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Exponential Quantum One-Wayness and EFI Pairs

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Security and Cryptography for Networks (SCN 2024)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14973))

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Abstract

In classical cryptography, one-way functions are widely considered to be the minimal computational assumption. However, when taking quantum information into account, the situation is more nuanced. There are currently two major candidates for the minimal assumption: the search quantum generalization of one-way functions are one-way state generators (OWSG), whereas the decisional variant are EFI pairs. A well-known open problem in quantum cryptography is to understand how these two primitives are related. A recent breakthrough result of Khurana and Tomer (STOC’24) shows that OWSGs imply EFI pairs, for the restricted case of pure states.

In this work, we make progress towards understanding the general case. To this end, we define the notion of inefficiently-verifiable one-way state generators (IV-OWSGs), where the verification algorithm is not required to be efficient, and show that these are precisely equivalent to EFI pairs, with an exponential loss in the reduction. Significantly, this equivalence holds also for mixed states. Thus our work establishes the following relations among these fundamental primitives of quantum cryptography:

$$ \text {(mixed) OWSGs} \implies \text {(mixed) IV-OWSGs} \equiv _\text {exp} \text {EFI pairs}, $$

where \(\equiv _\text {exp}\) denotes equivalence up to exponential security of the primitives.

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Notes

  1. 1.

    Without loss of generality, the \(\textsf{KeyGen}\) algorithm takes the following form: apply a QPT unitary to generate a superposition \(\sum _k \sqrt{\Pr [k\leftarrow \textsf{KeyGen}(1^\lambda )]}|k\rangle |\textrm{junk}_k\rangle \), measure the first register, and output the measurement result.

  2. 2.

    Without loss of generality, \(\textsf{StateGen}\) takes the following form: on input k, apply a QPT unitary \(U_k\) on \(|0...0\rangle \) to generate a pure state \(|\varPhi _k\rangle _{\textbf{A},\textbf{B}}=U_k|0...0\rangle \) and output the first register \(\textbf{A}\), which is in state \(\phi _k = \textrm{Tr}_{\textbf{B}}(|\varPhi _k\rangle \!\langle \varPhi _k|)\). Then the \(\textbf{A}\) registers of \(|\varPhi _k^{\otimes t}\rangle \) make up \(\textbf{R}_2\), while the \(\textbf{B}\) registers make up \(\textbf{C}_2\).

  3. 3.

    In fact, any any function \(f(\lambda )=\omega (\log \lambda )\) suffices.

  4. 4.

    A very recent work [4] shows that canonical quantum bit commitment schemes that satisfy computational hiding and computational \((1-1/{\textsf{poly}}(\lambda ))\)-binding are sufficient for constructing EFI pairs. We do not need this result.

  5. 5.

    The original OWSGs introduced in [20] were defined to have pure state outputs, but this was later generalized to allow for mixed state outputs [19].

  6. 6.

    It works even for \(t=0\).

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Acknowledgements

We thank anonymous reviewers for their valuable comments. GM was supported by the European Research Council through an ERC Starting Grant (Grant agreement No. 101077455, ObfusQation). GM and MW acknowledge support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy - EXC 2092 CASA - 390781972. MW also acknowledges support by the European Research Council through an ERC Starting Grant (grant agreement No. 101040907, SYMOPTIC), by the NWO through grant OCENW.KLEIN.267, and by the BMBF through project Quantum Methods and Benchmarks for Resource Allocation (QuBRA). TM is supported by JST CREST JPMJCR23I3, JST Moonshot JPMJMS2061-5-1-1, JST FOREST, MEXT QLEAP, the Grant-in Aid for Transformative Research Areas (A) 21H05183, and the Grant-in-Aid for Scientific Research (A) No.22H00522.

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Authors and Affiliations

  1. Bocconi University, Milan, Italy

    Giulio Malavolta

  2. Max Planck Institute for Security and Privacy, Bochum, Germany

    Giulio Malavolta

  3. Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto, Japan

    Tomoyuki Morimae & Takashi Yamakawa

  4. Faculty of Computer Science, Ruhr University Bochum, Bochum, Germany

    Michael Walter

  5. NTT Social Informatics Laboratories, Tokyo, Japan

    Takashi Yamakawa

  6. NTT Research Center for Theoretical Quantum Information, Atsugi, Japan

    Takashi Yamakawa

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  1. Giulio Malavolta
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  2. Tomoyuki Morimae
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  4. Takashi Yamakawa
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Correspondence to Takashi Yamakawa .

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Editors and Affiliations

  1. Università degli Studi di Salerno, Fisciano, Italy

    Clemente Galdi

  2. Télécom Paris, Paris, France

    Duong Hieu Phan

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Malavolta, G., Morimae, T., Walter, M., Yamakawa, T. (2024). Exponential Quantum One-Wayness and EFI Pairs. In: Galdi, C., Phan, D.H. (eds) Security and Cryptography for Networks. SCN 2024. Lecture Notes in Computer Science, vol 14973. Springer, Cham. https://doi.org/10.1007/978-3-031-71070-4_6

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