# A Novel Fractional Sine Chaotic Map and Its Application to Image Encryption and Watermarking
###### Abstract
Chaotic systems have gained popularity due to their unique characteristics such as sensitivity to initial conditions, unpredictability, and randomness. Their implementation in cryptography offers advantages such as cost-effectiveness, computational efficiency, and complexity. The study of chaos is not a recent development; it was first introduced by [PERSON] in the 1880s while investigating the three-body problem. However, at that time, there was no formal theory to support the observation of non-periodic oscillations, which hindered the progress of the field. It was in the later part of the twentieth century that [PERSON] discovered the sensitivity of chaotic systems to initial conditions, where even a small change can lead to a significantly different outcome. The advent of digital computers played a crucial role in advancing the field. The computational complexity of manually calculating repeated iterations for chaos theory was simplified by electronic computers. As a result, the field of chaos theory experienced significant growth, leading to the development of numerous chaotic systems with strange attractors and hidden attractors. The inherent randomness of chaotic systems makes them well-suited for image encryption processes. When it comes to encrypting images using chaotic maps, the selection of the system used to construct the algorithm becomes a critical task. During the initial phase of the study, logistic map [6] and tent map [7] are used for encryption. The limitations in implementing the system are that the key space is so small that a breach of security is possible. Later, to increase the level of security, higher-dimensional systems, such as the Henon map [8], Tinkerbell map [9], 5D hyperchaotic map [10], are introduced for better key space. An image encryption scheme with a combined 1D map was discussed in [11] and a coupled sine-logistic map was illustrated in [12]. The generalisation of integer-order chaotic systems with arbitrary real or complex orders provides enhanced complexity and non-linearity that highly suit the process of image encryption. Fractional-order derivatives, which are non-local, have the ability to bring the factor of memory into effect. The fractional Lorenz system was considered for the encryption scheme by [PERSON] et al. in [13]. Application of the fractional-order neural network model was presented in [13] and the authors analysed the characteristics of the complex chaotic system with fractional-order in the image encryption process in [14]. A chaotic system with improper fractional-order is considered for image encryption in [15] and the investigation with improper fractional laser systems was carried out by [PERSON] et al. in [16]. Encryption based on the coexisting attractors of the memristive fractional system was discussed in [17] and the fractional system together with the Fisher-Yates algorithm was employed in [18]. Consequently, during the last decade, discrete time fractional-order systems have gained importance due to their diverse applications. The development of the theory for the field of discrete fractional calculus was due to the pioneering works of [PERSON] and [PERSON] in [19], [PERSON] in [20], and [PERSON] in [21]. Application of discrete fractional calculus to tumour immune systems was discussed in [22], and heat transfer fins and pantograph equations were investigated in [23; 24]. Dynamic analysis of the fractional difference chemical reaction model was performed in [25]. The implementation of fractional order in the discrete maps enhances the chaotic nature of the maps. Some recent research contributions based on fractional difference chaotic systems include image encryption with compression based on Bayesian sensing and the 2D fractional discrete time chaotic map in [26]. The application of discrete fractional neural networks to encryption was presented in [27]. The elliptic curve cryptosystem and the 2D fractional-order discrete map are employed for the process of encryption in [28]. [PERSON] et al.