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PreprintCC BY-SA 4.0arXiv

Post-quantum commutative group actions from oriented superspecial abelian varieties

  • Wouter Castryck
  • Bogdan Adrian Dina
  • Péter Kutas
  • Gioella Lorenzon
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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.

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Cite

BibTeXRIS
BibTeX
@misc{castryck2026postquantum,
  title = {{Post-quantum commutative group actions from oriented superspecial abelian varieties}},
  author = {Wouter Castryck and Bogdan Adrian Dina and Péter Kutas and Gioella Lorenzon},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.27922},
  eprint = {2610.27922},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.27922},
  note = {CC BY-SA 4.0},
}
RIS
TY  - UNPB
TI  - Post-quantum commutative group actions from oriented superspecial abelian varieties
AU  - Wouter Castryck
AU  - Bogdan Adrian Dina
AU  - Péter Kutas
AU  - Gioella Lorenzon
PY  - 2026
DA  - 2026/10/20
PB  - arXiv
DO  - 10.48550/arxiv.2610.27922
UR  - https://arxiv.org/abs/2610.27922
AB  - 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.
ER  -