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A Note on the Security of CSIDH

  • Jean-François Biasse
  • Annamaria Iezzi
  • Michael J. JacobsonJr.
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11356)

Abstract

We propose a quantum algorithm for computing an isogeny between two elliptic curves \(E_1,E_2\) defined over a finite field such that there is an imaginary quadratic order \(\mathcal {O}\) satisfying \(\mathcal {O}\simeq {\text {End}}(E_i)\) for \(i = 1,2\). This concerns ordinary curves and supersingular curves defined over \(\mathbb {F}_p\) (the latter used in the recent CSIDH proposal). Our algorithm has heuristic asymptotic run time \(e^{O\left( \sqrt{\log (|\varDelta |)}\right) }\) and requires polynomial quantum memory and \(e^{O\left( \sqrt{\log (|\varDelta |)}\right) }\) quantumly accessible classical memory, where \(\varDelta \) is the discriminant of \(\mathcal {O}\). This asymptotic complexity outperforms all other available methods for computing isogenies.

We also show that a variant of our method has asymptotic run time \(e^{\tilde{O}\left( \sqrt{\log (|\varDelta |)}\right) }\) while requesting only polynomial memory (both quantum and classical).

Notes

Acknowledgments

The authors thank Léo Ducas for useful comments on the memory requirements of the BKZ algorithm. The authors thank Noah Stephens-Davidowitz for information on the resolution of the approximate CVP. The authors also thank Tanja Lange and Benjamin Smith for useful comments on an earlier version of this draft.

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Jean-François Biasse
    • 1
  • Annamaria Iezzi
    • 1
  • Michael J. JacobsonJr.
    • 2
  1. 1.Department of Mathematics and StatisticsUniversity of South FloridaTampaUSA
  2. 2.Department of Computer ScienceUniversity of CalgaryCalgaryCanada

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