP). This reduction is the core cryptographic analysis tool for studying the resistance of lattice problems against quantum attacks. Finally, we demonstrate that NTCF\(^1_{\kappa }\) can be further reduced to NTCF\(^1_2\), thus preserving its usefulness in proofs of quantumness.

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Notes

  1. QLWE assumes that the LWE problem is hard for any quantum polynomial-time algorithms.

  2. The term “proofs of quantumness” also known as “quantum supremacy,” is to demonstrate the quantum computational advantage.

  3. In fact, the bit c is evaluated by \(c=d\cdot (\mathcal {J}({\varvec{x}}_0) \oplus \mathcal {J}({\varvec{x}}_1))\) in [15], where \(\mathcal {J}(\cdot )\) is the binary representation function. For simplicity, we omit this function in the expression.

  4. Note that adding a polynomial-time quantum circuit in the quantum prover’s end is an intuitively reasonable assumption, which imposes no additional requirements on the computational power of the verifier and the prover compared to the original protocol in [15].

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Acknowledgements

The authors deeply thank Weiqiang Wen for many insightful exchanges and discussions. This work was supported by the National Key Research and Development Program of China (No. 2022YFB2702701), and the National Natural Science Foundation of China (NSFC) (No. 61972050).

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Authors and Affiliations

  1. State Key Laboratory of Networking and Switching Technology, Beijing University of Posts and Telecommunications, Beijing, 100876, China

    Xingyu Yan, Lize Gu & Jingwen Suo

  2. School of Cyberspace Science and Technology, Beijing Institute of Technology, Beijing, 100081, China

    Licheng Wang

  3. State Key Laboratory of Information Security, Institute of Information Engineering, UCAS, Beijing, 100049, China

    Ziyi Li

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Yan, X., Wang, L., Gu, L. et al. Post-quantum \(\kappa \)-to-1 trapdoor claw-free functions from extrapolated dihedral cosets. Quantum Inf Process 23, 188 (2024). https://doi.org/10.1007/s11128-024-04387-w

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