Paper 2026/2380

The Graded Monoidal Action for Cryptography

Abstract

We give a new framework for constructing post-quantum protocols based on graded monoidal actions. An essential difference between the graded monoidal action framework and the cryptographic group action framework is that our higher-rank problems give rise to infinite structures, where elements do not admit inverses, while cryptographic group actions are finite by definition. Nevertheless, we show that the hard problems for cryptographic group actions reduces to the corresponding family of hard problems for graded monoidal actions. However, the main motivation is that for general graded monoidal actions, the relevant problems admit no known subexponential quantum attacks, while still having enough structure to formulate some protocols based on cryptographic group actions, such as the Diffie-Hellman key-exchange. Our framework can, to the extent we make clear in the paper, be instantiated with the module-action on oriented principally polarized abelian varieties (PPAV's), generalizing the class group action on oriented elliptic curves. Although the theory involving the module-action is quite involved, the formulation of a graded monoidal action is strikingly simple, making it a suitable abstraction for constructing protocols in, which can then be instantiated with the module-action.