Fully Homomorphic Encryption for Matrix Arithmetic

We propose an efficient fully homomorphic encryption (FHE) scheme tailored for matrix arithmetic based on the Ring-Learning with Errors (RLWE) problem. The proposed scheme naturally supports matrix multiplication, addition, and Hadamard multiplication for batched matrices of various sizes over both complex numbers and integers. Encrypted matrix multiplication is reduced to four matrix multiplications of ciphertext elements, without the need for expensive operations such as slot-to-coefficient conversion or ring switching. In addition, the scheme efficiently supports matrix transformations, including general and conjugate transpositions, as well as matrix rotations: inter-matrix rotations across batched matrices and intra-matrix rotations within rows and columns. Moreover, the proposed FHE scheme can be directly combined with existing bootstrapping algorithms. By eliminating the need for expensive operations such as repeated slot rotations and conversion between slot- and coefficient-encoding, the proposed construction achieves significant performance improvements. In our construction, encrypted multiplications of $n\times n$ matrices under slot encoding are decomposed into two parts: (1) matrix multiplication — four $n\times n$ matrix multiplications of ciphertext coefficients, and (2) key switching — with a total cost approximately 2–4 times that of Hadamard multiplication. We implemented the proposed scheme and utilized the FLINT library for the matrix multiplication component. Experimental results demonstrate that, even when leveraging highly optimized implementations, matrix multiplication remains the major cost, indicating that our construction substantially reduces auxiliary overheads and achieves strong overall efficiency.