## 6 Cryptography
### Theoretical Algorithms  
The literature includes two articles in which the Lozi map is at the center of theoretical cryptographic algorithms. In the first considered article, Sira-Ramirez et al. [120] proposed the use of nonlinear exact chaotic system state reconstructors for the fast and efficient decoding of multiple discrete-time chaotic encrypted digital messages. After a general description of their method, in order to test the effectiveness of the exact reconstructor scheme in an actual message decoding problem, they applied it using the Lozi map. They considered transmission in a noise-free environment such as the internet. They transmitted to a remote computer containing a reconstructor digital computer program, a message consisting of a digitized file of Van Gogh's self portrait. The encoding of the messages was carried out using the states \(z\) and \(w\) of the following unobservable but constructible Lozi system:  
\[\left\{ \begin{array}{l}x_{k+1}=z_{k},\\ z_{k+1}=-a|z_{k}|+bx_{k}+1,\\ w_{k+1}=x_{k}z_{k},\\ y_{k}=x_{k}.\end{array} \right. \tag{80}\]The exact reconstructor of system (80) for the non measured variable \(z_{k}\) may be obtained trivially, as \(x\) is a one-step delay of \(z\). This was readily derived to be  
\[\left\{ \begin{array}{l}x_{k}=y_{k},\\ z_{k}=-a|y_{k}|+by_{k-1}+1,\\ w_{k+1}=y_{k-1}(-a|y_{k-1}|+by_{k-2}+1).\end{array} \right. \tag{81}\]  
The image and text files were chaotically encoded in a byte-by-byte fashion after a simple normalization of the carrier chaotic states by an appropriate constant factor. Thus, they defined  
\[I_{k}=K_{1}z_{k}+i_{k},T_{k}=K_{2}w_{k}+t_{k},i_{k}=I_{k}-K_{1}\hat{z}_{k},t_{k }=T_{k}-K_{2}\hat{w}_{k},\]  
where \(i_{k}\) and \(t_{k}\) represent the \(k\)th byte of the image file, \(I_{k}\) and \(T_{k}\) are the corresponding \(k\)th bit of the chaotic encrypted image, and the factors \(K_{1}\) and \(K_{2}\) are suitable normalizing parameters, both taken as equal to 1000. The reconstructed states \(\hat{z}\) and \(\hat{w}\) exactly coincide with the original states \(z\) and \(w\). The encryption and decryption programs were written in the C programming language and compiled with Borland \(C^{++}\) version 1.01. The size of the digital image file was approximately 120 KB. The process of encryption was completed in approximately two seconds, as was the decoding process. In the second article, Aliwi and Ajeena [121] used an unconventional generalization of (3) based on the chaotic cosine map recently published in [122]. The Cosine Chaotic Map (CCM) is a chaotification method that enhances the chaotic complexity of existing chaotic maps. This method performs the cosine function alongside a chaotic map that cascades in the system under consideration. Thus, the results provide a new chaotic map having a wide chaotic range within the closed interval \([-1,+1]\). Theoretically, the CCM has properties based on the properties of the underlying seed maps. In the case of the Lozi map, Aliwi and Ajeena considered (3), changing 1 to 3 in the first component  
\[f\left( \begin{array}{c}x\\ y\end{array} \right)=\left( \begin{array}{c}3-a|x|+y,\\ bx,\end{array} \right) \tag{82}\]  
with \(a=-1.8\) and \(b=0.25\). Then, they inserted (82) in CCM:  
\[f\left( \begin{array}{c}x\\ y\end{array} \right)=\left( \begin{array}{c}cos\left(2^{(k+3-a|x|+y)}\right), \\ cos\left(2^{(k+bx)}\right),\end{array} \right) \tag{83}\]  
where \(k\in[10,24]\). Starting with any initial point \((x_{0},y_{0})\) belonging to the basin of attraction of the Lozi map, the iterates randomly fulfill the square \([-1,+1]^{2}\) (see Figure 9). Based on a Knapsack problem approach, a fuzzy chaos-based cryptography method for a chaotic cosine map was proposed in which the system works by designing the Takagi-Sugeno (TS) fuzzy model (see Section 4.1.1) on a chaotic map as a seed map in the cosine map.