Plaintext-Ciphertext Matrix Multiplication and FHE Bootstrapping: Fast and Fused

Homomorphically multiplying a plaintext matrix with a ciphertext matrix (PC-MM) is a central task for the private evaluation of transformers, commonly used for large language models. We provide several RLWE-based algorithms for PC-MM that consist of multiplications of plaintext matrices (PP-MM) and comparatively cheap pre-processing and post-processing steps: for small and large dimensions compared to the RLWE ring degree, and with and without precomputation. For the algorithms with precomputation, we show how to perform a PC-MM with a single floating-point PP-MM of the same dimensions. This is particularly meaningful for practical purposes as a floating-point PP-MM can be implemented using high-performance BLAS libraries. The algorithms rely on the multi-secret variant of RLWE, which allows to represent multiple ciphertexts more compactly. We give algorithms to convert from usual shared-secret RLWE ciphertexts to multi-secret ciphertexts and back. Further, we show that this format is compatible with homomorphic addition, plaintext-ciphertext multiplication, and key-switching. This in turn allows us to accelerate the slots-to-coeffs and coeffs-to-slots steps of CKKS bootstrapping when several ciphertexts are bootstrapped at once. Combining batch-bootstrapping with efficient PC-MM results in MaMBo(Matrix Multiplication Bootstrapping), a bootstrapping algorithm that can perform a PC-MM for a limited overhead.