# A Zero-Watermarking Algorithm Based on Scale-Invariant Feature Reconstruction Transform
## 2 Related Work
### Logistic Chaotic Encryption  
Zero-watermarking algorithms typically need to consider security issues. Therefore, this paper adds encryption steps to the proposed algorithm to encrypt both the binary feature matrix and the watermarking information. In the process of constructing the binary feature matrix, this paper ensures that the statistical properties of the binary feature matrix comply with cryptographic statistical principles by introducing texture feature constraints and multiple threshold determinations. Building upon this foundation, this section further incorporates the logistic chaos encryption algorithm to encrypt the watermarking information and feature matrix of the image multiple times. In subsequent security experiments, this paper simulates exhaustive attacks under extreme conditions, and the experimental data confirm the security of the algorithm. During the process of layer-by-layer encryption, even if the specific steps of this algorithm are publicly disclosed, attackers would find it difficult to decrypt these copyrighted digital carrier images and tamper with the bound watermarking information without the key.  
The logistic mapping formula originates from mathematical models describing the growth trends of biological populations in natural environments. Subsequent scholars utilized the Euler method to discretize the continuous logistic growth model and introduced a dimensionless population density parameter \(\upmu\). This yields the logistic mapping expression that is widely used in the field of digital image encryption, as depicted in Formula (4).  
\[x_{k+1}=\mu x_{k}(1-x_{k}) \tag{4}\]where the control parameter \(\mu\in(0,\,4)\), \(x_{k}\in(0,\,1)\) represents the current value, and \(x_{k+1}\) is the value after iteration calculation. The mathematical definition of the logistic chaos model is based on equations of nonlinear dynamical systems. Its behavior exhibits irregular trajectories in phase space, demonstrating high sensitivity to initial conditions and unpredictability. When the control parameter of the chaos model reaches a certain value range, specifically \(3.5699456<\mu\leq 4\), the system transitions from a stable state to an unpredictable chaotic state. From a geometric perspective, this property is specifically manifested by the distribution of output values mapped using input values, which creates fractal structures within the coordinate system, as illustrated in Figure 3.  
Therefore, in image information protection, the logistic mapping can be used to achieve the bitwise encryption of pixel values [12]. This involves performing XOR operations between the image matrix and the sequence generated by the logistic mapping. Consequently, each pixel value in the image undergoes random changes. This pixel-level encryption makes both local and global features of the image difficult to discern and predict, thereby enhancing the security of watermarking embedding algorithms.  
This paper utilizes multiple chaotic sequences generated by the logistic chaos mapping for the following encryption operations: pixel value encryption of the feature matrix, position mapping of the watermarking information, and iterative calculation of the control parameter \(\mu\). To disrupt the correlation of the watermarking information image in two-dimensional space, pixel positions undergo Arnold mapping for position scrambling [13], followed by the introduction of chaotic keys for pixel value transformation. For each pixel, the algorithm applies different keys for encryption processing at both the spatial position distribution and pixel amplitude levels, ensuring the security and imperceptibility of the original copyright information and significantly reducing the probability of decryption by attackers.