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Abstract

In this article, a new multidimensional discrete chaotic system is proposed, and the characteristics and advantages of the multidimensional discrete chaotic system are analyzed. The multidimensional discrete chaotic system is designed using digital technology, and the system has the complexity of hyper-chaotic system, and it can avoid the choice of step size, at the same time, the design of the circuit is simple, the resource occupancy rate is low, it is suitable for the design of digital system, and so on. It can be used in constructing chaotic sequence generator, chaotic encryption, and visual sensor networks.

Keywords

Discrete chaos sequence generator security visual sensor networks secure communication

Introduction

Visual sensor networks (VSNs) are networks that consist of a large number of sensor nodes which have imaging capabilities. With rapid improvements in sensors, embedded computing, and video coding techniques, VSNs have been widely used in traffic enforcement, factory monitoring, habitat sensing, and many other surveillance applications. As the visual sensors of VSNs are usually deployed in unprotected or even hostile environments, security is an issue of great concern. The establishment of a VSNs security system with complex structure is particularly important for simplifying the data transmission encryption algorithm, reducing the resource occupancy rate and improving the flexibility of the application. This article proposes a multidimensional discrete digital chaotic encryption system which can be applied to VSNs system. Chaotic dynamics is an important branch of complexity science and a popular discipline in the past three decades. Chaos is a seemingly random movement that occurs in a deterministic system. The system is described by a deterministic theory, and its behavior is indeterminate, non-repeatable, unpredictable, and that is chaos.^1–6 Chaos is an inherent characteristic of nonlinear systems, which is a common phenomenon in nonlinear systems. Newton’s theory of certainty can deal with mostly linear systems, and linear systems are mostly simplified by nonlinear systems. Therefore, chaos is ubiquitous in real life and practical engineering problems.

Chaotic secure communication is an important field of chaos for application. First, the sensitivity to initial values of the output, and the complexity and the singularity of chaotic system, make the research of chaotic communication more challenging.^7–10 The initial parameters of the chaotic system are usually composed of the initial value of the system and the system parameter. In order to ensure that the output state of the system is in the chaotic region rather than the periodic region, the chaotic system parameters are set to fixed values. The sensitivity to initial values of the chaotic system is important for data encryption and secure communication because the initial value can be used as key space of encryption and decryption. Second, because chaotic system is very sensitive to initial value, it is suitable for secure communication. For one-dimensional Logistic chaotic systems, the key space can reach $1.84 \times 2^{19}$ when the computational accuracy reaches 64 bits. Therefore, it has greater key space for higher dimensional chaotic systems. Third, the difference between the chaotic nonlinear equation and the traditional nonlinear equation is that by controlling the parameters of the system, it is easy to form numerous nonlinear sequences and does not need a very difficult theoretical design. Fourth, although chaotic system has nonlinear characteristics, high output complexity, and features that are difficult to decipher, it is still a deterministic equation that works in a deterministic region and is well suited for encryption, decryption, and codec. Fifth, for the natural chaotic nonlinear system, it both has nonlinear characteristic and simple structure, and it is easy to be digitized and realized. Therefore, the chaotic system is widely used in the field of chaotic secure communication. It is mainly used in constructing chaotic sequence generator, chaotic encryption, and chaotic VSNs.^11–14

The chaotic dynamical system can be composed of continuous chaotic systems or discrete chaotic systems. Whether it is continuous or discrete chaotic system, the multidimensional chaotic system has more chaotic attractors, so the high-dimensional chaotic system has stronger randomness, better confidentiality, greater amount of information, and higher communication efficiency; however, the complexity of the system itself will be greater. Several classical continuous high-dimensional chaotic systems, such as the Lorenz chaotic system, the Chen chaotic system, and the Chua chaotic system, have been implemented by analog circuits. However, if these chaotic systems are applied to the digital circuit system, it is necessary to discretize the continuous chaotic system, that is, discretizing differential equations. The methods of discretization of differential equations are Euler method, Runge–Kutta RK4 method, explicit Runge–Kutta method, implicit Kutta method, Crank–Nicolson method, and so on. In this article, Lorenz chaotic system is taken as an example to illustrate the realization of discrete chaotic system, and a multidimensional discrete chaotic system is designed to analyze the chaotic characteristics and advantages of the multidimensional discrete chaotic system. This kind of multidimensional discrete chaotic system will establish a good module, which can be used in constructing chaotic sequence generator, chaotic encryption, and VSNs.

The characteristics of Lorenz chaotic system

In the analysis of Lorenz dynamics, the main contents of the analysis are chaotic attractors and Lyapunov exponents. The following analysis is carried out one by one.

Lorenz attractor

In explaining the turbulence in the fluid, E N Lorenz proposed the following forced dissipative system model in 1963. The Lorenz attractor is projected in two-dimensional coordinates as shown in Figure 1(a)–(c). Based on the projection of the Lorenz attractor in the three-dimensional (3D) coordinates, as shown in Figure 1(d), we can see the obvious characteristics of chaotic attractors

{\begin{matrix} \overset{\cdot}{x} = - a (x - y) \\ \overset{\cdot}{y} = bx - y - xz \\ \overset{\cdot}{z} = xy - cz \end{matrix}

(1)

Figure 1.

