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arXiv · 2610.27922

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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Wouter Castryck, Bogdan Adrian Dina, Péter Kutas, Gioella Lorenzon. 2026-10-20. Post-quantum commutative group actions from oriented superspecial abelian varieties. https://arxiv.org/abs/2610.27922

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