## 1 Introduction  
Visual cryptographic scheme (VCS) is a secret image-sharing method. In 1994, [PERSON] and [PERSON] proposed a common type of \((t,n)\)-threshold VCS [1]. In this threshold \((t,n)\)-VCS, a secret binary image is encrypted to get \(n\) images (referred to as shares), in such a way that it is impossible to learn anything about the original secret image, and then distributed sequentially to \(n\) participants. The secret image recovery method is straightforward; after collecting any \(t\) (or more) shares and stacking them together, and performing a simulated stacking operation by Boolean 'OR' operation, secret information can be revealed directly by the human visual system to identify the secret information without relying on complex calculations or cryptography knowledge. If any \((t-1)\) or fewer shares are available, then no secret will be revealed. Early encryption methods for VCS generally used pixel expansion techniques, meaning each pixel on a secret image is expanded into \(m\) pixels (\(m\geq 2\)) on the shares. Thus, the share size will be \(m\) times the size of the secret image. However, this will cause the restored image to be deformed and expanded. [PERSON] and [PERSON] proposed a random grid-based visual cryptographic scheme (RGVCS) that can encrypt black and white (binary) images, which can solve the problem of pixel expansion [2]. In other words, an RGVCS creates the shares and restored image without pixel expansion. Inspired by [PERSON] and [PERSON], [PERSON] proposed an RGVCS for grayscale/color images [3]. Contrast is the criterion for evaluating the visual quality of the recovered secret image. Its value is between \(-1\) and \(1\), with higher values indicating higher visual quality of the restored image. When the contrast value is \(0\), the recovered image is a meaningless image; and if the contrast value is negative, the image is a black-and-white reversed image of the original secret image. The traditional stacking operation in VCS is an OR operation, and the contrast of the restored image in an OR-based RGVCS can only reach 0.5 at most. Such low reduction quality limits many development possibilities. To resolve the problem of poor contrast in OR-based RGVCS, XOR-based RGVCS is proposed. The visual quality of the recovered image by the XOR operation can be greatly improved because the contrast value can theoretically be equal to 1. In 2003, [PERSON] et al. introduced a new visual cryptosystem using light polarization, the operation of which is mathematically described by a XOR operation (a modulo-two addition) [4]. So that the XOR-based VCS can be implemented by using a small, cheap and lightweight decryption display, and it is more suitable for practical situations. In 2007, [PERSON] et al. proposed a XOR-based (\(n\), \(n\)) RGVCS [5]. Although the generated share in [PERSON] et al. 's paper is not pixel-expanded and perfectly displays the secret image, some issues are still refined. One of them is that the shares are meaningless, which will attract the attention of cybercriminals and make it hard to manage. In 2013, [PERSON] and [PERSON] proposed a (2, 2) generalized XOR-based RGVCS [6], where the average light transmission of a share becomes adjustable. In their study, the visual quality of the shares and recovered image are still not good enough due to the limitations of the design method. In order to facilitate management and avoid unnecessary suspicion and attacks, VCS has added a new concept to make shares meaningful in recent years. Its main purpose is to make shares that look like a random grid no longer meaningless, and users can easily identify who is who with the naked eye. In 2015, [PERSON] et al. proposed a (\(n\), \(n\)) XOR-based RGVCS with meaningful shares [7] for increasing the image quality. They define a variable \(\beta\) to balance the visual quality of the restored image with shares. As the value of \(\beta\) becomes larger, the visual quality of the restored image will increase, while the camouflage result of the share will decrease.