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Sigma Protocols from Verifiable Secret Sharing and Their Applications

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Advances in Cryptology – ASIACRYPT 2023 (ASIACRYPT 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14439))

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Abstract

Sigma protocols are one of the most common and efficient zero-knowledge proofs (ZKPs). Over the decades, a large number of Sigma protocols are proposed, yet few works pay attention to the common design principal. In this work, we propose a generic framework of Sigma protocols for algebraic statements from verifiable secret sharing (VSS) schemes. Our framework provides a general and unified approach to understanding Sigma protocols. It not only neatly explains the classic protocols such as Schnorr, Guillou-Quisquater and Okamoto protocols, but also leads to new Sigma protocols that were not previously known. Furthermore, we show an application of our framework in designing ZKPs for composite statements, which contain both algebraic and non-algebraic statements. We give a generic construction of non-interactive ZKPs for composite statements by combining Sigma protocols from VSS and ZKPs following MPC-in-the-head paradigm in a seamless way via a technique of witness sharing reusing. Our construction has advantages of requiring no “glue” proofs for combining algebraic and non-algebraic statements. By instantiating our construction using Ligero++ (Bhadauria et al., CCS 2020) and designing an associated Sigma protocol from VSS, we obtain a concrete ZKP for composite statements which achieves a tradeoff between running time and proof size, thus resolving the open problem left by Backes et al. (PKC 2019).

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Notes

  1. 1.

    For the sake of convenience, we will not distinguish between computational and information-theoretic soundness, and thus refer to both proofs and arguments as “proofs”.

  2. 2.

    Actually, it is hard to give a more efficient ZKP for composite statement using Ligero/Ligero++ than those using ZKBoo/ZKB++, since the former two protocols reduce the proof size, at the cost of increasing the computation.

  3. 3.

    The value n could even be exponential in security parameter \(\lambda \) (e.g. the size of a finite field).

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Acknowledgements

We thank the anonymous reviewers for their valuable comments. This work was supported by the National Key Research and Development Program of China (Grant No. 2021YFA1000600), the National Natural Science Foundation of China (Grant No. 62272269, No. 61932019, and No. 62372447), Taishan Scholar Program of Shandong Province, and Beijing Natural Science Foundation (Grant No. M22003).

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Authors and Affiliations

  1. School of Cyber Science and Technology, Shandong University, Qingdao, 266237, China

    Min Zhang, Yu Chen & Chuanzhou Yao

  2. State Key Laboratory of Cryptology, P.O. Box 5159, Beijing, 100878, China

    Min Zhang, Yu Chen & Chuanzhou Yao

  3. Key Laboratory of Cryptologic Technology and Information Security of Ministry of Education, Shandong University, Qingdao, 266237, China

    Min Zhang, Yu Chen & Chuanzhou Yao

  4. State Key Laboratory of Information Security, Institute of Information Engineering, Chinese Academy of Sciences, Beijing, 100085, China

    Zhichao Wang

  5. School of Cyber Security, University of Chinese Academy of Sciences, Beijing, 100049, China

    Zhichao Wang

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  1. Min Zhang
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Correspondence to Yu Chen .

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Editors and Affiliations

  1. Nanyang Technological University, Singapore, Singapore

    Jian Guo

  2. Monash University, Melbourne, VIC, Australia

    Ron Steinfeld

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Zhang, M., Chen, Y., Yao, C., Wang, Z. (2023). Sigma Protocols from Verifiable Secret Sharing and Their Applications. In: Guo, J., Steinfeld, R. (eds) Advances in Cryptology – ASIACRYPT 2023. ASIACRYPT 2023. Lecture Notes in Computer Science, vol 14439. Springer, Singapore. https://doi.org/10.1007/978-981-99-8724-5_7

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