Generalized class group actions on oriented elliptic curves with level structure

Sarah Arpin; Wouter Castryck; Jonathan Komada Eriksen; Gioella Lorenzon; Frederik Vercauteren

Paper 2024/1172

Generalized class group actions on oriented elliptic curves with level structure

Sarah Arpin

, Leiden University

Wouter Castryck

, KU Leuven

Jonathan Komada Eriksen

, Norwegian University of Science and Technology

Gioella Lorenzon

, KU Leuven

Frederik Vercauteren

, KU Leuven

Abstract

We study a large family of generalized class groups of imaginary quadratic orders $O$ and prove that they act freely and (essentially) transitively on the set of primitively $O$-oriented elliptic curves over a field $k$ (assuming this set is non-empty) equipped with appropriate level structure. This extends, in several ways, a recent observation due to Galbraith, Perrin and Voloch for the ray class group. We show that this leads to a reinterpretation of the action of the class group of a suborder $O' \subseteq O$ on the set of $O'$-oriented elliptic curves, discuss several other examples, and briefly comment on the hardness of the corresponding vectorization problems.

Metadata

Available format(s): PDF
Category: Public-key cryptography
Publication info: Preprint.
Keywords: elliptic curves level structure orientations class group action
Contact author(s): sarpinmath @ gmail com
History: 2024-07-22: approved; 2024-07-19: received; See all versions
Short URL: https://ia.cr/2024/1172
License: CC BY

BibTeX

@misc{cryptoeprint:2024/1172,
      author = {Sarah Arpin and Wouter Castryck and Jonathan Komada Eriksen and Gioella Lorenzon and Frederik Vercauteren},
      title = {Generalized class group actions on oriented elliptic curves with level structure},
      howpublished = {Cryptology ePrint Archive, Paper 2024/1172},
      year = {2024},
      note = {\url{https://eprint.iacr.org/2024/1172}},
      url = {https://eprint.iacr.org/2024/1172}
}