# Joint Image Compression and Encryption Using IWT with SPIHT, Kd-Tree and Chaotic Maps
## 2 Related Works
### Image Encryption with Chaotic Maps  
Many encryption algorithms, specifically for digital images, have been proposed to meet the requirements of robust image encryption applications. Among them, the chaos-based encryption algorithms are popular due to several excellent properties, such as ergodicity, randomness, unpredictability, and high sensitivity to initial conditions and various parameters [5; 6; 7].  
However, most of these schemes encounter various problems that limit their robustness and security [19; 20; 21; 22; 23; 24]. It is well known that some chaotic maps completely lose their chaotic behavior and become periodic when they are discretized [25]. In addition, the security of chaotic systems depends on control parameters and shows specific behavior on these values, which can be determined in some cases. Therefore, one of the main problems in chaos-based cryptography is the selection of the chaotic system [21].  
Various types of chaotic maps have been proposed, such as the Logistic map, Sine map, Cubic map, Tent map, Baker map, Arnold map, Chen map and many more. Among these, the most primitive and complex is the logistic map, which arises from a nonlinear dynamical equation [24]. However, all these maps suffer from security deficiencies. The security of chaotic maps depends upon the number of parameters that are used in these chaotic cryptosystems. To increase the number of these parameters, multi-dimensional and multiple iterative chaotic maps have been used [26]. However, security deficiencies remain [27]. The control parameters in chaotic systems act as a key on which security of the system is based. It is well known that the secret key is of fundamental importance in the security of all cryptosystems. There exist several tools and methods for estimating the initial values of  
Figure 2: (**a**) SOT parent-child relationship (**b**). Scanning sequence of wavelet coefficients in SPIHT.  
chaotic systems by determining the behavior of the systems [28]. Therefore, some researchers used chaotic and non-chaotic encryption schemes together [29].  
Recently, another type of chaotic system was reported, namely, the hyper-chaotic system, which has superior properties compared to general chaotic systems. It has two or more Lyapunov exponents and more-complex structures. These characteristics are considered to provide a strong entropy source. However, cryptanalysis studies demonstrate that hyper chaotic schemes can be broken by chosen-plaintext attack; thus, these are also insecure and not suitable for secure image communication [30].  
For a fixed chaotic system, with the development of chaos theory, the chaotic orbits may be estimated, and their parameters or initial values may be predicted [31]. Therefore, it is not secure to use a fixed chaotic system in the encryption algorithm. Compared with fixed chaotic maps, time-varying chaotic maps are more complex and the output sequences are non-stationary, which makes them much more difficult to predict and analyze [31]. The varying parameters disrupt the phase space of the system; hence, the system can resist phase-space reconstruction attacks and chaotic signal estimation technologies effectively [32]. Thus, an image encryption algorithm that is based on a time-varying chaotic system is more secure. A direct and effective method for constructing a time-varying chaotic system is through varying parameters of the chaotic system. Recently, several chaotic image encryption algorithms that are based on varying parameters have been proposed [31;32]. In [32], a parameter-varying logistic map, which can overcome the weaknesses of the logistic map and resist phase-space reconstruction attacks, is used to shuffle the plain image.