Merits of geometric algebra applied to cryptography and machine learning.

We present Clifford fully homomorphic encryption (FHE), the first ring-learning with errors (RLWE)-based FHE scheme with native support for all Clifford algebra operations, enabling privacy-preserving computation on geometric data. Unlike conventional FHE schemes that flatten geometric structure into scalar operations, our approach maintains the algebraic properties of multi-vectors through homomorphic geometric products. Building on this foundation, we introduce geometric neural networks that operate directly on encrypted multi-vectors, achieving the first-ever demonstration of privacy-preserving geometric deep learning. Our construction encrypts three-dimensional point clouds as Cl(3,0) multi-vectors and performs encrypted classification with less than 1% accuracy loss compared to plaintext, completing inference in under 60 s. These results demonstrate that geometric algebra (GA) provides unique advantages for both cryptographic constructions (enabling structure-preserving encryption) and machine learning (ML) (enabling privacy-preserving geometric learning), opening new pathways at the intersection of cryptography, ML and applied mathematics. This article is part of the theme issue 'Modern applications of geometric algebra'.