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Generalized Hybrid Search with Applications to Blockchains and Hash Function Security

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Advances in Cryptology – ASIACRYPT 2024 (ASIACRYPT 2024)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 15492))

Abstract

In this work we first examine the hardness of solving various search problems by hybrid quantum-classical strategies, namely, by algorithms that have both quantum and classical capabilities. We then construct a hybrid quantum-classical search algorithm and analyze its success probability.

Regarding the former, for search problems that are allowed to have multiple solutions and in which the input is sampled according to arbitrary distributions, we establish their hybrid quantum-classical query complexities—i.e., given a fixed number of classical and quantum queries, determine what is the probability of solving the search task. At a technical level, our results generalize the framework for hybrid quantum-classical search algorithms recently proposed by Rosmanis [Ros22]. Namely, for an arbitrary distribution D on Boolean functions, the probability that an algorithm equipped with \(\tau _c\) classical queries and \(\tau _q\) quantum queries succeeds in finding a preimage of 1 for a function sampled from D is at most \(\nu _{D}\cdot (2\sqrt{\tau _c} + 2\tau _q+ 1)^2\), where \(\nu _{D}\) captures the average (over D) fraction of preimages of 1.

Regarding our second contribution, we design a hybrid algorithm which first spends all of its classical queries and in the second stage runs a “modified Grover” in which the initial state depends on the target distribution \({D}\). We then show how to analyze its success probability for arbitrary target distributions and, importantly, its optimality for the uniform and the Bernoulli distribution cases.

As applications of our hardness results, we first revisit and generalize the formal security treatment of the Bitcoin protocol called the Bitcoin backbone [Eurocrypt 2015], to a setting where the adversary has both quantum and classical capabilities, presenting a new hybrid honest majority condition necessary for the protocol to properly operate. Secondly, we re-examine the generic security of hash functions [PKC 2016] against quantum-classical hybrid adversaries.

The full version of the paper can be found at [CGS23].

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Notes

  1. 1.

    We remark that normalization is required as \(\omega \) might not be 1; for example, in the case of the Bernoulli distribution, \(\omega = m \eta \). Hence, normalization is needed so as to view \(w_i/w\) as a probability distribution.

  2. 2.

    Dimensions may grow depending on the arrangement of the pseudo-classical queries.

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Acknowledgements

J.G. was partially supported by NSF SaTC grants no. 2001082 and 2055694. F.S. was partially supported by NSF grant no. 1942706 (CAREER). J.G. and F.S. were also partially support by Sony by means of the Sony Research Award Program. A.C. acknowledges support from the National Science Foundation grant CCF-1813814, from the AFOSR under Award Number FA9550-20-1-0108 and the support of the Quantum Advantage Pathfinder (QAP) project, with grant reference EP/X026167/1 and the UK Engineering and Physical Sciences Research Council.

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Authors and Affiliations

  1. School of Informatics, University of Edinburgh, Edinburgh, UK

    Alexandru Cojocaru

  2. Department of Computer Science and Engineering, Texas A&M University, College Station, USA

    Juan Garay

  3. Department of Computer Science, Portland State University, Portland, USA

    Fang Song

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  1. Alexandru Cojocaru
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Correspondence to Alexandru Cojocaru .

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Editors and Affiliations

  1. Academia Sinica, Taipei, Taiwan

    Kai-Min Chung

  2. NTT Social Informatics Laboratories, Tokyo, Japan

    Yu Sasaki

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Cojocaru, A., Garay, J., Song, F. (2025). Generalized Hybrid Search with Applications to Blockchains and Hash Function Security. In: Chung, KM., Sasaki, Y. (eds) Advances in Cryptology – ASIACRYPT 2024. ASIACRYPT 2024. Lecture Notes in Computer Science, vol 15492. Springer, Singapore. https://doi.org/10.1007/978-981-96-0947-5_3

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