Secure Dimensionality Reduction: Applying Linear Discriminant Analysis over the TFHE Homomorphic Encryption Scheme

In the context of the proliferating AI landscape, driven by entities like ChatGPT, the demand for extensive data utilization for training has surged, raising concerns about unauthorized data aggregation and privacy breaches. Paradoxically, this surge in data consumption has raised a critical concern - the breach of privacy stemming from unauthorized data aggregation. Sensitive data categories, including credit card details, medical records, and geographical locations, are particularly vulnerable to misuse. Homomorphic Encryption (HE), a post-quantum attack-resistant cryptographic technique, addresses this concern by enabling secure computations on encrypted data. However, HE's potential is hindered by limitations in evaluation speed, particularly evident in high-dimensional data analysis. This paper introduces contributions, including efficient inverse matrix computation, tailored eigenvector extraction via the power method for the TFHE scheme, and eigenvalue calculation using the Rayleigh quotient within TFHE. The feasibility of applying LDA in the encrypted domain is demonstrated using Fast Fully Homomorphic Encryption over the Torus (TFHE) scheme.