# Chaotic Image Encryption: State-of-the-Art, Ecosystem, and Future Roadmap
## 3 State-of-the-Art
### State-of-the-Art in Chaos Aspect
#### 3.1.2 Chaos Sources  
Chaos can be created using mathematical or physical sources. In the following, we discuss the role of both types in the state-of-the-art of chaotic image encryption. Well-known mathematical chaos sources commonly used for image encryption purposes are studied below:  
* Chaotic systems and maps These are functions originally designed for creating chaos. Chaotic systems and maps play a critical role in chaotic image encryption. To mention a few, one may refer to the following:
* Fractional-order chaotic system Fractional calculus goes back more than 300 years, with modern studies focusing on systems such as the fractional-order Chen, Lorenz, and Liu systems [42]. A novel switching fractional-order chaos system was proposed by [PERSON] [42] and utilizes controlling switches to switch between its sub-systems and achieve a strong chaos source for applying the exclusive Or (XOR) operation against the plaintext image. Another algorithm utilizing fractional-order systems was proposed by [PERSON] [43], which opts to use a more standard third-order fractional system, as well as a novel Josephus scrambling algorithm and circular diffusion to achieve desirable encryption properties and resilience against common attacks. * Arnold cat map Arnold mapping is a well-know transposition chaotic map that, in the context of cryptography, was used by [PERSON] and [PERSON] [44], as well as [PERSON] et al. [45] to provide a method of permutating and de-correlating adjacent pixels in their proposed encryption algorithms. Both algorithms were proven to exhibit a large keyspace with high key sensitivity and be capable of resisting common attacks such as brute force, entropy, CPT, and KPT. * Coupled map lattice A Coupled Map Lattice (CML) is a form of spatiotemporal chaos map efficient for random number stream generation. In one use case, [PERSON] [46] proposed a novel implementation of the CML to create encryption streams dependent not only on initial values, but also on intermediate cipher images by using said ciphertexts to modify the CML parameters. This adds a layer of plaintext dependency, which aids in the defense against several attacks. * Lorenz map A Lorenz system is a type of differential equation that is highly susceptible to initial conditions. [PERSON] and [PERSON] [47] proposed an image encryption procedure in which the key is composed of the three inputs to a 3D Lorenz system and utilizes the chaotic nature of said system to provide strong security. * Logistic map A logistic map is a relativity simplistic mathematical mapping function, which when influenced by particular control values, acts chaotically. An algorithm proposed by [PERSON] and [PERSON] [48] utilizes a two-step interactive logistic map, where the next input is dependent on the previous two outputs, as a source of chaos. Similar work was performed by [PERSON] et al. [49], where they used a more standard logistic map and paired it with a hyper-chaos system to improve key generation effectiveness. Likewise, [PERSON] and [PERSON] [50] also found success utilizing the logistic map for key generation. * Tent map [PERSON] et al. [51] proposed an image encryption algorithm using the CTM, and the rectangular transform was later analyzed by [PERSON] et al. [52] and improved upon to better protect against plaintext attacks such as CPT and KPT. The Chaotic Tent Map (CTM) is a mapping function that, when configured with control values in a particular range, behaves chaotically. * Lotka-Volterra A Lotka-Volterra chaotic system is a third-order differential equation in a similar family to other systems such as Lorenz, Rossler, Shua, and Chen. In a particular case study by [PERSON] et al. [53], an encryption procedure was proposed that utilizes the Lotka-Volterra chaotic system to aid in the creation of Substitution boxes (S-boxes) with strong confusion properties.