# A Novel Fractional Sine Chaotic Map and Its Application to Image Encryption and Watermarking
## 3 Discrete Time Fractional-Order Sine Chaotic Map (DFSCM)  
Many existing discrete fractional maps and their modified versions do not exhibit long-term chaotic behaviour, which is a desirable characteristic for implementing chaotic maps in cryptographic applications. Chaotic maps used in cryptosystems should possess features such as maximum randomness, high non-linearity, and long-range chaotic behaviour. However, several discrete fractional-order maps employed in cryptographic applications have limitations such as limited chaotic range, a small key space, and weak complexity. To address these issues, researchers have developed modified versions of chaotic maps specifically designed for encryption applications. One drawback of maps with short-term chaos is that the probability of selecting secret keys beyond this region that will result in a chaotic regime becomes very low. This limitation compromises the security of the cryptosystem. To overcome these challenges associated with short-term chaotic behaviour and minimal non-linearity, this article presents the construction of a new discrete time fractional map capable of exhibiting long-term chaotic responses.  
\[\begin{split} x(n+1)=&\sin(y(n)(1+b\ x(n)))+x_{n},\\ y(n+1)=&\sin\Bigl{(}x(n)+cy^{2}(n)\Bigr{)}+y_{n}. \end{split} \tag{7}\]The block diagram for the proposed discrete (7) is given in Figure 1, where \(D\) represents one delay in discrete time, \(\otimes\) and \(\oplus\) represents the multiplier and addition, respectively. A discrete time fractional-order sine chaotic map (DFSCM) obtained from (7) is  
\[\begin{split}\Delta_{\kappa}^{\varnothing}x(n)=&\sin( y(\varpi+n-1)(1+b\ x(\varpi+n-1))),\\ \Delta_{\kappa}^{\varnothing}y(n)=&\sin\Bigl{(}x( \varpi+n-1)+cy^{2}(\varpi+n-1)\Bigr{)},\end{split} \tag{8}\]  
where \(\Delta^{\varnothing}\) is the Caputo fractional difference operator with \(0<\varpi\leq 1\), \(n\in\mathbb{N}_{\kappa+1-\varpi}\), \(b,c\) are real parameters. The system is now converted to a numerically feasible form employing Theorem 1 with \(\kappa=0\) as follows  
\[\begin{split} x(n)=& x(0)+\frac{1}{\Gamma(\varpi)} \sum_{\theta=1}^{n}\frac{\Gamma(n-\theta+\varpi)}{\Gamma(n-\theta+1)}\bigg{(} \sin(y(\theta-1)(1+b\ x(\theta-1)))\bigg{)},\\ y(n)=& y(0)+\frac{1}{\Gamma(\varpi)}\sum_{\theta= 1}^{n}\frac{\Gamma(n-\theta+\varpi)}{\Gamma(n-\theta+1)}\Bigl{(}\sin\Bigl{(}x (\theta-1)+cy^{2}(\theta-1)\Bigr{)}\Bigr{)}.\end{split} \tag{9}\]  
The behaviour analysis is performed with DFSCM (8) by fixing \(\varpi=0.5\), \(b=12\) and initial condition \((0.01,0.01)\) for various \(c\in(0,75)\).  
Some discrete fractional chaotic maps in the literature are considered for comparing their complexity with newly constructed discrete time fractional map by simulation of bifurcation diagrams and maximum Lyapunov exponents based on the Jacobian matrix method proposed in [41].  
**Fractional Tinkerbell map:** Let us consider the fractional discrete time Tinkerbell map [42] of the form  
\[\begin{split}\Delta_{\kappa}^{\varnothing}x(n)=& x^{2}(\varpi+n-1)-y^{2}(\varpi+n-1)+(\zeta_{1}-1)x(\varpi+n-1)+ \zeta_{2}x(\varpi+n-1),\\ \Delta_{\kappa}^{\varnothing}y(n)=& 2x(\varpi+n-1)y(\varpi+n-1)+ \zeta_{3}x(\varpi+n-1)+(\zeta_{4}-1)y(\varpi+n-1),\end{split} \tag{10}\]  
**2D Lorentz map:** The chaotic behaviour of the 2D Lorentz map [43] of the form  
Figure 1: Block diagram for 2D discrete sine map (7).  
\[\begin{split}\Delta_{\kappa}^{\omega}x(n)=&(1+\zeta_{1} \zeta_{2})x(\omega+n-1)-\zeta_{2}x(\omega+n-1)-x(\omega+n-1)[\zeta_{2}y(\omega+n -1)+1],\\ \Delta_{\kappa}^{\omega}y(n)=&(1-\zeta_{2})y( \omega+n-1)+\zeta_{2}x^{2}(\omega+n-1)-y(\omega+n-1),\end{split} \tag{11}\]  
**2D fractional discrete time chaotic map:** The discrete time fractional-order chaotic map investigated in [44] is given by  
\[\begin{split}\Delta_{\kappa}^{\omega}x(n)=&\zeta_{1} sin(y(\omega+n-1))-x(\omega+n-1),\\ \Delta_{\kappa}^{\omega}y(n)=&\zeta_{2}sin(x(\omega+ n-1))-y(\omega+n-1),\end{split} \tag{12}\]