Models of Kummer lines and Galois representations

Razvan Barbulescu; Damien Robert; Nicolas Sarkis

Paper 2025/543

Models of Kummer lines and Galois representations

Razvan Barbulescu

, Institut de Mathématiques de Bordeaux

Damien Robert

, Institut de Mathématiques de Bordeaux, Inria Bordeaux - Sud-Ouest Research Centre

Nicolas Sarkis

, Institut de Mathématiques de Bordeaux

Abstract

In order to compute a multiple of a point on an elliptic curve in Weierstrass form one can use formulas in only one of the two coordinates of the points. These -only formulas can be seen as an arithmetic on the Kummer line associated to the curve. In this paper, we look at models of Kummer lines, and define an intrinsic notion of isomorphisms of Kummer lines. This allows us to give conversion formulas between Kummer models in a unified manner. When there is one rational point of -torsion on the curve, we also use Mumford's theory of theta groups to show that there are two type of models: the “symmetric” ones with respect to and the “anti-symmetric“ ones. We show how this recovers the Montgomery model and various variants of the theta model. We also classify when curves admit these different models via Galois representations and modular curves. When an elliptic curve is viewed inside a -isogeny volcano, we give a criteria to say if it has a given Kummer model based solely on its position in the volcano. We also give applications to the ECM factorization algorithm.

Note: Clarify the abstract

Metadata

Available format(s): PDF
Category: Foundations
Publication info: Preprint.
Keywords: Elliptic curve cryptography Kummer lines theta functions modular curves Galois representation
Contact author(s): razvan barbulescu @ math u-bordeaux fr
damien robert @ inria fr
nicolas sarkis @ math u-bordeaux fr
History: 2025-03-25: revised; 2025-03-24: received; See all versions
Short URL: https://ia.cr/2025/543
License: CC BY

BibTeX

@misc{cryptoeprint:2025/543,
      author = {Razvan Barbulescu and Damien Robert and Nicolas Sarkis},
      title = {Models of Kummer lines and Galois representations},
      howpublished = {Cryptology {ePrint} Archive, Paper 2025/543},
      year = {2025},
      url = {https://eprint.iacr.org/2025/543}
}