HOMOMORPHIC ENCRYPTION AND ALGEBRAIC GEOMETRY FOR PRIVACY-PRESERVING MACHINE LEARNING

Homomorphic encryption combined with algebraic geometry is emerging as one of the most mathematically powerful strategies for enabling privacy-preserving machine learning in environments where data confidentiality cannot be compromised. Traditional cryptographic methods protect data only at rest or in transit, but expose it during computation, creating substantial vulnerability in modern AI pipelines. Homomorphic encryption enables computation directly on encrypted inputs, while algebraic geometry provides the structural foundation for constructing efficient polynomial representations, ciphertext rings, and error-tolerant operations required by encrypted learning algorithms. This paper examines the integration of lattice-based homomorphic schemes with algebraic-geometric tools such as ideal lattices, algebraic curves, and Gröbner-basis methods to support encrypted inference and training. The analysis focuses on three core challenges: minimizing noise growth during encrypted computation, reducing model complexity for polynomial-friendly transformations, and preserving accuracy while enforcing strict privacy guarantees. The study argues that algebraic-geometric optimisation significantly improves the feasibility of encrypted neural networks, encrypted linear models, and encrypted gradient updates, especially for cloud-hosted and multi-party learning environments. By demonstrating how these mathematical frameworks interact, the paper positions homomorphic encryption and algebraic geometry as a critical foundation for future secure AI systems capable of operating without exposing sensitive information at any stage of computation.