Abelian surfaces in Hesse form and explicit isogeny formulas

Thomas Decru; Sabrina Kunzweiler

Paper 2026/039

Abelian surfaces in Hesse form and explicit isogeny formulas

Thomas Decru

, KU Leuven

Sabrina Kunzweiler

, Inria Bordeaux - Sud-Ouest Research Centre

Abstract

We develop a new method for the computation of $(3,3)$-isogenies between principally polarized abelian surfaces. The idea is to work with models in $\mathbb P^8$ induced by a symmetric level-$3$ theta structure. In this setting, the action of three-torsion points is linear, and the isogeny formulas can be described in a simple way as the composition of easy-to-evaluate maps. In the description of these formulas, the relation with the Burkhardt quartic threefold plays an important role. Furthermore, we discuss generalizations of the idea to higher dimensions as well as different isogeny degrees.

Metadata

Available format(s): PDF
Category: Public-key cryptography
Publication info: Preprint.
Keywords: isogenies post-quantum cryptography abelian surfaces theta structures
Contact author(s): thomas decru @ kuleuven be
sabrina kunzweiler @ math u-bordeaux fr
History: 2026-04-10: revised; 2026-01-09: received; See all versions
Short URL: https://ia.cr/2026/039
License: CC BY

BibTeX

@misc{cryptoeprint:2026/039,
      author = {Thomas Decru and Sabrina Kunzweiler},
      title = {Abelian surfaces in Hesse form and explicit isogeny formulas},
      howpublished = {Cryptology {ePrint} Archive, Paper 2026/039},
      year = {2026},
      url = {https://eprint.iacr.org/2026/039}
}