## 1 Introduction  
Images require metadata that describe their existence and summarize the underlying image data. That is, details on the author, creation date, modification date, file size, etc. The purpose of this paper was to support image security by encrypting the compressed image with AMBTC, one of the compression methods of gray images, and creating a model that supports Data Hiding (DH) [1, 2, 3]. DH can be used for covert communication, authentication and copyright based on a variety of digital media (e.g., digital content such as images, video, audio, etc.). The receiving end is used to extract data from the encrypted image and support the recovery of the original cover image. The data hidden in the image should not be detectable by an attacker. Reversible DH(RDH) [4, 5] is a kind of DH. It is possible to recover the original image after extracting data from the marked image, and various RDHs have been proposed so far. RDH can be classified into three specialities. That is, pixel extension (DE) [6, 7, 5] histogram shift (HS) [8], prediction error extension (PE) [9]. DE is a method of first doubling the difference between two adjacent pixels, and then hiding one bit in the LSB (least significant bit). In HS, the zero and peak points in the histogram of the cover image are used to hide secret metadata. PE is a method of hiding data by using the difference in the prediction error between the original pixel and the predicted pixel. On the other hand, image encryption [10, 11] is a method used to protect digital information by encrypting the image so that only the user who knows the encryption key can restore the image, thereby protecting the privacy of the image. In particular, in order to share secret images, owners, assistants, and administrators must encrypt the image and then distribute it using a trusted service provider (SP). For example, hospital administrators can embed personal information, authentication, and diagnostic opinion data into encrypted X-ray, CT, or MRI images. In this case, encrypted image-based RDH combines the advantages of RDH and image encryption. It can be useful as it is a compromise between advantages. If the original image is encrypted and delivered to the SP, the SP cannot estimate the image before it is encrypted. Because the SP does not have the encryption key, it cannot recover this image. The role of the SP is to hide additional data in the encrypted image and transmit the encrypted image to the receiver. After decoding the image on the receiver side, the recovered cover image is lossless, and metadata can also be extracted on the receiver side. Since [PERSON] [12] introduced an RDH encrypted image (RDH-EI)-based RDH using the fluctuation principle, the extended RDH-EI has been actively studied. RDH-EI has a similar process to RDH, except that the cover image is encrypted before hiding the metadata. In fact, RDH-EI can be utilized for various applications such as authentication, copyright and personal security protection [13; 14; 15; 16; 17; 18]. [PERSON] [19] demonstrated that data extraction and image recovery are separable and that hidden metadata can be extracted without errors. The process is called Separable RDHEI (SRDH-EI) [19; 20; 21]. That is, the receiving side can use EK (encryption key) and DK (data secret key) to decrypt and extract the encrypted image regardless of the application order. In [20], [PERSON] et al. proposed n-ary SRDH-EI using HS and solved the original image recovery problem, which is the problem of [PERSON]'s method. [PERSON] et al. [21] proposed SRDH-EI with a general payload and error-free data extraction by introducing multi-granularity permutation. [PERSON] et al. (2018) [22] first proposed an AMBTC-based RDH-EI method. The first step is to encrypt the upper and lower averages of triples in the AMBTC-compressed video using a stream cipher. The prediction error histogram correction technique can then be used to insert additional data into the redundant space. [PERSON] et al.