Recent Development of Theory and Application on Homomorphic Encryption
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摘要:
随着云计算、云存储等各类云服务的普及应用，云环境下的隐私保护问题逐渐成为业界关注的焦点，同态密码成为解决该问题的关键手段，其中，如何构造高效的全同态加密方案是近年来同态加密研究的热点之一。首先，该文介绍了同态密码的发展情况，从不同角度对同态加密方案进行了分类分析，着重描述了可验证全同态加密方案的研究进展。通过分析近年来公开的同态加密领域知识产权文献，对同态加密在理论研究和实际应用中所取得的进展进行了归纳总结。其次，对比分析了目前主流全同态加密库Helib, SEAL以及TFHE的性能。最后，梳理了同态加密技术的典型应用场景，指出了未来可能的研究与发展方向。
Abstract:With the popularization of various cloud services such as cloud computing and cloud storage, privacy preservation issues in the cloud environment have gradually become the focus of industrial applications. Homomorphic encryption has become an important method to solve this issue. Among them, how to construct an efficient fully homomorphic encryption scheme is one of the hotspots at present. Firstly, the development of homomorphic encryption is introduced. The homomorphic encryption schemes are analyzed and classified from different perspectives. The research progress of verifiable fully homomorphic encryption schemes is discussed in detail. By analyzing the property rights literature on homomorphic encryption that has been published in recent years, the progress in the theoretical research and application about homomorphic encryption are summarized. Secondly, the working performances of three typical homomorphic encryption libraries, Helib, SEAL and TFHE, are compared and analyzed. Finally, various application scenarios of homomorphic encryption technology are sorted out, and possible research and development directions in the future are proposed.
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表 1 提高全同态加密效率的解决方案
方式 解决方案 优化Bootstrapping Gentry09[2]：首次提出Bootstrapping Ducas15[12]：运行时间从6 min缩短至1 s以内 Chillotti16[13]：运行时间从1 s以内缩短至0.1 s以内，密钥大小从1 GB减小至24 MB Chen18[14]：自举深度从${\log _{\rm{2}}}h + 2{\log _{\rm{2}}}t$降至${\log _{\rm{2}}}h + {\log _{\rm{2}}}t$ Batching Smart14[15]：构造可支持SIMD操作的FHE方案 Castryck18[16]：提高了明文封装容量和参数优化灵活性 无噪声FHE Kipnis12[18]：基于矩阵和多项式的无噪声FHE方案MORE和PORE Gentry14[20]：基于完备群概念的无噪声FHE框架 FPGA设计 Shi18[24]：16×24 bit有限域FFT算法的FPGA设计 Xie19[25]：768 kbit大整数乘法器FPGA设计 
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表 2 全同态加密在整数域和实数域上的研究进展
类型 解决方案 明文空间为整数的FHE Gentry10[27]：第1个基于整数的FHE方案DGHV方案 Cheon13[28]：将批处理技术引入DGHV方案 Nuida15[29]：将DGHV方案的明文空间从${Z_2}$扩展至${Z_Q}$ Cheon15[31]：将LWE问题归约为AGCD问题的一个变体 明文空间为实数的FHE Jaschke16[33]：通过与2的幂迭代相乘近似将有理数表示为整数 Dowlin17[34]：将定点小数编码为整系数多项式，但明文模随电路深度的增加呈指数增大 Cheon17[35]：可进行浮点数近似计算的CKKS方案，但仅为层次型FHE方案 Cheon18[36]：将文献[35]中的层次型同态加密方案扩展为全同态加密方案
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表 3 可验证同态加密研究进展
解决方案 研究进展 存在问题 Johnson02[41] 首次提出同态签名的概念 – Boneh11[42] 首个可执行确定阶数多项式运算的同态签名方案 – Gneearo13[43] 形式化定义了同态消息认证的概念 – Catalano13[44] 支持低次多项式运算的同态MAC方案 不能同时满足简洁性和复合性 Catalano14[46] 引入了一个新的密码学原语LAEPuV – Joo14[48] 首次给出在HAE中IND-CPA和IND-CCA的定义 – Bai18[45] 基于默克尔哈希树的同态认证方案 复合度上有所不足 Fiore16[50] 提出多密钥同态认证(M-HS)方案 可能存在不可信签名者 Lai18[51] 基于零知识证明提出了一种M-HS通用结构 没有分析其认证安全性和实用性 Alagic17[52] 可验证的量子全同态加密方案 –
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表 4 同态加密相关的知识产权聚焦的不同应用领域
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表 5 Test_Timing效率测试结果(μs)
m 密钥生成 加密 解密 同态加 同态乘 4051 317037 6786 4039 97 26701 4369 571082 8041 5173 98 31477 4859 664138 10497 7554 193 41354
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表 6 SEAL中BFV方案效率测试(μs)
Poly Coeff Plain 加密 解密 同态加 同态乘 重线性化 4096 109 786433 93464 32306 314 390192 63711 8192 218 786433 267727 112898 1074 1510281 319876 16384 438 786433 884862 439232 4341 6146131 1846517
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表 7 SEAL中CKKS方案效率测试(μs)
Poly Coeff 加密 解密 同态加 同态乘 重线性化 4096 109 87557 3359 309 12476 63459 8192 218 274215 12748 1071 47599 314501 16384 438 964821 51465 4317 200248 1850888
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