Post-quantum commutative group actions from oriented superspecial abelian varieties

Wouter Castryck; Bogdan Adrian Dina; Péter Kutas; Gioella Lorenzon

Paper 2026/2555

Post-quantum commutative group actions from oriented superspecial abelian varieties

Wouter Castryck

, KU Leuven

Bogdan Adrian Dina

, Eötvös Loránd University, University of Haifa

Péter Kutas

, Eötvös Loránd University, University of Birmingham

Gioella Lorenzon

, KU Leuven

Abstract

We study the cryptographic use of Shimura class group actions on superspecial principally polarized abelian varieties equipped with a CM orientation, within the context of post-quantum cryptography. Besides giving an accessible introduction to the topic, illustrated with many examples, we provide tools for representing CM orientations through higher-dimensional isogenies and for computing the Shimura class group action in the restricted effective group action model. Here, key auxiliary results are a version of Kani's lemma "with CM" and a trick that often allows to halve the ambient dimension. We also give a method for finding superspecial principally polarized abelian surfaces that are oriented by a given CM order, which is extracted from an algebraic framework for studying CM orientations using the Ibukiyama-Katsura-Oort correspondence. We also analyze obstructions to transitivity of the Shimura class group action, related to the characteristic-zero notion of CM type.

Metadata

Available format(s): PDF
Category: Public-key cryptography
Publication info: Preprint.
Keywords: isogeny principally polarized abelian variety superspecial group action complex multiplication key exchange
Contact author(s): wouter castryck @ esat kuleuven be
bogdanadrian dina @ gmail com
kutasp @ gmail com
gioella lorenzon @ esat kuleuven be
History: 2026-10-20: approved; 2026-10-19: received; See all versions
Short URL: https://ia.cr/2026/2555
License: CC BY

BibTeX

@misc{cryptoeprint:2026/2555,
      author = {Wouter Castryck and Bogdan Adrian Dina and Péter Kutas and Gioella Lorenzon},
      title = {Post-quantum commutative group actions from oriented superspecial abelian varieties},
      howpublished = {Cryptology {ePrint} Archive, Paper 2026/2555},
      year = {2026},
      url = {https://eprint.iacr.org/2026/2555}
}