Characteristics Of Chaos And Cryptography

I. Chaos and Cryptography
Chaos theory has been established since the 1970s in many different research areas, such as physics, mathematics, engineering, biology, and others. The most well-known characteristics of chaos are the sensitivity to initial conditions and have random-like behaviors. Many fundamental properties of chaotic systems have their corresponding counterparts in traditional cryptosystems. Chaotic systems have several significant features favorable to secure communications, such as ergodicity, sensitivity to initial conditions and control parameters, and random-like behavior \cite{ch1,ch2,ch3}. With all these advantages, scientists expected to introduce new and powerful tools of chaotic cryptography. So, chaos has become a new

Chaos describes a system that is sensitive to initial conditions to generate an apparently random behavior but at the same time it is completely deterministic. These properties of chaos have much potential for applications in cryptography as it is hard to make long-term predictions on chaotic systems. First, being completely deterministic means that we can always obtain the same set of values provided we have exactly the same mapping function and initial conditions. Because chaotic functions are sensitive to initial conditions, any slight difference in the initial values used means that the ciphertext produced using chaos will be completely different. This means that the system will be strong against brute-force attacks as the number of possible keys is large.
There is a set of properties that summarize the characteristics observed in chaotic systems. These are considered the mathematical criteria that define chaos. The most relevant are:
1. Dynamic instability: also referred as butterfly effect, it is the property of sensitivity to initial conditions, where two arbitrarily closed initial conditions evolve with significantly different and divergent trajectories \cite{ch6}.
2. Topological mixing: intuitively depicted as mixing coloured dyes, it means that the system will evolve in time so that any given region of states is always transformed or overlaps with

b. Chebyshev map

The Chebyshev map is a dynamical system defined as \cite{ch14}:

(2) Where , is an initial seed, and denotes the degree of the Chebyshev map. When the seed and , Eq. (2) is a nonlinear ergodic map. Figure 2 shows the bifurcation diagram of the Chebyshev map. The reason for choosing Chebyshev map is its simplicity and higher security level compared to some other chaotic systems.

Figure 2: Bifurcation diagram of the Chebyshev map.

c. The asymmetric tent map

The asymmetric tent map is one-dimensional dynamical system and is essentially a distorted version of the tent map. It is defined by Eq. (3): (3)
Where and are the initial condition and the control parameter, respectively. More details of this chaos map can be found in