International Association for Cryptologic Research

International Association
for Cryptologic Research

CryptoDB

SCALLOP-HD: group action from 2-dimensional isogenies

Authors:
Mingjie Chen , University of Birmingham
Antonin Leroux , DGA-MI, Bruz, France; IRMAR, UMR 6625, Université de Rennes
Lorenz Panny , Technical University of Munich
Download:
Search ePrint
Search Google
Presentation: Slides
Conference: PKC 2024
Abstract: We present SCALLOP-HD, a novel group action that builds upon the recent SCALLOP group action introduced by De Feo, Fouotsa, Kutas, Leroux, Merz, Panny and Wesolowski in 2023. While our group action uses the same action of the class group Cl(O) on O-oriented curves where O=Z[f−d] for a large prime f and small d as SCALLOP, we introduce a different orientation representation: The new representation embeds an endomorphism generating O in a 2e-isogeny between abelian varieties of dimension 2 with Kani's Lemma, and this representation comes with a simple algorithm to compute the class group action. Our new approach considerably simplifies the SCALLOP framework, potentially surpassing it in efficiency — a claim supported by preliminary implementation results in SageMath. Additionally, our approach streamlines parameter selection. The new representation allows us to select efficiently a class group Cl(O) of smooth order, enabling polynomial-time generation of the lattice of relation, hence enhancing scalability in contrast to SCALLOP. To instantiate our SCALLOP-HD group action, we introduce a new technique to apply Kani's Lemma in dimension 2 with an isogeny diamond obtained from commuting endomorphisms. This method allows one to represent arbitrary endomorphisms with isogenies in dimension 2, and may be of independent interest.
BibTeX
@inproceedings{pkc-2024-33812,
  title={SCALLOP-HD: group action from 2-dimensional isogenies},
  publisher={Springer-Verlag},
  author={Mingjie Chen and Antonin Leroux and Lorenz Panny},
  year=2024
}