Fully Homomorphic Encryption (FHE) is a cryptographic scheme that enables computations to be performed directly on encrypted data, as if the data were in plaintext. After all computations are performed on the encrypted data, it can be decrypted to reveal the result. The decrypted value matches the result that would have been obtained if the same computations were applied to the plaintext data. FHE supports basic operations such as addition and multiplication on encrypted numbers. Using these fundamental operations, more complex computations can be constructed, including subtraction, division, logic gates (e.g., AND, OR, XOR, NAND, MUX), and even advanced mathematical functions such as ReLU, sigmoid, and trigonometric functions (e.g., sin, cos). These functions can be implemented either as exact formulas or as approximations, depending on the trade-off between computational efficiency and accuracy. FHE enables privacy-preserving machine learning by allowing a server to process the client's data in its encrypted form through an ML model. With FHE, the server learns neither the plaintext version of the input features nor the inference results. Only the client, using their secret key, can decrypt and access the results at the end of the service protocol. FHE can also be applied to confidential blockchain services, ensuring that sensitive data in smart contracts remains encrypted and confidential while maintaining the transparency and integrity of the execution process. Other applications of FHE include secure outsourcing of data analytics, encrypted database queries, privacy-preserving searches, efficient multi-party computation for digital signatures, and more. A dynamic website version is available at (https://fhetextbook.github.io). Please report any bugs or errors to the Github issues board.

翻译：完全同态加密（FHE）是一种密码学方案，它允许直接在加密数据上执行计算，就如同数据是明文状态一样。在加密数据上完成所有计算后，可以解密以揭示结果。解密后的值与对明文数据应用相同计算所得到的结果一致。FHE 支持对加密数字执行加法和乘法等基本运算。利用这些基础运算，可以构建更复杂的计算，包括减法、除法、逻辑门（例如 AND、OR、XOR、NAND、MUX），甚至高级数学函数，如 ReLU、Sigmoid 和三角函数（例如 sin、cos）。这些函数既可以作为精确公式实现，也可以作为近似实现，具体取决于计算效率与准确性之间的权衡。FHE 通过允许服务器以加密形式处理客户端的数据（通过机器学习模型）来实现隐私保护的机器学习。使用 FHE，服务器既无法获知输入特征的明文版本，也无法获知推理结果。只有客户端在服务协议结束时才能使用其密钥解密并访问结果。FHE 还可应用于机密区块链服务，确保智能合约中的敏感数据保持加密和机密状态，同时维护执行过程的透明性和完整性。FHE 的其他应用包括安全外包数据分析、加密数据库查询、隐私保护搜索、用于数字签名的高效多方计算等。动态网站版本可在 (https://fhetextbook.github.io) 获取。请将任何错误或问题报告至 Github 问题列表。