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Generating Supersingular Elliptic Curves over \(\mathbb {F}_p\) with Unknown Endomorphism Ring

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Progress in Cryptology – INDOCRYPT 2023 (INDOCRYPT 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14459))

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Abstract

A number of supersingular isogeny based cryptographic protocols require the endomorphism ring of the initial elliptic curve to be either unknown or random in order to be secure. To instantiate these protocols, Basso et al. recently proposed a secure multiparty protocol that generates supersingular elliptic curves defined over \(\mathbb {F}_{p^2}\) of unknown endomorphism ring as long as at least one party acts honestly. However, there are many protocols that specifically require curves defined over \(\mathbb {F}_p\), for which the Basso et al. protocol cannot be used. Also, the simple solution of using a signature scheme such as CSI-FiSh or SeaSign for proof of knowledge either requires extensive precomputation of large ideal class groups or is too slow for everyday applications.

In this paper, we present CSIDH-SCG, a new multiparty protocol that generates curves of unknown endomorphism ring defined over \(\mathbb {F}_p\). This protocol relies on CSIDH-ROIP, a new CSIDH based proof of knowledge. We also present CSIDH-CR, a multiparty algorithm that be used in conjunction with CSIDH-SCG to generate a random curve over \(\mathbb {F}_p\) while still keeping the endomorphism ring unknown.

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Acknowledgments

We would like to thank Andrea Basso for helpful comments on a draft version of this paper. This research was supported by NSERC Alliance Consortia Quantum Grant ALLRP 578463–2022.

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Authors and Affiliations

  1. Department of Combinatorics and Optimization, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

    Youcef Mokrani & David Jao

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  1. Youcef Mokrani
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  2. David Jao
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Correspondence to Youcef Mokrani .

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Editors and Affiliations

  1. Nanyang Technological University, Singapore, Singapore

    Anupam Chattopadhyay

  2. Nanyang Technological University, Singapore, Singapore

    Shivam Bhasin

  3. Radboud University, Nijmegen, The Netherlands

    Stjepan Picek

  4. Indian Institute of Technology Madras, Chennai, India

    Chester Rebeiro

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Mokrani, Y., Jao, D. (2024). Generating Supersingular Elliptic Curves over \(\mathbb {F}_p\) with Unknown Endomorphism Ring. In: Chattopadhyay, A., Bhasin, S., Picek, S., Rebeiro, C. (eds) Progress in Cryptology – INDOCRYPT 2023. INDOCRYPT 2023. Lecture Notes in Computer Science, vol 14459. Springer, Cham. https://doi.org/10.1007/978-3-031-56232-7_8

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  • DOI: https://doi.org/10.1007/978-3-031-56232-7_8

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