New Permutation Decomposition Techniques for Efficient Homomorphic Permutation

TL;DR

This paper introduces novel permutation decomposition techniques that optimize homomorphic permutations, significantly improving the efficiency of privacy-preserving computations in homomorphic encryption, especially for neural network inference.

Contribution

It proposes new decomposition algorithms for homomorphic permutations, proves their effectiveness for specific matrix operations, and demonstrates practical speed-ups in encrypted neural network inference.

Findings

Up to 3.9× latency reduction in encrypted neural network inference.

Achieved asymptotic speed improvements in homomorphic matrix transposition and multiplication.

Outperformed state-of-the-art permutation methods with up to 1.69× speed-up.

Abstract

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.