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具体有效的安全多方计算协议：调查和更多

安全的多方计算（MPC）允许一组当事人在他们的隐私输入上联合计算一个函数，并且除了函数的输出外什么都不透露。在过去的十年中，MPC 已经从一个纯粹的理论研究迅速转变成一个具有实际意义的对象，人们对实际应用的兴趣越来越大，如保护隐私的机器学习（PPML）。在本文中，我们全面调查了在不诚实多数和诚实多数情况下，具有半诚实和恶意安全的具体有效的 MPC 协议的现有工作。我们重点考虑有中止的安全概念，也就是说，腐败的一方可以阻止诚实的一方在收到输出后收到输出。我们提出了设计不同风格的 MPC 协议的基本和关键方法的高层次想法，以及 MPC 的关键构建模块。对于 MPC 的应用，我们比较了已知的建立在 MPC 上的 PPML 协议，并描述了最先进的 PPML 协议的私有推理和训练的效率。此外，我们总结了几个挑战和开放性问题，以突破 MPC 协议的效率，以及一些有趣的未来工作，值得被解决。这项调查的目的是向那些对了解、改进和应用具体的高效 MPC 协议感兴趣的研究人员提供 MPC 的最新发展和关键方法。

1 介绍

2 预备工作

3 基于秘密共享的 MPC 协议

4 基于混淆电路的恒圆 MPC

5 不经意传输和不经意线性函数评估

在本节中，我们描述了不经意传输（OT）及其重要变体（即随机 OT 和相关 OT）的最新发展和技术。此外，我们还介绍了 OT 的算术泛化，即不经意线性函数评估（OLE）及其重要变体（即 VOLE）。OT 主要用于布尔电路的 MPC 协议，而 OLE 主要应用于算术电路的 MPC 协议。在这项调查中，我们主要回顾了构建（相关）OT 的最先进技术，并对设计（矢量）OLE 的技术进行了简明的概述。请注意，OLE 与 OT 具有相同的重要性。此外，对于基于带噪学习奇偶性（LPN）的最先进技术来说，矢量 OLE 可以在与相关 OT 相同的框架内设计。我们注意到，同态加密（HE）是产生（矢量）OLE 相关性的关键技术，尽管在本节中没有详细描述它。与基于 HE 的线性通信复杂性相比，最近基于 LPN 变体的技术允许获得亚线性通信复杂性。

5.1 不经意传输

不经意传输（OT）[95，96] 是发送方 和接收方 之间的一个基本加密原语，它使 只能获得 的两个输入信息中的一个，而 对 的选择位一无所知。它不仅可以用来构建姚氏两方安全计算协议，还可以用来构建很多其他具有半诚实和恶意安全的 MPC 协议。此外，OT 还可以用来设计很多其他种类的加密协议。OT 协议可以从不同的密码学假设中构建，包括决策性 Diffie-Hellman（DDH）[241-245]、计算性 DiffieHellman（CDH）[244, 246-248]、带错误学习（LWE）[242, 245, 249-251]、带噪声学习奇偶性（LPN）[247] 和交换性超同构 Diffie-Hellman（CSIDH）[252]。然而，当需要产生大量的 OT 关联时（特别是对于 MPC 应用），这些基于公钥操作的 OT 协议非常昂贵。为了处理这个问题，Beaver [253] 引入了 OT 扩展的概念，在这个概念中，使用快速操作将少量的基础 OT 有效地扩展到大量的 OT（甚至是任意多项式数的 OT）。Beaver [253] 的第一个 OT 扩展协议以非黑箱方式使用了伪随机发生器（PRG），因此只在理论上有意义。目前，具体有效的 OT 扩展协议分为两种风格：一种是基于 IKNP 框架 [130]，另一种是 PCG 框架 [162, 254]。IKNP 风格的协议采用对称密钥的原始 PRG 来进行扩展，并支持选择位，而 PCG 风格的协议则利用 LPN 问题中噪声的稀疏特征 [255] 来实现扩展，并且只允许随机选择位。两种风格的 OT 扩展都采用以下结构，从弱 OT 到标准 OT： 其中 COT 要求发送方的两个消息 满足固定的相关关系（即，对于固定的字符串 ， ），ROT 只允许输出两个均匀的随机消息，而 OT 允许输入任意的两个消息。这两种转换（即 和 ）都是标准的，回顾如下：

……

因此，我们可以专注于设计具体有效的 COT 协议，然后将其转化为标准的 OT 协议。此外，COT 协议能够被用来使用类似 TinyOT 的协议 [104, 106-110] 产生 BDOZ 式的认证共享，以及使用比特分解思想产生 SPDZ 式的认证共享 [43, 150]。对于具有自由 XOR 的 GCs，电路中每条线的乱码标签都满足 COT 的相关性，因此可以使用 COT 协议从乱码者向评估者无意识地传输，也就是说，COT 也可以直接用于 MPC 协议。

