Post-quantum cryptography from special abelian varieties with complex multiplication
Post-quantum commutative group actions from oriented superspecial abelian varieties
Cryptography and Security
Summary
Some cryptographic methods rely on mathematical structures called abelian varieties, which are shapes defined by a lot of symmetry. This paper studies a special kind of these shapes, called superspecial abelian varieties, equipped with extra symmetry from complex multiplication (CM). By exploring how certain group actions act on these shapes, the authors develop new techniques potentially useful for making cryptography secure against quantum computers. They provide tools to represent these structures and compute the group actions efficiently, which could help build better post-quantum cryptographic systems.
What this means in practice
- •For post-quantum cryptography engineers: Use new representations and computational tools for Shimura class group actions to design quantum-resistant cryptographic schemes based on abelian varieties.$Commercial implications: Enables development of novel quantum-resistant cryptographic products aimed at securing sensitive data infrastructures against quantum attacks.
- •For cryptographic implementation teams: Implement efficient algorithms for managing complex multiplication orientations and isogeny computations to improve post-quantum cryptographic protocol performance.
Authors
Wouter Castryck, Bogdan Adrian Dina, Péter Kutas, Gioella Lorenzon
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.