## 1 Introduction  
The security of data communication plays a crucial role with the rapid development of technology such as high-speed optical communications, wireless mobile communication, and high-performance/low-cost computing, along with the continuous emergence of innovative applications. Cryptography, which deals with different data encryption methodologies, provides a secure way for data communication as the encrypted data are incomprehensible to any third party. Data hiding, also known as steganography, is another way to secure messages during data communication [1], by embedding encrypted secret data in meaningful cover contents, such as a digital image, video, or 3D model, before delivering the stego contents conveying the hidden message to the recipient. Since a data-hiding technique does not change the format of the carrier, it offers additional security to cryptographic techniques. With the combination of both aspects, it provides an additional layer of security for the message during data communication [1]. The ultimate goal of data hiding is to conceal a great number of secret messages within a cover medium and deliver this innocuous stego medium to the recipient in a secured approach without arousing the suspicion of eavesdroppers. Digital images, among a variety of cover contents, are of great interest in the data-hiding community due to their high popularity and availability; up to three billion images are shared on the Internet daily. Algorithms developed for image data hiding are broadly classified into two categories: irreversible and the reversible methods [1]. The former schemes permit the recipient to extract secret messages conveyed in the stego images. The latter ones, however, enable the recipient not only to extract secret messages but also to restore the stego image, thereby producing the restored image identical to the original cover one. The least significant bits (LSB) substitution method [2] is a well-known irreversible data-hiding method, which replaces the least significant bits with the hidden secret messages. Matrix embedding [3] and exploiting modification direction (EMD) [4] are two popular irreversible data-hiding algorithms offering high embedding efficiency, and are thereby less vulnerable to steganalytic detection [5; 6]. In these irreversible methods, although the original cover images cannot be restored, they actually provide a great amount of embedding capacity, representing the total number of bits that a cover image can carry. In contrast, reversible data hiding (RDH) enables the receiver to retrieve the original cover image without any image degradation after extracting secret messages, and a number of RDH methods have been proposed in the extant literature. [PERSON] [7] proposed an RDH method based on difference expansion (DE). [PERSON] et al. [8] proposed a histogram shifting-(HS) based data-hiding method. [PERSON] et al. [9] proposed a pixel values order (PVO) method. These approaches have boosted the development of RDH algorithms [10; 11; 12; 13; 14; 15], aiming to increase the payload or to enhance the image quality. A conventional RDH scheme usually generates a single stego image. In 2007, [PERSON] et al. [16] proposed an RDH algorithm, where dual stego images were produced to carry two 5-ary secret digits in a cover pixel pair. Reversible data hiding based on such a dual-image approach both increases the payload and distributes the distortion to dual images, therefore attracting attention from researchers [17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29]. Using a larger size of the magic matrix, [PERSON] et al. [17; 18] improved the payload reported in [16]. Furthermore, taking advantage of the orientation combinations, [PERSON] and [PERSON] [19] increased the embedding capacity. [PERSON] et al. [20] made use of the EMD method to generate the first stego image and adjusted the modification direction to render the second one. [PERSON] et al.