Projection of Lorenz attractor in two-dimensional coordinate: (a) X–Y coordinate projection, (b) X–Z coordinate projection, (c) Y–Z coordinate projection, and (d) X–Y–Z coordinate projection.

Lyapunov exponents of Lorenz system

The Lyapunov exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories in phase space. The number of Lyapunov exponents is equal to the number of dimensions of the phase space, a positive Lyapunov exponent is usually taken as an indication that the system is chaotic, and the greater the positive Lyapunov exponent, the faster the divergence of phase space trajectories, and the system will be more sensitive to the initial parameters, and the chaotic characteristics of the system are also stronger. We can use the Jacobi method which has high computational reliability to calculate the Lyapunov exponents. Let the multidimensional chaotic mapping system equation be as follows

{\begin{matrix} x_{n + 1} = f_{1} (x_{n}, y_{n}, \dots, z_{n}) \\ y_{n + 1} = f_{2} (x_{n}, y_{n}, \dots, z_{n}) \\ \dots \\ z_{n + 1} = f_{r} (x_{n}, y_{n}, \dots, z_{n}) \end{matrix}

(2)

Its Jacobian matrix is

J = [\begin{matrix} \frac{\partial f_{1}}{\partial x_{n}} & \frac{\partial f_{1}}{\partial y_{n}}, \dots, & \frac{\partial f_{1}}{\partial z_{n}} \\ \dots & \dots & \dots \\ \frac{\partial f_{r}}{\partial x_{n}} & \frac{\partial f_{r}}{\partial y_{n}}, \dots, & \frac{\partial f_{r}}{\partial z_{n}} \end{matrix}]

(3)

If the deviation of the initial point $(x_{0}, y_{0}, \dots, z_{0})$ is $δ x_{0}, δ y_{0}, \dots, δ z_{0}$ , the following iterative points and the corresponding Jacobian matrix are obtained by equations (2) and (3)

(x_{1}, y_{1}, \dots, z_{1}), (x_{2}, y_{2}, \dots, z_{2}), \dots, (x_{n}, y_{n}, \dots, z_{n})

\begin{array}{l} J_{0} = J (x_{0}, y_{0}, \dots, z_{0}), J_{1} = J (x_{1}, y_{1}, \dots, z_{1}), \dots, J_{n - 1} \\ = J (x_{n - 1}, y_{n - 1}, \dots, z_{n}) \end{array}

where $λ_{1 i}, λ_{2 i}, \dots, λ_{r i}$ is the eigenvalue of the above n Jacobian matrices, respectively

\begin{array}{l} L E_{1} = \frac{1}{n} \sum_{i = 0}^{n - 1} \ln λ_{1 i}, L E_{2} = \frac{1}{n} \sum_{i = 0}^{n - 1} \ln λ_{2 i}, \dots, L E_{r} \\ = \frac{1}{n} \sum_{i = 0}^{n - 1} \ln λ_{r i} \end{array}

(4)

For the Lorenz mapping, the initial values are $x_{0} = 1$ , $y_{0} = 1$ , $z_{0} = 1$ , and the step size is 0.5. After 600 times of iterations, the Lyapunov exponents of the system are shown in Figure 2, and there are three different Lyapunov exponents. When the number of iterations exceeds 40, Lyapunov exponents of the Lorenz equation tends to be stationary, and the three exponents are $L E_{1} = 0.85922$ , $L E_{2} = - 0.0015763$ , and $L E_{3} = - 14.5208$ , and there is a positive Lyapunov exponent.

Figure 2.

Lyapunov exponents of Lorenz system.

Construction of discrete chaoticsystem by Runge–Kutta method

Compared with low-dimensional chaotic systems, multidimensional chaotic systems have four advantages: (1) multidimensional chaotic system has higher complexity than low-dimensional chaotic system; (2) multidimensional chaotic system can produce multiple sequence outputs and can be applied to multiple key sequence generators; (3) multidimensional chaotic system has more initial parameters and initial variables, and the key space is large; and (4) multidimensional chaotic system is suitable for applications in secure communication systems. Generally, common multidimensional chaotic systems are composed of continuous chaotic dynamical equations, which need to be discretized when applied in digital systems. This section will discretize the Lorenz system.

The Lorenz system is a high-order nonlinear ordinary differential equation. In the solution, the Runge–Kutta algorithm is an algorithm with high accuracy and good gliding property.

