Optimized Homomorphic Permutation From New Permutation Decomposition Techniques

2024-10-29

AI Summary

This work addresses the low efficiency of homomorphic permutations and the high overhead of rotation keys in batched homomorphic encryption. We propose a novel permutation optimization framework based on ideal decomposition. Methodologically: (1) we formally define ideal decomposition and design a depth-1 search algorithm; (2) we prove that arbitrary permutations in homomorphic matrix transpose (HMT) and homomorphic matrix multiplication (HMM) are fully depth-ideally decomposable; (3) we construct a lightweight decomposition network architecture, breaking from conventional butterfly/shuffle paradigms. Our key contribution is the first systematic formulation of permutation decomposition as a provable, constructible, and optimizable theoretical pathway. Experiments show that replacing HMM components reduces neural network inference latency by 7.9×, accelerates weak-structured permutation computation by 1.69×, and minimizes the number of required rotation keys.

Technology Category

• Search and Optimization: Learning to Search
• Machine Learning: Hardware-aware ML
• Computer Vision: Learning & Optimization for CV

Application Category

• Security and Privacy: Applications of cryptography
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

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 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 $7.9 imes$ 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 $1.69 imes$ under a minimal rotation key requirement.

Problem

• Optimizing homomorphic permutations for faster privacy-preserving computations
• Improving homomorphic matrix operations via novel decomposition techniques
• Enhancing encrypted neural network efficiency with reduced latency

Innovation

• Novel decomposition techniques optimize homomorphic permutations
• Depth-1 ideal decomposition algorithm improves HMT and HMM
• Network structure speeds up weak-structure permutations