Welcome to the resource topic for 2023/1488
Title:
SCALLOP-HD: group action from 2-dimensional isogenies
Authors: Mingjie Chen, Antonin Leroux
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 \textnormal{Cl}(\mathfrak{O}) on \mathfrak{O}-oriented curves where \mathfrak{O} = \mathbb{Z}[f\sqrt{-1}] for a large prime f as SCALLOP, we introduce a different orientation representation: The new representation embeds an endomorphism generating \mathfrak{O} in a 2^e-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 to be confirmed by implementation results. Additionally, our approach streamlines parameter selection. The new representation allows us to select efficiently a class group \textnormal{Cl}(\mathfrak{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.
ePrint: https://eprint.iacr.org/2023/1488
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