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The Abe-Okamoto Partially Blind Signature Scheme Revisited

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Advances in Cryptology – ASIACRYPT 2022 (ASIACRYPT 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13794))

Abstract

Partially blind signatures, an extension of ordinary blind signatures, are a primitive with wide applications in e-cash and electronic voting. One of the most efficient schemes to date is the one by Abe and Okamoto (CRYPTO 2000), whose underlying idea—the OR-proof technique—has served as the basis for several works.

We point out several subtle flaws in the original proof of security, and provide a new detailed and rigorous proof, achieving similar bounds as the original work. We believe our insights on the proof strategy will find useful in the security analyses of other OR-proof-based schemes.

J. Kastner—Work done while supported by ERC Project PREP-CRYPTO 724307.

J. Loss—Work done while at University of Maryland.

J. Xu—Work done while at George Mason University.

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Notes

  1. 1.

    Due to the WI property of the scheme, for any \((\textbf{I},\textsf{rand},\overrightarrow{h})\), there exists a corresponding triple \((\textbf{I}',\textsf{rand},\overrightarrow{h})\) that contains the other witness and produces the same transcript as \((\textbf{I},\textsf{rand},\overrightarrow{h})\). This means that the same witness w could have been extracted from a pair of partnering runs \((\textbf{I},\textsf{rand},\overrightarrow{h})\), \((\textbf{I},\textsf{rand},\overrightarrow{h}')\), or from \((\textbf{I}',\textsf{rand},\overrightarrow{h})\), \((\textbf{I}',\textsf{rand},\overrightarrow{h}')\), where one of \(\textbf{I}\) and \(\textbf{I}'\) contains w, and the other instance does not.

  2. 2.

    We stress that simply replacing a triple \((\textbf{I},\textsf{rand},\overrightarrow{h})\) with an indistinguishable triple \((\textbf{I}',\textsf{rand},\overrightarrow{h})\) is not sufficient to solve this problem. Indeed, one might hope that since the adversary can not detect this change, an unsuccessful side may become successful when switching from \(\textbf{I}\) to \(\textbf{I}'\), as the desired witness would flip. However, a successful forking pair \(((\textbf{I}',\textsf{rand},\overrightarrow{h}),(\textbf{I}',\textsf{rand},\overrightarrow{h}'))\) need only exist if \(((\textbf{I},\textsf{rand},\overrightarrow{h}),(\textbf{I},\textsf{rand},\overrightarrow{h}'))\) is a base. The same is not true, in general, for sides, as their endpoints may not (and generally do not) yield the same transcript. Because of this, an unsuccessful side \(((\textbf{I},\textsf{rand},\overrightarrow{h}),(\textbf{I},\textsf{rand},\overrightarrow{h}'))\) might not even be part of a triangle side when switching witnesses from \(\textbf{I}\) to \(\textbf{I}'\).

  3. 3.

    See the top of [4, p. 285], where the reduction’s advantage includes a term \(\eta _1^2\), where \(\eta _1 = \eta /2Q_h^{\ell +1}\) and \(\eta \) is the adversary’s advantage.

  4. 4.

    This is because \(E(\textbf{I}_1,\textsf{rand}_1,\overrightarrow{h}_1) \cap E(\textbf{I}_2,\textsf{rand}_2,\overrightarrow{h}_2) \ne \emptyset \) implies that \(\textbf{I}_1 = \textbf{I}_2\), \(\textsf{rand}_1 = \textsf{rand}_2\), and \(\overrightarrow{e}(\textbf{I}_1,\textsf{rand}_1,\overrightarrow{h}_1) = \overrightarrow{e}(\textbf{I}_2,\textsf{rand}_2,\overrightarrow{h}_2)\), which in turn implies that \(E(\textbf{I}_1,\textsf{rand}_1,\overrightarrow{h}_1) = E(\textbf{I}_2,\textsf{rand}_2,\overrightarrow{h}_2)\).

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Author information

Authors and Affiliations

  1. Department of Computer Science, ETH Zurich, Zurich, Switzerland

    Julia Kastner

  2. CISPA Helmholtz Center for Information Security, Saarbrücken, Germany

    Julian Loss

  3. School of Electrical Engineering and Computer Science, Oregon State University, Corvallis, OR, USA

    Jiayu Xu

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  1. Julia Kastner
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  2. Julian Loss
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Correspondence to Julia Kastner .

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Editors and Affiliations

  1. Indian Institute of Technology Madras, Chennai, India

    Shweta Agrawal

  2. Chinese Academy of Sciences, Beijing, China

    Dongdai Lin

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Kastner, J., Loss, J., Xu, J. (2022). The Abe-Okamoto Partially Blind Signature Scheme Revisited. In: Agrawal, S., Lin, D. (eds) Advances in Cryptology – ASIACRYPT 2022. ASIACRYPT 2022. Lecture Notes in Computer Science, vol 13794. Springer, Cham. https://doi.org/10.1007/978-3-031-22972-5_10

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