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Optimal Tightness for Chain-Based Unique Signatures

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Advances in Cryptology – EUROCRYPT 2022 (EUROCRYPT 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13276))

Abstract

Unique signatures are digital signatures with exactly one unique and valid signature for each message. The security reduction for most unique signatures has a natural reduction loss (in the existentially unforgeable against chosen-message attacks, namely EUF-CMA, security model under a non-interactive hardness assumption). In Crypto 2017, Guo et al. proposed a particular chain-based unique signature scheme where each unique signature is composed of n BLS signatures computed sequentially like a blockchain. Under the computational Diffie-Hellman assumption, their reduction loss is \(n\cdot q_H^{1/n}\) for \(q_H\) hash queries and it is logarithmically tight when \(n=\log {q_H}\). However, it is currently unknown whether a better reduction than logarithmical tightness for the chain-based unique signatures exists.

We show that the proposed chain-based unique signature scheme by Guo et al. must have the reduction loss \(q^{1/n}\) for q signature queries when each unique signature consists of n BLS signatures. We use a meta reduction to prove this lower bound in the EUF-CMA security model under any non-interactive hardness assumption, and the meta-reduction is also applicable in the random oracle model. We also give a security reduction with reduction loss \(4\cdot q^{1/n}\) for the chain-based unique signature scheme (in the EUF-CMA security model under the CDH assumption). This improves significantly on previous reduction loss \(n\cdot q_H^{1/n}\) that is logarithmically tight at most. The core of our reduction idea is a non-uniform simulation that is specially invented for the chain-based unique signature construction.

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Notes

  1. 1.

    In 2012, Kakvi and Kiltz [37] introduced a conceptual level RSA-FDH scheme with unique signatures and a tight security reduction.

  2. 2.

    We meant reductions against general adversaries. It is worth noting that BLS-like unique signatures can be proved tight security in the Algebraic Group Model [22] when adversaries are restricted in algebraic operations.

  3. 3.

    To implement such a non-uniform choice, we firstly randomly choose an integer \(w\in [1,2^{n+1}]\). Then we find the integer i satisfying \(2^{i}\le w < 2^{i+1}\) and set \(c_m=i\). It is not hard to verify that \( \Pr [w\leftarrow _R [1,2^{n+1}]: 2^{i}\le w < 2^{i+1}]={2^i}/{2^{n+1}}=\Pr [c_m=i]\).

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Acknowledgement

We would like to thank Tibor Jager for insightful discussions on the first version of this work in 2020. We would also like to thank the anonymous reviewers from Eurocrypt 2021, Crypto 2021, and Eurocrypt 2022 for their constructive comments.

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  1. Institute of Cybersecurity and Cryptology (iC²), School of Computing and Information Technology, University of Wollongong, Wollongong, NSW, Australia

    Fuchun Guo & Willy Susilo

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  1. Fuchun Guo
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  1. University of Haifa, Haifa, Haifa, Israel

    Orr Dunkelman

  2. University of Warsaw, Warsaw, Poland

    Stefan Dziembowski

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Guo, F., Susilo, W. (2022). Optimal Tightness for Chain-Based Unique Signatures. In: Dunkelman, O., Dziembowski, S. (eds) Advances in Cryptology – EUROCRYPT 2022. EUROCRYPT 2022. Lecture Notes in Computer Science, vol 13276. Springer, Cham. https://doi.org/10.1007/978-3-031-07085-3_19

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