Paper 2025/543
Models of Kummer lines and Galois representations
, Institut de Mathématiques de Bordeaux, Institut de Mathématiques de Bordeaux, Inria Bordeaux - Sud-Ouest Research Centre, Institut de Mathématiques de BordeauxAbstract
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 $x$-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 $T$ of $2$-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 $T$ 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 $2$-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 cryptographyKummer linestheta functionsmodular curvesGalois 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}
}
