Secure Outsourced Matrix Multiplication with Fully Homomorphic Encryption

Fully Homomorphic Encryption (FHE) is a powerful cryptographic tool that enables the handling of sensitive encrypted data in untrusted computing environments. This capability allows for the outsourcing of computational tasks, effectively addressing security and privacy concerns. This paper studies the secure matrix multiplication problem, a fundamental operation used in various outsourced computing applications such as statistical analysis and machine learning. We propose a novel method to solve the secure matrix multiplication \(A_{m\times l}\times B_{l\times n}\) with arbitrary dimensions, which requires only O(l) rotations and \(\min (m,l,n)\) homomorphic multiplications. In comparison to the state-of-the-art method [14], our approach stands out by achieving a remarkable reduction in the number of rotations by a factor of \(O(\log \max (l,n))\) , as well as a reduction in the number of homomorphic multiplications by a factor of \(O(l/\min (m,l,n))\) . We implemented [14, 21], and our method using the BGV scheme supported by the HElib library. Experimental results show that our scheme has the best performance for matrix multiplication of any dimension. For example, for \(A_{16\times 128}\times B_{128\times 4}=C_{16\times 4}\) , the runtime of our method is 32 s, while both [14, 21] take 569 seconds.