k-min是计算机科学中常见的数据选择问题。作为k-min问题的变形，k-min阈值问题是指从大规模数据集中选择符合用户阈值标准的第k个最小值。但在解决k-min阈值问题时常常涉及大量的个人敏感信息或商业机密信息，这对包含这些敏感信息或机密信息的文件造成严重的隐私威胁。因此，在处理k-min阈值问题的同时，如何确保数据安全是一个重要的研究课题。围绕该问题，本文针对2个具体保护隐私应用场景中的k-min阈值问题做了以下研究：

        ⑴针对保护隐私商业合作中的企业业绩评估问题，提出一种高效的k-min阈值谓词加密方案。首先，设计一种新的编码，并结合内积思想，将第k个最小值与阈值比较大小问题转化为内积问题；其次，利用压缩技巧提高效率，同时实现常数级私钥和解密配对运算；最后，将方案的安全性规约到判定性双线性Diffie-Hellman指数数学难题假设，在标准模型下严格证明方案的安全性。由于方案生成的密钥存储开销和解密成本均为常数级，因此，将方案应用到商业合作中的企业业绩评估，可高效、安全地完成评估工作。

        ⑵针对保护隐私的电子交易中的产品质量评估问题，提出一种属性隐藏的多维k-min阈值的谓词加密方案。首先，设计一种新的编码，结合对偶向量空间上的内积加密技术，将多个第k个最小值和多个阈值的比较大小问题转化为多内积问题；其次，利用随机数等式实现多内积问题，并构造属性盲化方法实现属性隐藏；最后，将方案的安全性规约到相关及不相关对偶向量的子群判定变体数学难题假设，在标准模型下严格证明方案的安全性。该方案可以解决多个第k个最小值和阈值比较大小问题，相比其他支持比较大小的谓词加密方案，表达力更加丰富；此外，该方案还具有更高级别的隐私安全——属性隐藏。因此，将该方案应用于电子交易中的产品质量评估问题，可以同时对多个产品进行保护隐私的质量评估，且评估过程中既可以保护生产商商业文件的隐私，还可以保护与之相关的产品属性的隐私。

       K-min is a basic selection problem in computer science. As a variant of the k -min selection problem, the k-min threshold problem refers to finding out the k-th minimum value from a large-scale dataset, which is subject to a certain threshold criteria users required. K-min threshold problem has a wide range of applications in real world, such as search engine systems, personalized recommendations, electronic transactions, business investment and so on. In some application scenarios, however, a large amount of sensitive information or confidential information will be involved in the process of conducting k-min threshold problem, which raises a serious security threat to the files containing these sensitive or confidential information. Therefore, it has become a vital issue how to ensure the security of sensitive or confidential information when dealing with the k-min threshold problem. Around this issue, this paper studies the k-min threshold problem in two specific privacy-preserving application scenarios as follows.

       Aiming at the problem of enterprise performance evaluation in privacy-preserving business cooperation, an efficient k-min threshold predicate encryption scheme is proposed. First, with a new design of coding in combination with the idea of inner product, the k-th minimum and threshold comparison problem is transformed into the inner product problem. Second, a compact technique is used for the sake of the constant size private key and decryption pairing operation at the same time. Finally, the security of the scheme is strictly proved under the standard model, which is reduced to the mathematical problem assumption of decisional n-Bilinear Diffie-Hellman exponent problem. Enjoying the advantage -- shorter size key and lower decryption cost, it is able to guarantee the evaluation work can be proceeded efficiently and safely when our proposal is applied to manage enterprise performance evaluation.

       Aiming at the problem of product quality evaluation in privacy-preserving electronic transactions, a hidden attribute encryption scheme supporting multi-dimensional k-min threshold is proposed. First, with a new design of coding in combination with the technique of the inner product encryption on the dual-pairing vector space, the comparison problem of multiple k-th minimum values and multiple thresholds is transformed into a multi-inner product problem. Second, an equality for some random numbers is constructed to solve the multi-inner product problem, and the attribute blind approach is developed to realize attribute hiding. Finally, the security of the scheme is strictly proved under the standard model, which is reduced to the mathematical problem hypothesis of variants of decisional subspace problem with relevant dual vector and irrelevant dual vector. Our proposal is more expressive than other predicate encryption schemes in the sense that it can solve multiple k-th minimum values and thresholds. In addition, our proposal also achieves a higher-level security: attribute hiding. Enjoying these advantages, therefore, it is able to ensure the quality assessment of multiple products carried out at the same time while learn nothing about the privacy of the manufacturer commercial documents as well as the related product attributes when the scheme is applied to the product quality evaluation in electronic transactions.