Finding Orientations of Supersingular Elliptic Curves and Quaternion Orders

Sarah Arpin; James Clements; Pierrick Dartois; Jonathan Komada Eriksen; Péter Kutas; Benjamin Wesolowski

Paper 2023/1268

Finding Orientations of Supersingular Elliptic Curves and Quaternion Orders

Sarah Arpin, Leiden University

James Clements, University of Bristol

Pierrick Dartois, Inria Bordeaux - Sud-Ouest Research Centre

Jonathan Komada Eriksen, Norwegian University of Science and Technology

Péter Kutas, Eötvös Loránd University, University of Birmingham

Benjamin Wesolowski, École Normale Supérieure de Lyon

Abstract

Orientations of supersingular elliptic curves encode the information of an endomorphism of the curve. Computing the full endomorphism ring is a known hard problem, so one might consider how hard it is to find one such orientation. We prove that access to an oracle which tells if an elliptic curve is -orientable for a fixed imaginary quadratic order provides non-trivial information towards computing an endomorphism corresponding to the -orientation. We provide explicit algorithms and in-depth complexity analysis. We also consider the question in terms of quaternion algebras. We provide algorithms which compute an embedding of a fixed imaginary quadratic order into a maximal order of the quaternion algebra ramified at and . We provide code implementations in Sagemath which is efficient for finding embeddings of imaginary quadratic orders of discriminants up to , even for cryptographically sized .

Metadata

Available format(s): PDF
Category: Public-key cryptography
Publication info: Preprint.
Keywords: isogeny-based cryptography public-key cryptography cryptanalysis
Contact author(s): s a arpin @ math leidenuniv nl
james clements @ bristol ac uk
pierrick dartois @ u-bordeaux fr
jonathan k eriksen @ ntnu no
p kutas @ bham ac uk
benjamin wesolowski @ math u-bordeaux fr
History: 2023-08-24: approved; 2023-08-22: received; See all versions
Short URL: https://ia.cr/2023/1268
License: CC BY

BibTeX

@misc{cryptoeprint:2023/1268,
      author = {Sarah Arpin and James Clements and Pierrick Dartois and Jonathan Komada Eriksen and Péter Kutas and Benjamin Wesolowski},
      title = {Finding Orientations of Supersingular Elliptic Curves and Quaternion Orders},
      howpublished = {Cryptology {ePrint} Archive, Paper 2023/1268},
      year = {2023},
      url = {https://eprint.iacr.org/2023/1268}
}