Computing Isomorphisms between Products of Supersingular Elliptic Curves

Pierrick Gaudry; Julien Soumier; Pierre-Jean Spaenlehauer

Paper 2025/136

Computing Isomorphisms between Products of Supersingular Elliptic Curves

Pierrick Gaudry

, Université de Lorraine, CNRS, Inria

Julien Soumier, Université de Lorraine, CNRS, Inria

Pierre-Jean Spaenlehauer

, Université de Lorraine, CNRS, Inria

Abstract

The Deligne-Ogus-Shioda theorem guarantees the existence of isomorphisms between products of supersingular elliptic curves over finite fields. In this paper, we present methods for explicitly computing these isomorphisms in polynomial time, given the endomorphism rings of the curves involved. Our approach leverages the Deuring correspondence, enabling us to reformulate computational isogeny problems into algebraic problems in quaternions. Specifically, we reduce the computation of isomorphisms to solving systems of quadratic and linear equations over the integers derived from norm equations. We develop -adic techniques for solving these equations when we have access to a low discriminant subring. Combining these results leads to the description of an efficient probabilistic Las Vegas algorithm for computing the desired isomorphisms. Under GRH, it is proved to run in expected polynomial time.

Metadata

Available format(s): PDF
Category: Public-key cryptography
Publication info: Preprint.
Keywords: Isogenies Superspecial abelian surfaces
Contact author(s): pierrick gaudry @ loria fr
julien soumier @ inria fr
pierre-jean spaenlehauer @ inria fr
History: 2025-03-28: last of 2 revisions; 2025-01-28: received; See all versions
Short URL: https://ia.cr/2025/136
License: CC BY

BibTeX

@misc{cryptoeprint:2025/136,
      author = {Pierrick Gaudry and Julien Soumier and Pierre-Jean Spaenlehauer},
      title = {Computing Isomorphisms between Products of Supersingular Elliptic Curves},
      howpublished = {Cryptology {ePrint} Archive, Paper 2025/136},
      year = {2025},
      url = {https://eprint.iacr.org/2025/136}
}