半诚实的 IKNP 协议 [130]（在 [113] 中改进）的工作原理大致如下。1）通过切换发送方和接收方的角色，在设置阶段执行一个 baseOT 协议（依赖于公钥操作）以产生 个 ROT 关联，然后 2）在扩展阶段将 个 ROT 关联扩展为大量的 COT 关联，使用 PRG 并将列向量切换为行向量。当使用相同的设置阶段时，扩展阶段可以迭代执行以产生无限数量的 COT 关联 [113]。后来，在 [135，136] 中提出了具有恶意安全的 IKNP 式 OT 扩展协议。使用随机线性组合方法，Keller, Orsini, and Scholl [136] 提出的恶意安全协议在 IKNP 框架中实现了最好的效率，并且其通信成本与最著名的半诚实协议 [113] 相匹配。虽然 IKNP 式的 OT 扩展协议享有快速计算，但它们需要线性通信成本（即每个 COT 相关的 比特）。

……

5.2 不经意线性函数评估

OLE：不经意线性函数评估（OLE）是 OT 的算术泛化，对于设计大领域算术电路的 MPC 协议特别有用 [120，146，262-264]。特别是，OLE 直接给出了两个秘密值的乘法的两方加法共享。因此，通过成对的 OLE 协议执行，我们可以使用 OLE 来生成 Beaver 乘法三元组，而无需在多方设置中进行认证。OLE 可以使用 OT 扩展和 Gilboa 乘法的方法来构建 [43, 150]，并且具有便宜的计算成本，但通信成本高得多。存在一种标准的方法来设计 OLE，即使用基于 RLWE 的加法同态加密（AHE），该方法已被用于 Overdrive [152] 和最近的工作 [265]，其中接收方 发送 给发送方 ，然后 计算 并将其发送给 ，后者解密得到 的大域 ，这里，AHE 需要满足电路隐私属性。此外，OLE 也可以从某种程度的同态加密中构建 [103, 149]，但将需要更大的通信。不依赖同态加密，OLE 也能够直接从 Ring-WE [266, 267] 构建。此外，我们还可以从 OT 和噪声 Reed-Solomon 编码 [97, 264, 268]，或 Paillier 加密 [269] 中构建 OLE 协议。在所有的 OLE 协议中，基于 AHE 的协议 [152, 265] 获得了最好的通信效率，而来自 RLWE 的协议 [266] 具有最佳的一轮通信。

最近，Boyle 等人 [162] 提出了一个直接基于 LPN 的 OLE 构造，它比上述 OLE 协议具有非常低的通信成本，但在生成 个 OLE 关联时需要至少 的计算成本。后来，他们 [161] 使用环 LPN 假设的变体解决了计算问题，并构建了一个用于计算大量 OLE 关联的 OLE 协议。这个 OLE 协议比基于 RLWE 的协议具有非常低的通信成本，并提供了 的计算复杂性。他们的基于环 LPN 的 PCG 方法是产生大量 OLE 相关的好方法（例如， ）。对于少量的 OLE 关联，基于 RLWE 的方法可能更好。基于环 LPN，产生的 OLE 相关性是随机的（即 、 、 是均匀随机的），但是足以设计 MPC 协议，其中只需要在预处理阶段产生随机的乘法三元组。

VOLE：向量不经意线性函数评估（VOLE）是 COT 在大域的算术泛化，定义如下：

  • 发送方持有一个统一的全局密钥 。
  • 对于每个 VOLE 的执行，发送方得到一个向量 ，接收方得到两个向量 ，这样 。

我们有一个使用 CRHFs 从 COT 到 OT 的标准转换 [130]。这对 VOLE 和 OLE 来说不是这样的，因为底层字段 很大，发送方不能列举所有可能的值 。与 OLE 类似，VOLE 可以基于 OT 扩展 [43] 或 AHE [152, 265] 建立，其中后者的通信量较低。基于 AHE 的 VOLE 协议 [152, 265] 的通信复杂性与 VOLE 的输出长度成线性关系。基于 LPN 假设，PCG 方法 [254] 可以构建具有亚线性通信的 VOLE 协议，并且是产生大量 VOLE 关联的最有希望的方法（例如，N≥105）。随后，这种方法在 [69, 132-134, 162, 256, 260] 中被进一步优化。这些基于 LPN 的 VOLE 协议之间的效率和安全性比较与上一小节中显示的 COT 情况类似。

最先进的 VOLE 协议 [69, 133] 与最著名的基于双 LPN 或原始 LPN 的 COT 协议 [133, 134] 采用相同的框架，只是在单点 VOLE 协议执行中需要生成一个额外的 VOLE 相关，因为单个非零元素在大场 F 中是均匀的而不是等于 1。此外，对于 VOLE，我们需要在大场 F 而不是 F2 上使用 LPN 假设。 我们能够使用基于 AHE 的 VOLE 协议 [152, 265] 在设置阶段产生 VOLE 相关。此外，我们可以使用 PCF 方法在 VDLPN 假设下生成 VOLE 相关 [256]，如果生成的 VOLE 相关的数量非常大，可能比 PCG 方法有效率优势。与 COT 的情况类似，我们可以使用最先进的一致性检查 [69, 134] 来构建恶意安全的 VOLE 协议。

6 MPC 在机器学习中的应用

7 结论和未来工作

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