有理数域上集合成员关系的保密判定协议
Privacy Preserving Protocol of Rational Set Membership’s Judge
摘要: 本文通过将有理数按位编码为矩阵，并结合ElGamal同态加密算法，设计了有理数域上集合成员关系的保密判定协议，其中点集成员关系的保密判定协议可适用于 维有理点。其次，应用模拟范例的方法严格证明了协议的安全性，同时协议能够保护集合的势。最后，比较分析表明当参与者的有理数满足一定条件时，本文设计的协议是高效的。
Abstract: By encoding the rational numbers into a matrix, combined with the ElGamal homomorphic encryption algorithm, this paper designs secure protocol for the rational set membership’s judge, in which the protocol for the point set membership can be applied to n(n≥2) dimensional rational points. Secondly, using the well accepted simulation paradigm proves that the proposed protocols are secure, and at the same time, the protocol can protect the number of elements in the partici2243pant set. Finally, the comparison and analysis show that the efficiency of the proposed protocol when the rational numbers meet certain conditions.
文章引用：李亚伟. 有理数域上集合成员关系的保密判定协议[J]. 计算机科学与应用, 2021, 11(8): 2080-2087. https://doi.org/10.12677/CSA.2021.118213

参考文献

[1] Yao, A.C. (1982) Protocols for Secure Computations. The 23rd Annual IEEE Symposium on Foundations of Computer Science, Chicago, 3-5 November 1982, 160-164. [Google Scholar] [CrossRef] 
[2] Benor, M., Goldwasser, S., Widerson, A., et al. (1988) Completeness Theorems for Non-Crypto-Graphic Fault-Tolerant Distributed Computation. The Twentieth Annual ACM Symposium on Theory of Computing, Chicago, 2-4 May 1988, 1-10. [Google Scholar] [CrossRef] 
[3] Dan, B., Giovanni, D.C., Rafail, O., et al. (2004) Public Key Encryption with Keyword Search. International Conference on the Theory and Applications of Cryptographic Techniques, Interlaken, 2-6 May 2004, 506-522.
[4] Sandhu, R.S., Coyne, E.J., Feinstein, H.L., et al. (1996) Role-Based Access Control Models. Computer, 29, 38-47. [Google Scholar] [CrossRef] 
[5] Ruj, S., Stojmenovic, M., Nayak, A. (2014) Decentralized Access Control with Anonymous Authentication of Data Stored in Clouds. IEEE Transactions on Parallel and Distributed Systems, 25, 384-394. [Google Scholar] [CrossRef] 
[6] 秦静, 张振峰, 冯登国, 李宝. 一个特殊的安全双方计算协议[J]. 通信学报, 2004(11): 35-42.
[7] 秦静, 张振峰, 冯登国, 李宝. 无信息泄漏的比较协议[J]. 软件学报, 2004(3): 421-427.
[8] Pinka, B., Schneider, T., Weinert, C. and Wieder, U. (2018) Efficient Circuit-Based PSI via Cuckoo Hashing. In: Nielsen, J.B. and Rijmen, V., Eds., Advances in Cryptology—EUROCRYPT 2018, LNCS, Vol. 10822, Springer, Berlin, 125-157. [Google Scholar] [CrossRef] 
[9] 窦家维, 刘旭红, 周素芳, 等. 高效的集合安全多方计算协议及应用[J]. 计算机学报, 2018, 41(8): 1844-1860.
[10] 周素芳, 李顺东, 郭奕旻, 窦家维, 陈振华. 保密集合相交问题的高效计算[J]. 计算机学报, 2018, 41(2): 464-480.
[11] 陈振华, 李顺东, 黄琼, 丁勇, 刘娅茹. 非加密方法安全计算两种集合关系[J]. 软件学报, 2018, 29(2): 473-482.
[12] 唐春明, 林旭慧. 隐私保护集合交集计算协议[J]. 信息网络安全, 2020, 20(1): 9-15.
[13] 李荣花, 武传坤, 张玉清. 判断集合包含关系的安全计算协议[J]. 计算机学报, 2009, 32(7): 1337-1345.
[14] Dou, J.W., Gong, L.M., Li, S.D., et al. (2016) Ef-ficient Private Subset Computation. Security and Communication Networks, 9, 5965-5976. [Google Scholar] [CrossRef] 
[15] Zhou, S.F., Li, S.D., Dou, J.W., et al. (2017) Efficient Secure Multiparty Subset Computation. Security and Communication Networks, 2017, Article ID: 9717580. [Google Scholar] [CrossRef] 
[16] 程楠, 赵运磊. 一种高效的关于两方集合并/交集基数的隐私计算方法[J]. 密码学报, 2021, 8(2): 352-364.
[17] Li, S.D., Wang, D.S. and Dai, Y.Q. (2009) Symmetric Cryptographic Protocols for Extended Millionaires’ Problem. Science China Information Sciences, 52, 974-982. [Google Scholar] [CrossRef] 
[18] 张茜, 苏烨, 秦静. 集合成员关系判定的安全多方计算协议[J]. 山东大学学报(理学版), 2020, 55(4): 118-126.
[19] 窦家维, 刘旭红, 王文丽. 有理数域上两方集合的高效保密计算[J]. 计算机学报, 2020, 43(8): 1397-1413.
[20] 李顺东, 杜润萌, 杨颜璟, 魏琼. 有理数相等的保密判定[J]. 电子学报, 2020, 48(10): 1933-1937.
[21] Goldreich, O. (2004) Foundation of Cryptography: Volume 2, Basic Appli-cations. Cambridge University Press, London, 599-764. [Google Scholar] [CrossRef] 
[22] Rivest, R.L., Adleman, L.M. and Dertouzos, M.L. (1978) On Data Banks and Privacy Homeomorphisms. In: Foundations of Secure Computation, Academia Pres, Cambridge, 169-177.
[23] Gamal, T. (1984) A Public Key Cryptosystem and a Signature Scheme Based on Discrete Logarithms. IEEE Transactions on Information Theory, 31, 469-472. [Google Scholar] [CrossRef]