Paper 2025/1274

Improved Matrix Inversion with Packed Ciphertexts using Fully Homomorphic Encryption

Abstract

Matrix inversion is a fundamental operation, but performing it over encrypted matrices remains a significant challenge. This is mainly due to the fact that conventional inversion algorithms—such as Gaussian elimination—depend heavily on comparison and division operations, which are computationally expensive to perform under homomorphic encryption. To mitigate this, Ahn et al. (ESORICS 2023) introduced an inversion method based on iterative matrix multiplications. However, their approach encrypts matrices entry-wise, leading to poor scalability. A key limitation of prior work stems from the absence of an efficient matrix multiplication technique for matrix-packed ciphertexts, particularly one with low multiplicative depth. In this paper, we present a novel homomorphic matrix multiplication algorithm optimized for matrix-packed ciphertexts, requiring only a multiplicative depth of two. Building on this foundation, we propose an efficient algorithm for homomorphic matrix inversion. Experimental results show that our method outperforms the state-of-the-art: for $8\times 8$ matrices, it achieves a $6.8\times$ speedup over the method by Ahn et al., and enables inversion of larger matrices that were previously infeasible. We further compare our homomorphic matrix multiplication technique against existing matrix-packed homomorphic matrix multiplication algorithms. When used for iterative inversion, our method consistently outperforms prior approaches. In particular, for $16\times 16$ and $32\times 32$ matrices, it achieves $1.88\times$ and $1.43\times$ speedups, respectively, over the algorithm by Aikata and Roy. Finally, we demonstrate the practical benefits of our method by applying it to privacy-preserving linear regression. For a dataset of $64$ samples with $8$ features, our approach achieves a $1.13\times$ speedup in training time compared to the state-of-the-art homomorphic matrix inversion solution.