FHE4DMM: A Low-Latency Distributed Matrix Multiplication With Fully Homomorphic Encryption

Fully Homomorphic Encryption (FHE) is a promising technology for secure, non-interactive outsourced computation. One notable method to increase the throughput of FHE-based outsourcing is batching, which typically involves large-scale matrix-matrix multiplications (MM). However, the substantial overhead inherent in existing FHE schemes poses a major challenge for processing these large-scale tasks, often resulting in insufficient memory or prolonged delays on a single machine, making it practically unviable. Utilizing multi-machine parallelism in cloud clusters for outsourced computation offers a natural solution to these obstacles. In this work, we propose FHE4DMM, a distributed algorithm that provides a unified view on encrypted matrices, accommodating various FHE schemes and any matrix dimensions, to accelerate large-scale encrypted MM. A key innovation is its reuse optimizations for parallelized homomorphic computations, which can offer valuable insights for broader FHE-based applications. We utilized FHE4DMM to conduct large-scale square ($4096\times 4096$) and rectangular ($32768\times 32768,32768\times 16$ ) matrix multiplications on 256 machines, achieving computation time of 172.2 s and 76.1 s, respectively, while ensuring a 128-bit security level. For scalability, the experiments demonstrate that FHE4DMM achieves linear speedup for $2^{i}$ ($i$ is from 0 to 6) machines across various matrix dimension cases. In addition, within the range of matrix dimensions that the state-of-the-art (SOTA) distributed FHE-MM algorithm (Huang et al. 2023) can handle, FHE4DMM attains a maximum speedup of 16.62x. To assess its practical performance, FHE4DMM is applied in a basic multi-layer feedforward network. We used 64 machines to perform secure outsourced inference on MNIST and CIFAR-10 datasets with encrypted models and data. Compared to using the SOTA, our method achieved speedups of up to 3.54x and 4.22x respectively, with the MM module obtaining a 4.09x and 4.87x speedup.