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Top1. Introduction
Combinatorics is a branch of mathematics studying the enumeration, combination, and permutation of finite or countable discrete structures. It has applications in Mathematics Optimization, Computer Science, and Statistical Physics. Enumerative combinatorics is the most important area of combinatorics and mainly concentrates on counting the number of certain objects. (Gross, J. L., 2007)
The chessboard is a type of checkerboard that contains 64 square and arranged in 8 × 8 blocks. It contains 32 movable chess pieces. They are king, queen, rook, bishop, knight, and pawn. Each chess piece has its unique way to move. There are some similarities between the moves of various pieces except the knight move in the straight line horizontally, vertically, and diagonally. All the pieces except the knight cannot jump over other pieces (Watkins, J. J., 2007).
Steganography is an art of hiding the secret information in another transmission medium known as cover to achieve secret communication. The important requirements (Taleby Ahvanooey et. al, 2019, Cheddad, Condell, Curran & Mc Kevitt, 2010) for steganographic methods are imperceptibility, capacity, and robustness to some types of attacks. However, it is not practical to simultaneously satisfy all of the above requirements. Steganographic methods are mainly divided into two groups: spatial domain (Malik, A. et al., 2019, Dhawale, C. A., & Jambhekar, N. D., 2017, Chen & Cheng, 2004; Provos & Honeyman, 2003) and frequency domain (Aqeel, I et al., 2018, Chantrapornchai, C., & Preechasuk, J., 2017, Kumar & Muttoo, 2009, 2011, 2013). Information hiding in the spatial domain is a popular steganographic technique that embeds the bits of the secret message directly into the least significant bits (LSB) plane of the cover. LSB based approaches just modify the image pixels without considering the position of the pixel. Therefore, the distortion of the cover image is increased and it may be easily detectable by the steganalysis approaches such as RS method and sample-pair method (Dumitrescu, S., Wu, X., & Wang, Z., 2003). To overcome this problem, we need to develop such a method that may robust against steganalysis approaches. Mathematical chess problems can help to develop such robust steganographic approaches that have low distortion.
In the literature, some famous steganography approaches are based on chess problems. Brain Lang (2004) proposed a new steganographic approach to protecting sensitive information using a chessboard format portable game notation (PGN) standard. The PGN format is a universally accepted text-based format for documenting chess games. Muttoo et al. (2012) proposed a data hiding method based on eight queen solutions. This method helps in randomizing the bit selection in a cover image for hiding purpose and further hiding those randomize 8 × 1 blocks, which satisfy the matching between secret information and eight queen solutions. Chakraborty et. al. (2012) proposed a steganographic approach in LSB substitution where the n-Queen matrix acts as a key. Kumar et al. (2014) also proposed a data hiding technique based on LSB using eight queen solutions. The proposed method provides better security and high payload capacity. Further, another improves technique based on eight queens proposed by Bansal et al. (2014). This method is based on eight queen solutions and pixel mapping for embedding a high payload of secret information. Bansal et al. (2016) also proposed a steganography technique for improving the security in exploiting the modification direction method using knight tour. Bhatia M. K. (2017) proposed a steganography method which is based on the solution of the rook. In this method, the selection of pixel is based on the solutions of 8-Rooks problem of placing 8-non-attacking Rooks on an 8×8 chessboard. Some other steganography techniques proposed by the Zeyed S. Y. et al., 2019 and Z. S. Y. Alsaffawi, 2016 are based on knight tour. These techniques are mainly for video steganography. These proposed methods improve the capacity and security by a selection of randomizing bits using various chess problems but distortion may compromise. Therefore, we propose a new steganographic approach inspired by rook which can deal with distortion of the image while embedding. The rook is one of the chess pieces which moves in a straight line either horizontally or vertically through any number of unoccupied squares until it reaches the end of the board or it is blocked by another piece (Eade, J., 2016) as shown in Figure 1. The rook captures on the same path it moves, by occupying the square on which an enemy piece stands.