Efficient and Verifiable Ciphertext Retrieval Scheme Based on Trusted Execution Environment
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摘要: 密文检索(Ciphertext Retrieval)机制能够实现密态数据上的检索功能。对称可搜索加密(Symmetric Searchable Encryption, SSE)是密文检索的一个重要分支。然而，出于节省算力等特定因素的考虑，云服务器有可能会返回错误或不完整的结果。此外，攻击者也能利用搜索与访问模式的泄露信息还原出关键字内容。因此，在实现搜索结果可验证性的同时，保护搜索模式和访问模式的隐私，是很有必要和有意义的。但现有支持搜索模式和访问模式隐私的可验证SSE方案普遍需要关键字遍历机制且其验证机制的运行效率不尽理想。这些情况使得数据用户面临较高的计算成本和通信开销。针对上述性能瓶颈，该文提出一种基于可信执行环境(Trusted Execution Environment, TEE)的高效可验证密文检索方案。为提升密文检索等效率，该方案借助硬件级安全隔离与不经意数据重排的协同实现关键字陷门尺寸独立于关键字字典的规模。同时，通过嵌入随机数以及盲化多项式常数项方式验证返回结果的正确性。得益于上述设计，该方案在效率层面取得了显著提升。具体而言，其一，该方案使得关键字陷门规模仅与查询关键字数量相关，而与全局字典规模无关，有效降低计算和通信成本；其二，该方案仅需存储两个随机数即可实现可验证功能，大幅降低用户本地存储开销。其三，数据用户与单服务器单轮交互中获取检索结果以及对称同态加密机制的应用等技术进一步提升了运行效率。此外，TEE中的密态计算弱化了对TEE的安全假设和信任程度。在通过模拟游戏方式完成对方案安全性的证明之后，该文对所提出的方案进行了综合的性能评估，评估结果证实了该方案效率上显著优于其他具备相同功能的方案。Abstract: The ciphertext retrieval mechanism enables retrieval functionality over encrypted data. Symmetric Searchable Encryption (SSE) is a critical branch of ciphertext retrieval. However, due to considerations such as saving computing power, cloud servers may return incorrect or incomplete results. Moreover, attackers can also exploit these leaked information from search and access patterns to reconstruct the keyword details. Therefore, it is necessary and meaningful to protect the privacy of search and access patterns while achieving result verifiability. Nevertheless, existing verifiable SSE schemes that support search and access pattern privacy typically rely on keyword traversal mechanisms and their verification mechanisms are inefficient, which impose high computational and communication overheads on users. To address the above performance bottlenecks, this paper introduces an efficient and verifiable ciphertext retrieval scheme based on Trusted Execution Environment (TEE). To improve the efficiency of ciphertext retrieval, this scheme employs the collaborative implementation of hardware-level security isolation and oblivious data rearrangement to achieve keyword trapdoor size independent of the size of the keyword dictionary. Meanwhile, the correctness of the returned results is verified by embedding random numbers and blinding polynomial constant terms. Thanks to these designs, the scheme achieves significant efficiency improvements. Specifically, firstly, this scheme ensures that the size of keyword trapdoors depends solely on the number of query keywords, not the global dictionary size, effectively minimizing communication and computational costs. Secondly, this scheme requires storing only two random numbers to enable verifiability, substantially minimizing local storage overhead for users. Thirdly, the adoption of techniques, such as enabling data users to retrieve results via single-server and single-round interaction and leveraging symmetric homomorphic encryption, further enhances operational efficiency. Additionally, confidential computing within TEE weakens the security assumptions and trust level towards TEE. After formally proving the security of the proposed scheme using simulation-based methods, this paper has conducted a comprehensive performance evaluation. The evaluation results confirm that this scheme is significantly more efficient than other schemes with the same functionalities.
