Merits of geometric algebra applied to cryptography and machine learning

Area of Science:

• Cryptography
• Machine Learning
• Applied Mathematics

Background:

• Conventional fully homomorphic encryption (FHE) schemes flatten geometric data into scalar operations.
• This approach loses the inherent algebraic properties of geometric structures.
• There is a need for cryptographic methods that preserve geometric data structures.

Purpose of the Study:

• To introduce the first ring-learning with errors (RLWE)-based FHE scheme with native support for Clifford algebra operations.
• To enable privacy-preserving computation on geometric data.
• To demonstrate privacy-preserving geometric deep learning.

Main Methods:

• Developed Clifford FHE, a novel RLWE-based FHE scheme supporting Clifford algebra.
• Utilized homomorphic geometric products to maintain multi-vector algebraic properties.
• Integrated Clifford FHE with geometric neural networks for encrypted geometric deep learning.

Main Results:

• Achieved the first demonstration of privacy-preserving geometric deep learning.
• Encrypted 3D point clouds as Cl(3,0) multi-vectors.
• Performed encrypted classification with <1% accuracy loss and inference in under 60 seconds.

Conclusions:

• Geometric algebra (GA) offers unique advantages for structure-preserving encryption and privacy-preserving geometric learning.
• The Clifford FHE scheme opens new research avenues at the intersection of cryptography, ML, and applied mathematics.
• This work paves the way for secure geometric data analysis and machine learning applications.