Homomorphic vector permutation is fundamental to privacy-preserving computations based on batch-encoded homomorphic encryption, underpinning nearly all homomorphic matrix operation algorithms and predominantly influencing their complexity. A potential approach to optimize this critical component lies in permutation decomposition, a technique we consider as not yet fully explored. In this paper, we enhance the efficiency of homomorphic permutations through novel decomposition techniques, thus advancing privacy-preserving computations. We start by estimating the ideal performance of decompositions on permutations and proposing an algorithm that searches depth-1 ideal decomposition solutions. This enables us to ascertain the full-depth ideal decomposability of specific permutations in homomorphic matrix transposition (SIGSAC 18) and multiplication (CCSW 22), allowing these privacy-preserving computations to achieve asymptotic improvement in speed and rotation key reduction. We further devise a new method for computing arbitrary homomorphic permutations, aiming to approximate the performance of ideal decomposition, as permutations with weak structures are unlikely to be ideally factorized. Our design deviates from the conventional scope of permutation decomposition. It outperforms state-of-the-art techniques (EUROCRYPT 12, CRYPTO 14) with a speed-up of up to $\times2.27$ under the minimum requirement of rotation keys.

翻译：同态向量置换是基于批量编码同态加密的隐私保护计算的基础，支撑着几乎所有同态矩阵运算算法，并主要决定其复杂度。优化这一关键组件的一个潜在途径在于置换分解，我们认为该技术尚未得到充分探索。本文通过新颖的分解技术提升同态置换的效率，从而推动隐私保护计算的发展。我们首先评估分解在置换上的理想性能，并提出一种搜索深度为1的理想分解解的算法。这使得我们能够确定特定置换在同态矩阵转置（SIGSAC 18）和乘法（CCSW 22）中的全深度理想可分解性，从而使这些隐私保护计算在速度和旋转密钥减少方面实现渐进性改进。我们进一步设计了一种计算任意同态置换的新方法，旨在逼近理想分解的性能，因为具有弱结构的置换不太可能被理想分解。我们的设计偏离了置换分解的传统范畴。在满足旋转密钥最低要求的前提下，其性能优于现有先进技术（EUROCRYPT 12, CRYPTO 14），加速比最高可达$\times2.27$。