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1 加密数据的构建$ \boldsymbol{D}\boldsymbol{B}\boldsymbol{B}\boldsymbol{u}\boldsymbol{i}\boldsymbol{l}\boldsymbol{d} $
输入：$ \boldsymbol{\lambda } $和$ \boldsymbol{D}\boldsymbol{B} $ 输出：$ \boldsymbol{E}\boldsymbol{D}\boldsymbol{B}\boldsymbol{t}\boldsymbol{r}\boldsymbol{e}\boldsymbol{e} $,$ \boldsymbol{P}\boldsymbol{M} $,$ \boldsymbol{V}\boldsymbol{K} $和$ \boldsymbol{S}\boldsymbol{K} $ $ \boldsymbol{E}\boldsymbol{D}\boldsymbol{B}\leftarrow \boldsymbol{\varnothing } $ 参数选择 设关键字字典$ \boldsymbol{W} $的大小为$ \boldsymbol{m} $ 选择一个随机数$ \boldsymbol{\beta } $和一个随机数$ \boldsymbol{\theta } $ 随机选择两个哈希函数$ {\boldsymbol{H}\colon \{\mathbf{0},\mathbf{1}\}}^{\boldsymbol{*}}\rightarrow {\mathbb{Z}}_{\boldsymbol{p}} $ 随机抽样一个私钥为$ \boldsymbol{k} $的伪随机函数$ \boldsymbol{F} $ 初始化一个私钥为$ \boldsymbol{s}\boldsymbol{k} $的对称同态加密算法$ \boldsymbol{A}\boldsymbol{S}\boldsymbol{H}\boldsymbol{E}=(\boldsymbol{S}\boldsymbol{e}\boldsymbol{t}\boldsymbol{u}\boldsymbol{p},\boldsymbol{E}\boldsymbol{n}\boldsymbol{c},\boldsymbol{D}\boldsymbol{e}\boldsymbol{c}) $ 具体步骤 1 $ \boldsymbol{L}={\{|\boldsymbol{D}\boldsymbol{B}\left({\boldsymbol{w}}_{\boldsymbol{i}}\right)|\}}_{\boldsymbol{m}\boldsymbol{a}\boldsymbol{x}} $ 2 对于每个关键字$ {\boldsymbol{w}}_{\boldsymbol{i}}\in \boldsymbol{W} $，执行下面的算法： 3 $ {\boldsymbol{k}}_{{{\boldsymbol{w}}_{\boldsymbol{i}}}}\leftarrow \boldsymbol{F}\left(\boldsymbol{k},{\boldsymbol{w}}_{\boldsymbol{i}}\right) $ 4 $ {\boldsymbol{P}}_{{{\boldsymbol{w}}_{\boldsymbol{i}}}}\left(\boldsymbol{x}\right)={\boldsymbol{x}}^{\boldsymbol{L}-\left| \boldsymbol{D}\boldsymbol{B}\left({\boldsymbol{w}}_{\boldsymbol{i}}\right)\right| }{(\boldsymbol{x}-\boldsymbol{\beta })\prod }_{\boldsymbol{i}\boldsymbol{d}\in \boldsymbol{D}\boldsymbol{B}\left({\boldsymbol{w}}_{\boldsymbol{i}}\right)}(\boldsymbol{x}-(\boldsymbol{i}\boldsymbol{d}||\boldsymbol{H}\left(\boldsymbol{i}\boldsymbol{d}\right)))=\displaystyle\sum\nolimits _{\boldsymbol{j}=\mathbf{0}}^{\boldsymbol{j}=\boldsymbol{L}+\mathbf{1}}{{{\boldsymbol{\tau }}_{\boldsymbol{i}\boldsymbol{j}}}\boldsymbol{x}}^{\boldsymbol{j}} $ 5 $ {\boldsymbol{E}\boldsymbol{D}\boldsymbol{B}}_{{{\boldsymbol{w}}_{\boldsymbol{i}}}}={\boldsymbol{E}\boldsymbol{n}\boldsymbol{c}}_{\boldsymbol{s}\boldsymbol{k}}\left({\boldsymbol{P}}_{{{\boldsymbol{w}}_{\boldsymbol{i}}}}\left(\boldsymbol{x}\right)-\boldsymbol{H}({\boldsymbol{w}}_{\boldsymbol{i}}||\boldsymbol{\theta })\right)={{\displaystyle\sum\nolimits _{\boldsymbol{j}=\mathbf{0}}^{\boldsymbol{L}+\mathbf{1}}}\boldsymbol{E}\boldsymbol{n}\boldsymbol{c}}_{\boldsymbol{h}\boldsymbol{p}\boldsymbol{k}}\left({\boldsymbol{\tau }}_{\boldsymbol{i}\boldsymbol{j}}\right){\boldsymbol{x}}^{\boldsymbol{j}} $ 6 $ \boldsymbol{E}\boldsymbol{D}\boldsymbol{B}\leftarrow (({\boldsymbol{k}}_{{{\boldsymbol{w}}_{\mathbf{1}}}},{\boldsymbol{E}\boldsymbol{D}\boldsymbol{B}}_{{{\boldsymbol{w}}_{\mathbf{1}}}}),{\boldsymbol{k}}_{{{\boldsymbol{w}}_{\mathbf{2}}}},{\boldsymbol{E}\boldsymbol{D}\boldsymbol{B}}_{{{\boldsymbol{w}}_{\mathbf{2}}}}),\cdots ,({\boldsymbol{k}}_{{{\boldsymbol{w}}_{\boldsymbol{m}}}},{\boldsymbol{E}\boldsymbol{D}\boldsymbol{B}}_{{{\boldsymbol{w}}_{\boldsymbol{m}}}})) $和$ \boldsymbol{V}\boldsymbol{K}\leftarrow \boldsymbol{\beta },\boldsymbol{\theta } $ 7 初始化Path ORAM二叉树结构EDBtree； 8 PM$ \leftarrow $随机将每个关键字映射到一条路径； 9 对每个关键字对应的多项式执行： 10 将多项式加密为数据块； 11 将数据块插入到关键字映射路径上的对应桶中； 12 使用虚拟块(dummy blocks)将每个桶填充至大小S； 13 返回加密数据库$ \boldsymbol{E}\boldsymbol{D}\boldsymbol{B}\boldsymbol{t}\boldsymbol{r}\boldsymbol{e}\boldsymbol{e} $，$ \boldsymbol{P}\boldsymbol{M} $，$ \boldsymbol{V}\boldsymbol{K} $和私钥$ \boldsymbol{S}\boldsymbol{K}=\left(\boldsymbol{k},\boldsymbol{s}\boldsymbol{k}\right) $； 14 上传$ \boldsymbol{E}\boldsymbol{D}\boldsymbol{B}\boldsymbol{t}\boldsymbol{r}\boldsymbol{e}\boldsymbol{e} $到云服务器的不可信环境； 15 通过安全信道上传PM到云服务器的Enclave。 
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2 关键字陷门生成$ \boldsymbol{T}\boldsymbol{r}\boldsymbol{a}\boldsymbol{p}\boldsymbol{d}\boldsymbol{o}\boldsymbol{o}\boldsymbol{r} $
输入：$ \boldsymbol{s}\boldsymbol{k} $和被询问的关键字集合$ \boldsymbol{Q} $ 输出：关键字陷门$ \boldsymbol{T}\boldsymbol{o}\boldsymbol{k}\boldsymbol{e}\boldsymbol{n} $ 1 对于$ {\boldsymbol{w}}_{\boldsymbol{i}}\in \boldsymbol{Q} $，执行下面的循环： 2 $ {\boldsymbol{k}}_{{{\boldsymbol{w}}_{\boldsymbol{i}}}\boldsymbol{'}}\leftarrow \boldsymbol{F}\left(\boldsymbol{k},{\boldsymbol{w}}_{\boldsymbol{i}}\right) $。 