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Rational Isogenies from Irrational Endomorphisms

Part of the Lecture Notes in Computer Science book series (LNSC,volume 12106)

Abstract

In this paper, we introduce a polynomial-time algorithm to compute a connecting \(\mathcal {O}\)-ideal between two supersingular elliptic curves over \(\mathbb {F}_p\) with common \(\mathbb {F}_p\)-endomorphism ring \(\mathcal {O}\), given a description of their full endomorphism rings. This algorithm provides a reduction of the security of the CSIDH cryptosystem to the problem of computing endomorphism rings of supersingular elliptic curves. A similar reduction for SIDH appeared at Asiacrypt 2016, but relies on totally different techniques. Furthermore, we also show that any supersingular elliptic curve constructed using the complex-multiplication method can be located precisely in the supersingular isogeny graph by explicitly deriving a path to a known base curve. This result prohibits the use of such curves as a building block for a hash function into the supersingular isogeny graph.

Keywords

  • Isogeny-based cryptography
  • Endomorphism rings
  • CSIDH

Author list in alphabetical order; see https://www.ams.org/profession/leaders/culture/CultureStatement04.pdf. This work was supported in part by the Commission of the European Communities through the Horizon 2020 program under project number 643161 (ECRYPT-NET) and by the Research Council KU Leuven grants C14/18/067 and STG/17/019, and by CyberSecurity Research Flanders with reference number VR20192203. The first listed author was affiliated with the Department of Mathematics at KU Leuven during part of the preparation of this paper.

Date of this document: 2020-02-20.

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Fig. 1.

Notes

  1. 1.

    Unless \(\beta = 0\).

  2. 2.

    After we posted a version of this paper online, we learned that this was observed independently and quasi-simultaneously in [27], with a more elaborate discussion.

  3. 3.

    One could handle the purely inseparable part—powers of \(\pi _E\)—in a unified way by working with scheme-theoretic kernels. Since this issue is only tangential to our work, we will for simplicity avoid this technical complication and deal with \(\pi _E\) explicitly.

  4. 4.

    Unfortunately, the statement of [25, Prop. 3.2] wrongly attributes this description to the quadratic twist of \(E_1\).

  5. 5.

    Here we deviate from our convention that \(\mathbf {i}^2 = -1\) as soon as \(p \equiv 3 \pmod {4}\).

  6. 6.

    Under GRH, Bach [2] proved that \({\text {cl}}(\mathcal {O})\) is generated by prime ideals of norm less than \(C(\log p)^2\) for an explicitly computable small constant C. It is not known unconditionally whether a polynomial bound on the norms suffices.

  7. 7.

    Note that this does not require any assumptions on the output distribution of \(\varDelta (\mathsf a)\), other than that the returned vectors are correct. (The algorithm still takes polynomial time if the oracle \(\varDelta \) only succeeds on an inverse polynomial fraction of inputs.).

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Acknowledgements

The authors would like to thank Benjamin Wesolowski, Robert Granger, Christophe Petit, and Ben Smith for interesting discussions regarding this work, and Lixia Luo for pointing out an error in an earlier version of Lemma 22, as well as a few smaller mistakes. Thanks to Daniel J. Bernstein for providing key insights regarding the proof of Lemma 24.

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Correspondence to Wouter Castryck , Lorenz Panny or Frederik Vercauteren .

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Castryck, W., Panny, L., Vercauteren, F. (2020). Rational Isogenies from Irrational Endomorphisms. In: Canteaut, A., Ishai, Y. (eds) Advances in Cryptology – EUROCRYPT 2020. EUROCRYPT 2020. Lecture Notes in Computer Science(), vol 12106. Springer, Cham. https://doi.org/10.1007/978-3-030-45724-2_18

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