## 6 Conclusions  
Our proposed system can retrieve private data between hospitals through digital signature and HE schemes and prevent insider and man-in-the-middle attacks. In addition, the adopted optimization processes in our system proved the feasibility of private data retrieval using HE schemes while achieving a considerable speedup in execution time. Furthermore, we use the BFV HE scheme [27] in our implementation to encode the data as integers to ensure the correctness of operations. As one of the anonymized reviewers pointed out, our proposed system can apply not only to the case of Hospitals but also to Banks, Department Stores, Research Institutes, and so force. Our proposed protocol aims to preserve users' privacy meanwhile allows for secure data transmission. Based on this, we analyze many situations in practical usage, such as that the client set size is usually much smaller than the server data size and the flexibility of use of BFV homomorphic implementation to ensure 100% data security. In other words, if the data can be encoded or indexed as integers, detailed information about the symptoms can be accessed, such as diabetes and hypertension indexes. This consideration can be extended to other scenarios, such as the interactions between insurance companies and hospitals. According to our experiments, even if the Server's dataset size becomes huge, our protocol can complete one PSI task within 2.33 min with multi-threading execution (eight threads in our setup). In practice, the frequency of secure patient data exchanges among hospitals' EMRSs is limited daily due to various administration considerations. In other words, with the proposed protocol, each hospital can do secure patient data exchanges with other hospitals approximately 600 times a day. In more precious, let us take the longest execution time in our experiments into account (where polynomial degree = 16,384, client data size = 100, server data size = 224, and the time spent is 139.77 s). Then, considering we have 86,400 s per day, it takes 139.77 s for our proposed protocol. So, 86,400/139.77 \(\approx\) 621.58. That is the reason why we state that the hospital can perform 600 such operations per day. In the future, we will look for more possibilities to accelerate our protocol from the hardware perspective and analyze more on memory usage optimization. Most recently, [PERSON], [PERSON], [PERSON], and [PERSON] presented a non-homomorphic encryption-based protocol, PSImple [28], to solve the Multiparty Maliciously-Secure Private Set Intersection Problem. Instead of using homomorphic encryption schemes, the construction of PSImple is based on oblivious transfer and garbled Bloom filters. To demonstrate the practicality of PSImple, the authors implemented their protocol and ran experiments with up to 32 parties and 220 inputs. Experimental results showed that PSImple is competitive even with the state-of-the-art concretely efficient semi-honest multiparty PSI protocols. Inspired by [28], how to extend our protocol to a malicious threat model and integrate our protocol with non-homomorphic-based approaches, such as PSImple, to further enhance the values of PSI in practical applications are listed at the top of our future research list. The following are available online at [[https://www.mdpi.com/article/10.3390/a16050244/s1](https://www.mdpi.com/article/10.3390/a16050244/s1)]([https://www.mdpi.com/article/10.3390/a16050244/s1](https://www.mdpi.com/article/10.3390/a16050244/s1)). **Author Contributions:** Formal analysis, Y.-W.W.; Funding acquisition, J.-L.W. ; Investigation, Y.-W.W. and J.-L.W. ; Methodology, Y.-W.W.; Project administration, J.-L.W. ; Resources, J.-L.W. ; Software,Y.-W.W.; Supervision, J.-L.W. ; Writing--original draft, Y.-W.W.; Writing--review & editing, Y.-W.W. and [PERSON]-L.W. All authors have read and agreed to the published version of the manuscript. The Minister of Science and Technology, Taiwan: MOST 111-2221-E-002-134-MY3 and Taiwan Semiconductor Manufacturing Company: TSMC: 112H1002-D.  
Not applicable.