Post-quantum commutative group actions from oriented superspecial abelian varieties

Wouter Castryck, Bogdan Adrian Dina, Péter Kutas, Gioella Lorenzon · 2026 · arXiv

DOI: 10.48550/arXiv.2610.27922 · open-access PDF

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.

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