## 6 Cryptography
### Algorithms for Images
#### 6.2.2 Image Authentication Algorithm  
Suresh et al. [125] proposed a new approach to authenticate images using a combination of different chaotic maps along with a newly developed kind of chaotic neural network. This authentication code is carried by the transmitted image itself at near-lossless image content quality, and does not require any extra memory requirement. The initial parameter sensitivity and aperiodicity of chaotic maps and its integration with other chaotic maps and a chaos-based neural network were utilized to create uncertainty in the predictability of authentication code in association with image pixels.  
The proposed scheme for an image authentication scheme uses the Lozi map corresponding to (3) under the form  
\[\left\{\begin{array}{l}S_{n+1}=1-P|S_{n}|+y_{n},\\ y_{n+1}=QS_{n},\end{array}\right. \tag{85}\]  
where \(n\) is the iteration number and the logistic map  
\[X_{n+1}=\
u X_{n}(1-X_{n}). \tag{86}\]  
is as follows [125]:  
* _Step 1: Generate the random initial condition under the defined range for the Lozi and logistic maps._
* _Step 2: Generate more samples based on the number of image pixels using the Lozi map and logistic map (say, \(L_{i}\) and \(D_{i}\))._
* _Step 3: Remove the LSB value from all pixels \(P_{ri}\) and normalize the pixels (say, \(P_{i}\))._
* _Step 4: From the initial starting sample of the Lozi chaotic sequence, start adding samples with normalized pixels, i.e., \(S_{i}=P_{i}+L_{i}\)._
* _Step 5: Keep the range of \(S_{i}\) within \([0,1]\) by subtracting any values greater than 1 (say, \(S_{i}\))._
* _Step 6: From the initial starting sample of the logistic chaotic sequence, start adding samples with \(S_{ni}\), \(Z_{i}=S_{ni}+D_{i}\)._
* _Step 7: Keep the range of \(Z_{i}\) within \([0,1]\) by subtracting any values more than 1 (say, \(Z_{i}\))._
* _Step 8: Take each value of \(Z_{i}\) as an input to the ANN, which has weights trained using the Lozi map and generate the corresponding output (say, \(AN_{i}\))._
* _Step 9: Evaluate the mean value of \(AN_{i}\) (say, \(m\)) and apply thresholding with \(ANi\geq m\) to generate the authentication code (say, \(A_{i}\))._
* _Step 10: With \(P_{ri}m\) add the authentication code \(A_{i}\) to obtain the image with authentication code._For the Lozi map, the values of \(y\) are normalized in the range \([0,1]\) to each decision variable in the n-dimensional space of the optimization problem. This transformation is provided by  
\[Z_{k}=\frac{y_{k}-\lambda_{min}}{\lambda_{max}-\lambda_{min}} \tag{87}\]  
where \(y\in[-0.6418,0.6716]\), \([\lambda_{max},\lambda_{min}]\) equals \([0.6418,-0.6716]\), \(P=1.7\), and \(Q=0.5\) (this is similar to the normalization used by Araujo and Coelho in (30)). For the logistic map, \(\
u\) is defined with uniform randomness in the range of \([3.65,3.95]\).  
Various experiments were carried out with initial conditions defined randomly in the range of \([0,0.5]\). Extracting the right authentication code is only possible with the correct information about the initialization of both chaotic maps and neural network weights. It is impossible to extract this information from image pixels.  
The proposed schemes completely destroy the relationship among pixels in the generation of authentication code. Experimental results show that this scheme has efficient in authentication and localization and does minimal damage to the original image; furthermore, it can be applied to both grayscale and color images.