# A Hybrid Domain Color Image Watermarking Scheme Based on Hyperchaotic Mapping
## 2 Background  
### Lorenz Hyperchaotic System  
Chaotic phenomena are common in nonlinear systems that are sensitive to initial conditions and control parameters, and their orbits exhibit randomness and unpredictability. This feature makes chaotic systems have significant advantages in the field of security, such as randomness, complexity and reliability ([PERSON] et al., 2019). Therefore, chaotic systems are often used as encryption methods. As a classical chaotic model, the Lorenz hyperchaotic system is defined as follows:\[\begin{cases}\frac{\mathrm{d}x}{\mathrm{d}t}=a(y-x)+w,\\ \frac{\mathrm{d}y}{\mathrm{d}t}=cx-y-xz,\\ \frac{\mathrm{d}z}{\mathrm{d}t}=xy-bz,\\ \frac{\mathrm{d}w}{\mathrm{d}t}=-yz+rw.\end{cases} \tag{1}\]  
The hyperchaotic system needs to meet the following conditions. First, its phase space is at least four-dimensional. Secondly, it has at least two positive Lyapunov exponents [21; 22].  
In Equation (1), \(a\), \(b\), \(c\) and \(r\) are control parameters. When \(a\ =\ 10\), \(b\ =\ 8/3\), \(c\ =\ 28\), and \(-1.52\leq r\leq-0.06\), the system is in a hyperchaotic state. When \(r\ =\ -\ 1\), the four Lyapunov exponents of the above formula are \(\lambda 1\ =\ 0.3381\), \(\lambda 2\ =\ 0.1586\), \(\lambda 3\ =\ 0\), and \(\lambda 3\ =\ -15.1752\). The initial value range of the [PERSON] hyperchaotic system is \(x_{0}\in(-40,40)\), \(y_{0}\in(-40,40)\), \(z_{0}\in(1,81)\), \(w_{0}\in(-250,250)\). Figure 1 shows six strange attractors of the Lorenz hyperchaotic system, where the phase diagram of Figure 1c resembles a butterfly, and is also known as the butterfly attractor.  
Figure 2 shows the encryption of a \(256\times 256\) grayscale image using the chaotic sequence generated by the [PERSON] hyperchaotic system in the case of initial values \(x_{0}\ =\ 3.5739\), \(y_{0}\ =\ 3.7979\), \(z_{0}\ =\ 4.2980\), \(w_{0}\ =\ 3.9282\).  
From Figure 2, it can be found that the histogram of the original image and the decrypted image is almost the same, indicating that the encryption process has little effect on the image quality. However, the encrypted image histogram is significantly different from the original image, and its distribution is more uniform. As a visual display of image pixel frequency distribution, histogram is an important parameter to measure the anti-statistical analysis ability of the image encryption algorithm. In general, the higher the uniformity of the encrypted image histogram, the stronger its ability to resist statistical analysis attacks. The histogram of the original image often has specific rules and characteristics, while the histogram of the encrypted image tends to be flat or have uniform distribution, which is significantly different from the original image. This reflects that the image encryption algorithm based on the Lorenz hyperchaotic system significantly reduces the statistical  
Figure 1: Attractor projection of Lorenz hyperchaotic system. (**a**) x-y plane, (**b**) x-z plane, (**c**) x-w plane, (**d**) y-z plane, (**e**) y-w plane, (**f**) z-w plane.  
correlation between pixels, making it difficult to reveal the obvious rules of the image through histogram analysis, thereby improving the security of image encryption.