# Reversible Data Hiding Scheme in Homomorphic Encrypted Image Based on EC-EG
## 1 Introduction  
Data hiding (DH) and digital watermarking (DW) are two of the important contents of cyberspace security research, which have been greatly developed in recent years. Various new technologies and new methods for them emerge one after another. For example, [PERSON] et al. [1, 2] proposed some new algorithms for video watermarking. [PERSON] et al. [3] summarized the current robust video watermarking algorithms for copyright protection. However, most steganography algorithms cause permanent distortion of the original carrier after embedding the data, which is not allowed in some areas where data authentication is required, such as privacy protection in the cloud environment, military combat map transmission, etc. [4]. Reversible data hiding (RDH) [5] takes data hiding and distortion-free recovery of the original carrier into account. Currently, RDH algorithms mainly include: lossless compression (LC) [6, 7], difference expansion (DE) [8, 9] and histogram shifting (HS) [10, 11]. With the promotion of cloud services, for the purpose of privacy protection, users usually encrypt the uploaded data, so that the original data become incomprehensible ciphertext data. Reversible data hiding in encrypted image (RDH-EI) comes into being and has become the latest research hotspot. The RDH-EI requires that the carrier used for embedding be encrypted, and the carrier still be decrypted withouterror after extracting embedded data. RDH-EI is an important combination of signal processing technology and data hiding technology in encrypted domain. It has dual functions of privacy protection and secret information transmission for information security in data processing. There are two main applications in medical health. The first is the integrity protection of medical images. Medical images are mostly digital images that are easily copied, edited, or maliciously altered while being easy to store. A more mature technique for protecting image integrity information is digital signatures. The main principle is to generate non-repudiation hash values that are appended to original image and transmitted to the receiver. However, in reality, confidential communications often run the risk of being attacked by malicious attacks and analysis. The way to directly attach the hash value to the image is easily vandalized and replaced, thus increasing the difficulty of image authentication. On some special occasions, image reversible information hiding can be used as an important complementary technology to digital signatures, embedding image integrity information such as hash values into medical images. The receiver can not only recover the medical image reversibly, but also correctly extract the legal authentication information to protect the integrity of the medical image. The second is encrypted image management in telemedicine. Image-oriented telemedicine is widely used in the future information society. For the purposes of confidentiality and privacy protection, medical images sometimes need to be encrypted beforehand and sent to third parties such as the cloud for management. The sensitive information such as the medical record is embedded in encrypted image by means of RDH-EI, so that encrypted image management in telemedicine can be realized. [PERSON] [12] firstly proposed RDH-EI, which encrypts images with stream ciphers, and then divides encrypted images into blocks that do not overlap each other. Each block has two groups. By flipping each pixel's three least significant bits (LSBs) in the corresponding group, 1-bit information can be embedded. The receiver extracts extra data through the wave function. However, when the block is small, the error rate becomes high. [PERSON] et al. improved [PERSON]'s method by using a wave function capable of edge matching in [13] and using unbalanced bit flipping in [14]. In [15], [PERSON] proposed a separable RDH-EI algorithm, which can extract additional data in both the encrypted domain and the plaintext domain.