Homomorphic permutation is fundamental to privacy-preserving computations based on batch-encoding homomorphic encryption. It underpins nearly all homomorphic matrix operation algorithms and predominantly influences their complexity. Permutation decomposition as a potential approach to optimize this critical component remains underexplored. In this paper, we enhance the efficiency of homomorphic permutations through novel decomposition techniques, advancing homomorphic encryption-based privacy-preserving computations. We start by estimating the ideal effect of decompositions on permutations, then propose an algorithm that searches depth-1 ideal decomposition solutions. This helps us ascertain the full-depth ideal decomposability of permutations used in specific secure matrix transposition and multiplication schemes, allowing them to achieve asymptotic improvement in speed and rotation key reduction. We further devise a new method for computing arbitrary homomorphic permutations, considering that permutations with weak structures are unlikely to be ideally factorized. Our design deviates from the conventional scope of decomposition. But it better approximates the ideal effect of decomposition we define than the state-of-the-art techniques, with a speed-up of up to $\times 2.27$ under minimal rotation key requirements.

翻译：同态置换是基于批编码同态加密的隐私保护计算的基础。它支撑着几乎所有同态矩阵运算算法，并主要影响其复杂度。置换分解作为优化这一关键组件的潜在方法，目前仍未得到充分探索。本文通过新颖的分解技术提升同态置换的效率，从而推动基于同态加密的隐私保护计算。我们首先估计分解对置换的理想效果，随后提出一种搜索深度为1的理想分解解的算法。这帮助我们确定了特定安全矩阵转置与乘法方案中所用置换的满深度理想可分解性，使其在速度和旋转密钥减少方面实现渐进性改进。考虑到弱结构置换不太可能被理想分解，我们进一步设计了一种计算任意同态置换的新方法。我们的设计偏离了传统的分解范畴，但相比现有技术能更接近我们所定义的分解理想效果，在最小旋转密钥要求下可实现高达$\times 2.27$的加速。