New chaotic map and its application in encrypted color image

Introduction

Chaos is a phenomenon very close to randomness, which occupies an important position in cryptography, but determinism and dynamics distinguish it from randomness. In the past three decades, chaos has swept through most sciences (mathematics, physics, biology). It is defined by a nonlinear equation that is very sensitive to initial conditions. The expansion of chaos theory is closely related to the development of computer science and new mathematical advances (modeling, simulation, etc.). Like all new theories, chaos theory is still the subject of many controversies. Various forms of disputes have caused disputes over legal opinions and interpretations. Will science be able to explain it more and more, or is it impossible to understand the world by accident? In fact, for scientists, this is a matter of defining the complexity of the phenomenon they are studying. Chaos as understood by scientists does not mean that there is no order. In fact, this is related to unpredictability, because the final state is very sensitive to the initial state, so long-term evolution cannot be predicted. We believe that the difference between chaos and randomness is the most important point for understanding chaos. Indeed, there is always a tendency to believe that a phenomenon is unpredictable due to the large number of parameters involved. In his description, this prompted us to give a probabilistic method, which by definition can satisfy a certain degree of freedom completely satisfactorily. Randomness. As far as chaos is concerned, this is actually not the case, and the behavior of the chaotic system seems to be random. But in reality, this behavior is described in a deterministic way by fully deterministic nonlinear equations, that is, in particular using mathematics that allow accurate and deterministic methods. To explain with a famous advertisement, a person can write: "Looks like an opportunity, tastes like an opportunity, but not accidental. With the passage of time, people have made several attempts to build a chaotic graph, and realized the password A large number of chaotic graphs used in learning. Domain [1] . Other technologies use chaotic maps to construct hash functions [2] . On the other hand, other technologies use chaotic cards in symmetric encryption systems [3] .

There are also some the technology combines several chaotic maps to improve performance. Their system [4][5][6] . In the absence of any deterministic formula for generating random numbers, tables of such numbers appeared.

Pseudo-random number generator

Finding that they were unable to master random numbers, researchers quickly turned to the generation of pseudo-random numbers defined by mathematical relationships that produce the same sequence under the same conditions. Among all these technologies, we mentioned the most important ones

Von Neumann Generator

In 1946 Von Neumann proposed the following pseudo-random number generator  Take an integer (x 0 ) of n digits  Calculate (x 1 = x 0 2 )  Take the (n) middle digits  Restart

Linear congruence generator

The linear congruential generator was introduced by Lehmer, and is still popular in today's methods for generating pseudorandom numbers quickly. The sequence of random numbers (x n )is created as follows:

In order to be able to choose a seed x 0 without constraints between 0 and n -1, it is necessary to try to maximize the generator period. However, it turns out that the values of a and c are known, which makes it possible to obtain a maximum period (equal to n).

The period of a linear congruential generator is maximum if and only if:

II. Selected popular chaotic maps

Functions that generate chaotic sequences can be divided into two categories: one-dimensional sequences and multi-dimensional sequences. Chaotic functions are rare in the literature, and there are only a dozen functions used in cryptography.

One-dimensional chaotic map a. Logistic recurrence

Logistic recursion [7] is a simple example of nonlinear sequence. It is defined by a simple relation managed by a second order polynomial described by the following recurrence relation

b. The Skew Tent Map

The Skew tent map [8] will be redefined as the equation below

PWLCM Map

It is a real linear sequence [9] by pieces defined by the equation below

The simplicity and robustness of this card encourages researchers to use it in cryptography.

III. Recommended knowledge

Before revealing the structure of this new map, it is necessary to define some basic properties. Let (f) be a continuous function over the interval(I) and defined by the equation

We are going to give some definitions to clarify all the points of the article

Trajectory

Let (x) be an element of (I), we call the trajectory of x the set of iterates of(x) by the function (f). This set is defined by

Periodicity

We say that x ∈ I is periodic if there is an integer (r) such that

Period

The (l) period of an element x ∈ I is the smallest integer r such that

We notice that if (l) is the period of element (x) then

Derived from f

The function (f) is continuously derivable and its derivative is given by the expression

(𝐟) fixed points

The two stationary points of (f) are

V. Chaotic sequence building

The sequence (x n ) is defined by the following expression

Initial Condition Sensitivities

To measure the sensitivity to the initial conditions of the sequence (x n ) defined by function (f), we have to calculate the Lyapunov exponent

In our case, we notice that We can conclude from the value of the Lyapunov exponent that the sequence (x n ) defined by the function (f) is sensitive to the initial conditions. This value is higher than the value of the logistics diagram, indicating that it is highly sensitive to initial conditions and control parameters. The logarithmic scale plot shows that the distance between two very close initial conditions varies according to an exponential law.

Sarkovskii's Theorem

All-natural integers are represented in this Sarkovskii order.  The first line represents odd numbers  The row line (n) represents the numbers 2 n-1 (2k + 1)

a. Theorem

Let f: I → I continue. Suppose that (f ) has a periodic point of period( k). If (k ≻ ℓ) according to Sarkovskii's order, then f also has a periodic point of period (ℓ).

b. Corollary (Lie and York)

If (f) admits an item from period 3, then it admits an item of any order. As a result, the function has a chaotic appearance. searches for points (x 0 ) for which there is a k such that

If there is such a point (x n ) then the sequence (x n ) is stationary. For (k = 1), we have

The sequence (x n ) defines is a chaotic sequence under the specified conditions.

We are looking for an elementx 0 > 1 (1+p) such as

If such a point exists then two situations present themselves

In this case the sequence would be stationary from iteration (k)

Feigenbaum's Constants -Renormalization -

In 1975 the physicist Feigenbaum noticed that the general pattern of the logistic sequence was repeated at each bifurcation to within a factor of scale. He then used a process of renormalization. This involves enlarging smaller and smaller parts of the graph and comparing these magnifications to the original pattern. When the enlarged pattern reproduces the first pattern, it is called selfsimulation. As it grows to infinity, the general structure repeats itself. If globally, the duplications are not the same, they keep the same ratios

The first constant intervening horizontally

The second constant occurring vertically

Universality

The same sequence, but defined by another function (f) defines single hump type has the same properties 6. Some simulations { x 0 = 0,45 p = 1,5

Conclusion

Faced with various difficulties in constructing random numbers, researchers are committed to using generators that follow simple mathematical formulas to create pseudo-random numbers. With the passage of time, chaos theory suddenly appeared, and due to the need to use passwords with such numbers to create private encryption keys, chaos was born. We have witnessed the emergence of some chaotic maps in color image encryption. Our article contributed to the creation of a new chaotic map, and implemented the chaotic map in the color image encryption example by enhancing the classic Hill function. The proof of chaos is proved using Sarkovskii's theorem and the inference of Lie and York. The map will be added to the list of existing maps on the market to increase this range.

References

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