3 返回关键字陷门$ \boldsymbol{T}\boldsymbol{o}\boldsymbol{k}\boldsymbol{e}\boldsymbol{n}=\{{\boldsymbol{k}}_{{{\boldsymbol{w}}_{\mathbf{1}}}\boldsymbol{'}},{\boldsymbol{k}}_{{{\boldsymbol{w}}_{\mathbf{2}}}\boldsymbol{'}},\cdots ,{\boldsymbol{k}}_{{{\boldsymbol{w}}_{\boldsymbol{n}}}\boldsymbol{'}}\} $ 4 通过与TEE建立的安全信道上传到Enclave中
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3 搜索过程$ \boldsymbol{S}\boldsymbol{e}\boldsymbol{a}\boldsymbol{r}\boldsymbol{c}\boldsymbol{h} $
输入：$ \mathbf{EDBtree} $,和$ \boldsymbol{T}\boldsymbol{o}\boldsymbol{k}\boldsymbol{e}\boldsymbol{n} $ 输出：搜索结果$ \boldsymbol{R} $ 云服务器 // Enclave 1 $ \boldsymbol{R}\leftarrow \{\} $ 2 解析$ \boldsymbol{T}\boldsymbol{o}\boldsymbol{k}\boldsymbol{e}\boldsymbol{n}=\{{\boldsymbol{k}}_{{{\boldsymbol{w}}_{\mathbf{1}}}\boldsymbol{'}}{\boldsymbol{k}}_{{{\boldsymbol{w}}_{\mathbf{2}}}\boldsymbol{'}},\cdots ,{\boldsymbol{k}}_{{{\boldsymbol{w}}_{\boldsymbol{n}}}\boldsymbol{'}}\} $ 3 对于$ {\boldsymbol{k}}_{{{\boldsymbol{w}}_{\boldsymbol{i}}}\boldsymbol{'}} $执行： 4 如果$ {\boldsymbol{k}}_{{{\boldsymbol{w}}_{\boldsymbol{i}}}\boldsymbol{'}} $对应的多项式数据块不在STASH中，执行： 5 从PM中读取对应的路径PATH； 6 Enclave调用Path ORAM的Access (PATH, EDBtree)接口，从不可信环境中读取EDBtree中PATH路径上的密文数据块； 7 解密数据块并得到$ {\boldsymbol{k}}_{{{\boldsymbol{w}}_{\boldsymbol{i}}}\boldsymbol{'}} $对应的加密多项式； 8 计算$ {\boldsymbol{E}\boldsymbol{n}\boldsymbol{c}}_{\boldsymbol{s}\boldsymbol{k}}(\mathbb{P}\left(\boldsymbol{x}\right)\boldsymbol{'})={\boldsymbol{E}\boldsymbol{D}\boldsymbol{B}}_{{{\boldsymbol{w}}_{\mathbf{1}}}\boldsymbol{'}} $+···+$ {\boldsymbol{E}\boldsymbol{D}\boldsymbol{B}}_{{{\boldsymbol{w}}_{\boldsymbol{i}}}\boldsymbol{'}} $+···+$ {\boldsymbol{E}\boldsymbol{D}\boldsymbol{B}}_{{{\boldsymbol{w}}_{\boldsymbol{n}}}\boldsymbol{'}} $ $ \text{=}{\boldsymbol{E}\boldsymbol{n}\boldsymbol{c}}_{\boldsymbol{s}\boldsymbol{k}}(\left({\boldsymbol{P}}_{{{\boldsymbol{w}}_{\mathbf{1}}}\boldsymbol{'}}\left(\boldsymbol{x}\right)-\boldsymbol{H}({\boldsymbol{w}}_{\mathbf{1}}||\boldsymbol{\theta })\right)+\cdots +\left({\boldsymbol{P}}_{{{\boldsymbol{w}}_{\boldsymbol{n}}}\boldsymbol{'}}\left(\boldsymbol{x}\right)-\boldsymbol{H}({\boldsymbol{w}}_{\boldsymbol{n}}||\boldsymbol{\theta })\right)) $ 9 将搜索结果$ {\boldsymbol{E}\boldsymbol{n}\boldsymbol{c}}_{\boldsymbol{s}\boldsymbol{k}}\left(\mathbb{P}\left(\boldsymbol{x}\right)\boldsymbol{'}\right) $返回给数据用户。 