Optimized Homomorphic Vector Permutation From New Decomposition Techniques

Homomorphic permutations are fundamental to privacy-preserving computations based on word-wise homomorphic encryptions, which can be accelerated through permutation decomposition. This paper defines an ideal performance of any decomposition on permutations and designs algorithms to achieve this bound. We start by proposing an algorithm searching depth-1 ideal decomposition solutions for permutations. This allows us to ascertain the full-depth ideal decomposability of two types of permutations used in specific homomorphic matrix transposition (SIGSAC 18) and multiplication (CCSW 22), enabling these algorithms to achieve asymptotic improvement in speed and rotation key reduction. We further devise a new strategy for homomorphically computing arbitrary permutations, aiming to approximate the performance limits of ideal decomposition, as permutations with weak structures are unlikely to be ideally factorized. Our design deviates from the conventional scope of permutation decomposition and surpasses state-of-the-art techniques (EUROCRYPT 12, CRYPTO 14) with a speed-up of $\times 1.05 \sim \times 2.27$ under minimum requirement of rotation keys.