Data privacy concerns are increasing significantly in the context of the Internet of Things, cloud services, edge computing, artificial intelligence applications, and other applications enabled by next-generation networks. Homomorphic encryption addresses privacy challenges by enabling multiple operations to be performed on encrypted messages without decryption. This article comprehensively addresses homomorphic encryption from both theoretical and practical perspectives. This article delves into the mathematical foundations required to understand fully homomorphic encryption ( $\textsf {FHE}$ ). It consequently covers design fundamentals and security properties of $\textsf {FHE}$ and describes the main $\textsf {FHE}$ schemes based on various mathematical problems. On a more practical level, this article presents a view on privacy-preserving machine learning using homomorphic encryption and then surveys $\textsf {FHE}$ at length from an engineering angle, covering the potential application of $\textsf {FHE}$ in fog computing and cloud computing services. It also provides a comprehensive analysis of existing state-of-the-art $\textsf {FHE}$ libraries and tools, implemented in software and hardware, and the performance thereof.

中文翻译：

---

全同态加密、理论与应用综述

在物联网、云服务、边缘计算、人工智能应用程序和其他由下一代网络支持的应用程序的背景下，数据隐私问题正在显着增加。同态加密通过允许对加密消息执行多个操作而无需解密来解决隐私挑战。本文从理论和实践两个角度全面阐述了同态加密。本文深入研究了理解完全同态加密所需的数学基础（ $\textsf {FHE}$ ）。因此，它涵盖了设计基础和安全属性 $\textsf {FHE}$并描述了主要 $\textsf {FHE}$基于各种数学问题的方案。在更实际的层面上，本文介绍了使用同态加密的隐私保护机器学习的观点，然后进行了调查 $\textsf {FHE}$从工程角度详细介绍，涵盖了 $\textsf {FHE}$雾计算和云计算服务。它还提供了对现有最先进技术的全面分析 $\textsf {FHE}$以软件和硬件实现的库和工具及其性能。