数据用户 10 数据用户通过解密密钥$ \boldsymbol{s}\boldsymbol{k} $计算出明文$ \mathbb{P}{\left(\boldsymbol{x}\right)}^{\boldsymbol{'}}={\boldsymbol{D}\boldsymbol{e}\boldsymbol{c}}_{\boldsymbol{s}\boldsymbol{k}}\left({\boldsymbol{E}\boldsymbol{n}\boldsymbol{c}}_{\boldsymbol{s}\boldsymbol{k}}\left(\mathbb{P}{\left(\boldsymbol{x}\right)}^{\boldsymbol{'}}\right)\right)=\left({\boldsymbol{P}}_{{{\boldsymbol{w}}_{\mathbf{1}}}\boldsymbol{'}}\left(\boldsymbol{x}\right)-\boldsymbol{H}({\boldsymbol{w}}_{\mathbf{1}}||\boldsymbol{\theta })\right)+\cdots +$
$\left({\boldsymbol{P}}_{{{\boldsymbol{w}}_{\boldsymbol{n}}}\boldsymbol{'}}\left(\boldsymbol{x}\right)-\boldsymbol{H}({\boldsymbol{w}}_{\boldsymbol{n}}||\boldsymbol{\theta })\right) $11 对于$ {\boldsymbol{w}}_{\boldsymbol{i}}\in \boldsymbol{Q}=({\boldsymbol{w}}_{\mathbf{1}},{\boldsymbol{w}}_{\mathbf{2}},\cdots ,{\boldsymbol{w}}_{\boldsymbol{n}}) $，数据用户通过$ \boldsymbol{V}\boldsymbol{K} $获得$ \mathbb{P}\left(\boldsymbol{x}\right)=\mathbb{P}{\left(\boldsymbol{x}\right)}^{\boldsymbol{'}}+\boldsymbol{H}({\boldsymbol{w}}_{\mathbf{1}}||\boldsymbol{\theta })+\cdots +\boldsymbol{H}({\boldsymbol{w}}_{\boldsymbol{n}}||\boldsymbol{\theta }) $ 12 如果$ {\mathbb{P}}_{\boldsymbol{\alpha }}\left(\boldsymbol{\beta }\right)\neq \mathbf{0} $，数据用户拒绝搜索结果。如果验证通过，则数据用户计算出多项式$ \mathbb{P}\left(\boldsymbol{x}\right) $的根，然后从中找到符合格式$ \boldsymbol{i}\boldsymbol{d}||\boldsymbol{H}\left(\boldsymbol{i}\boldsymbol{d}\right) $的有效根并将其添加到结果$ \boldsymbol{R} $中。
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算法模拟 输入：$ {\mathcal{L}}_{\boldsymbol{I}}\boldsymbol{'}=(\boldsymbol{m},\boldsymbol{L},\boldsymbol{n}) $ 输出：$ \boldsymbol{E}\boldsymbol{D}\boldsymbol{B} $和$ \boldsymbol{T}\boldsymbol{o}\boldsymbol{k}\boldsymbol{e}\boldsymbol{n} $ 加密数据$ \mathit{E}\mathit{D}\mathit{B} $的生成： 1 $ \boldsymbol{E}\boldsymbol{D}\boldsymbol{B}\leftarrow \boldsymbol{\varnothing } $ 2 对于$ \boldsymbol{i}\in [\mathbf{1},\boldsymbol{m}] $，执行下面的算法: 3 选择随机数$ {\boldsymbol{k}}_{{{\boldsymbol{w}}_{\boldsymbol{i}}}} $并记录元组$ (\boldsymbol{i},{\boldsymbol{k}}_{{{\boldsymbol{w}}_{\boldsymbol{i}}}}) $到表格$ \boldsymbol{T} $中 4 选择$ \boldsymbol{L} $+1个随机数$ {\boldsymbol{\eta }}_{\boldsymbol{j}} $得到$ {\boldsymbol{E}\boldsymbol{D}\boldsymbol{B}}_{{{\boldsymbol{w}}_{\boldsymbol{i}}}}=\displaystyle\sum\nolimits_{\boldsymbol{j}=\mathbf{1}}^{\boldsymbol{L}+\mathbf{1}}{\boldsymbol{\eta }}_{\boldsymbol{j}}{\boldsymbol{x}}^{\boldsymbol{j}} $ 5 生成$ \boldsymbol{E}\boldsymbol{D}\boldsymbol{B}=(({\boldsymbol{k}}_{{{\boldsymbol{w}}_{\mathbf{1}}}},{\boldsymbol{E}\boldsymbol{D}\boldsymbol{B}}_{\mathbf{1}}),{\boldsymbol{k}}_{{{\boldsymbol{w}}_{\mathbf{2}}}},{\boldsymbol{E}\boldsymbol{D}\boldsymbol{B}}_{\mathbf{2}}),\cdots , $
$({\boldsymbol{k}}_{{{\boldsymbol{w}}_{\boldsymbol{m}}}},{\boldsymbol{E}\boldsymbol{D}\boldsymbol{B}}_{\boldsymbol{m}})) $生成陷门$ \mathit{T}\mathit{o}\mathit{k}\mathit{e}\mathit{n} $ 6 对于$ \boldsymbol{i}\in [\mathbf{1},\boldsymbol{n}] $，执行下面的循环： 7 从$ \boldsymbol{T} $中根据索引选出$ {\boldsymbol{k}}_{{{\boldsymbol{w}}_{\boldsymbol{i}}}} $ 8 返回关键字陷门$ \boldsymbol{T}\boldsymbol{o}\boldsymbol{k}\boldsymbol{e}\boldsymbol{n}=\{{\boldsymbol{k}}_{{{\boldsymbol{w}}_{\mathbf{1}}}},{\boldsymbol{k}}_{{{\boldsymbol{w}}_{\mathbf{2}}}},\cdots ,{\boldsymbol{k}}_{{{\boldsymbol{w}}_{\boldsymbol{n}}}}\} $
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表 1 搜索模式和访问模式隐藏的可验证多关键字可搜索加密机制比较
方案 计算代价 本地通讯代价 本地存储代价 加法同态加密机制 初始化 陷门生成 搜索阶段 初始化 搜索 Wu方案[18] $ \mathrm{O}(\mathrm{mL}) $ $ \mathrm{O}(\mathrm{m}) $ $ \mathrm{O}(\mathrm{mL}) $ $ \mathrm{O}(\mathrm{mL}) $ $ \mathrm{O}(\mathrm{mL}) $ $ \mathrm{O}(1) $ 公钥加密 Ji方案[19] $ \mathrm{O}(\mathrm{mL}) $ $ \mathrm{O}(\mathrm{m}) $ $ \mathrm{O}(\mathrm{mL}) $ $ \mathrm{O}(\mathrm{mL}) $ $ \mathrm{O}(\mathrm{mL}) $ $ \mathrm{O}(\mathrm{m}) $ 对称机制 我们的方案 $ \mathrm{O}(\mathrm{mL}) $ $ \mathrm{O}(\mathrm{n}) $ $ \mathrm{n*O}(\mathrm{log}\mathrm{N}) $ $ \mathrm{O}(\mathrm{mL}) $ $ \mathrm{O}(\mathrm{log}\mathrm{L}) $ $ \mathrm{O}(1) $ 对称机制 $ \mathrm{m} $是关键字字典集合的大小；$ \mathrm{L} $是包含关键字文档标识集合大小的最大值；$ \mathrm{n} $是要搜索关键字集合的大小；$ \mathrm{N} $是Path ORAM桶的数量。
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