## 1 Introduction  
### Background  
In recent years, the rapid development of electronic medicine has benefitted from the development of Internet of Things and mobile communication equipment. Personal health data can be collected in a comprehensive way by using wireless sensors. Through the communication network, personal health data can be transmitted to the data center and then transmitted to remote experts or doctors with some additional information for healing purposes through the cloud server. By this way, the communication overhead of the patient is reduced. With this benefit, an increasing number of patients outsource their personal health data to data centers. However, the crux of the matter is data privacy. As we all know, personal health data contains a lot of sensitive information, such as patients' identity, case report and inspection report. Malicious users or data centers can easily access this privacy data if stored in plaintext. To achieve privacy, these data must be stored in data centers with encrypted forms. Due to the encrypted data, sharing data with the experts or doctors would be a significant issue. A common solution is that patients download and decrypt the encrypted data locally, then encrypt the data with a corresponding public key. In contrast to inefficient methods, proxy re-encryption (PRE) has been proposed as a better solution. The concept of proxy re-encryption was introduced in 1998 by [PERSON] et al. [1]. Proxy re-encryption is used to deal with the ciphertext transformation in the same data encryption system. [PERSON] et al. [2] classified PRE schemes into four different types. If the ciphertext can be transformed more than once, i.e., from A to B, and then from B to C, we call this multi-hop; otherwise, it is a single-hop. If the ciphertext can be transformed from A to B, at the same time, from B to A, we call this bidirectional; otherwise, it is unidirectional. In a proxy re-encryption scheme, data are stored in encrypted form. In order to share the data with experts or doctors, the patient generates a re-encryption key and sends it to the cloud server. With the re-encryption key, the cloud server can transform the ciphertext under the patient's public key to that under the expert's or doctor's public key without knowing the plaintext. Solutions for proxy re-encryption have been proposed in the literature [2; 3; 4; 5]. In a multi-hop proxy re-encryption scheme, the delegatee \(i\) may further delegates the decryption privilege to another delegatee, say \(j\). In this case, it is possible that there is no relation between delegatee \(j\) and the delegator. This means, in a multi-hop proxy re-encryption scheme, the ciphertext may be re-encrypted to a delegatee who is unknown to the delegator. This may leak the privacy of the data, so a multi-hop PRE scheme is desired in which the delegator can control all of the delegatees. A cryptographic solution to fulfill the above requirements is autonomous path proxy re-encryption (AP-PRE) [6]. For a better understanding of the autonomous path proxy re-encryption, we illustrate it with the Figure 1 in the following scenario: suppose that \(A\) is a patient who wants to make an online treatment in hospital \(H\). Making an online treatment requires some information which is sensitive. Patient \(A\) just wants to share the information with his/her favourite doctors, so \(A\) makes a list of his/her favourite doctors. For the privacy of the information, \(A\) uses his/her public key to encrypt the information and generates the corresponding re-encryption keys of his/her doctor list. Then, \(A\) uploads the encrypted information and re-encryption keys to the proxy. We suppose that the priority of the doctors in the list are from high to low. Therefore, the proxy re-encrypts the information from \(D_{1}\) to \(D_{n}\). If one of the doctors has no time to deal with the online treatment, the process of re-encryption continues; otherwise, the process of re-encryption stops. In this scenario, [PERSON] et al.