New Permutation Decomposition Techniques for Efficient Homomorphic Permutation

Homomorphic permutation is fundamental to privacy-preserving computations based on batch-encoding homomorphic encryption. It underpins nearly all homomorphic matrix operations and predominantly influences their complexity. Permutation decomposition as a potential approach to optimize this critical component remains underexplored. In this paper, we propose novel decomposition techniques to optimize homomorphic permutations, advancing homomorphic encryption-based privacy-preserving computations. We start by defining an ideal decomposition form for permutations and propose an algorithm searching for depth-1 ideal decompositions. Based on this, we prove the full-depth ideal decomposability of permutations used in specific homomorphic matrix transposition (HMT) and multiplication (HMM) algorithms, allowing them to achieve asymptotic improvement in speed and rotation key reduction. As a demonstration of applicability, substituting the HMM components in the best-known inference framework of encrypted neural networks with our enhanced version shows up to $3.9\times$ reduction in latency. We further devise a new method for computing arbitrary homomorphic permutations, specifically those with weak structures that cannot be ideally decomposed. We design a network structure that deviates from the conventional scope of decomposition and outperforms the state-of-the-art technique with a speed-up of up to $1.69\times$ under a minimal rotation key requirement.