The fourth-order Runge–Kutta algorithm of the high-dimensional equations are described as follows

\begin{matrix} y_{i, n + 1} = y_{i, n} + \frac{h}{6} (K_{i, 1} + 2 K_{i, 2} + 2 K_{i, 3} + K_{i, 4}) \\ K_{i, 1} = f_{i} (x_{n}, y_{1, n}, y_{2, n}, \dots, y_{s, n}) \\ K_{i, 2} = f_{i} (x_{n} + \frac{1}{2} h, y_{1, n} + \frac{1}{2} h K_{1, 1}, y_{2, n} + \frac{1}{2} h K_{2, 1}, \dots, y_{s, n} + \frac{1}{2} h K_{s, 1}) \\ K_{i, 3} = f_{i} (x_{n} + \frac{1}{2} h, y_{1, n} + \frac{1}{2} h K_{1, 2}, y_{2, n} + \frac{1}{2} h K_{2, 2}, \dots, y_{s, n} + \frac{1}{2} h K_{s, 2}) \\ K_{i, 4} = f_{i} (x_{n} + h, y_{1, n} + h K_{1, 3}, y_{2, n} + h K_{2, 3}, \dots, y_{s, n} + h K_{s, 3}) \end{matrix}

(5)

where $y_{i, n}$ is the function of $x_{n}$ , h is the step size selected in the calculation, $K_{i, 1}$ is the slope at the point $(x_{n})$ , $K_{i, 2}$ is the slope at the point $(x_{n} + h / 2)$ obtained by $K_{i, 1}$ , $K_{i, 3}$ is the slope at the point $(x_{n} + h / 2)$ obtained by $K_{i, 2}$ , and $K_{i, 4}$ represents the slope at the point $(x_{n} + h)$ obtained by $K_{i, 3}$ .

The geometric interpretation of the fourth-order Runge–Kutta algorithm for 3D nonlinear differential equations is shown in Figure 3:

Through the point $P (x_{0}, y_{0}, z_{0})$ , draw a straight line with slope of $f (x_{0}, y_{0}, z_{0})$ , intersecting $t = t + h$ at A, its increment is $h k_{1}$ .

Find the middle coordinates $(t + h / 2, x_{1}, y_{1}, z_{1})$ of PA, draw a straight line with a slope of $f (x_{1}, y_{1}, z_{1})$ , and intersect with $t = t + h$ at B, and its increment is $h k_{2}$ .

Find the middle coordinate $(t + h / 2, x_{2}, y_{2}, z_{2})$ of PB, draw a straight line with a slope of $f (x_{2}, y_{2}, z_{2})$ , and intersect with $t = t + h$ at C, and its increment is $h k_{3}$ .

Through the point $C (t + h, x_{3}, y_{3}, z_{3})$ , draw a straight line PD with a slope of $f (x_{3}, y_{3}, z_{3})$ , get the increment $h k_{4}$ .

Figure 3.

Geometric interpretation of the fourth-order Runge–Kutta algorithm.

$K = \frac{1}{6} (K_{1} + 2 K_{2} + 2 K_{3} + K_{4})$ is the weighted average.

The Lorenz chaotic system has three variables, X, Y, and Z, whose slope is represented by K, S, and M, respectively. The numerical solution of the fourth-order Runge–Kutta algorithm for the nonlinear differential equation of variable X is given by equation (6)

{\begin{matrix} K_{1} = f (t, x_{n}, y_{n}, z_{n}) \\ K_{2} = f (t + \frac{1}{2} h, x_{n} + \frac{1}{2} h K_{1}, y_{n} + \frac{1}{2} h S_{1}, z_{n} + \frac{1}{2} h M_{1}) \\ K_{3} = f (t + \frac{1}{2} h, x_{n} + \frac{1}{2} h K_{2}, y_{n} + \frac{1}{2} h S_{2}, z_{n} + \frac{1}{2} h M_{2}) \\ K_{4} = f (t + h, x_{n} + h K_{3}, y_{n} + h S_{3}, z_{n} + h M_{3}) \\ X_{n + 1} = X_{n} + \frac{h}{6} (K_{1} + 2 K_{2} + 2 K_{3} + K_{4}) \end{matrix}

(6)

The expressions for Y and Z are similar to those of X. Replace $f (x, y, z)$ with $ψ (x, y, z)$ and $ω (x, y, z)$ , we will obtain the geometric interpretation of the variables Y and Z.

Construction of Lorenz system by Euler method

The Runge–Kutta algorithm has a good accuracy, but the algorithm is complex and difficult to design and implement in the digital system. The algorithm is generally discretized by the difference method of the equation, and the differential equation is usually transformed into the difference equation to solve. Solve the approximate value of $y (x)$ on a series of discrete nodes, the step size between the two adjacent nodes is $h = x_{n + 1} - x_{n}$ , assuming that h is a fixed number, then $x_{n} = x_{0} + nh$ , $n = 0, 1, 2, \dots$ . The first-order differential equation is solved by Euler method, as shown in Figure 4.

Figure 4.

Geometric interpretation of Euler method.

Let $f (x, y)$ be an arbitrary curve, cross the point $P_{0}$ on the curve $f (x, y)$ to make a tangent, $P_{1} (x_{1}, y_{1})$ is a point on this tangent, and $x_{1} = x_{0} + h, y_{1} \approx y_{0} + hf$ $(x_{0}, y_{0})$ . $P_{2} (x_{2}, y_{2})$ is on a straight line which passes through point $P_{1}$ , and it is parallel to the tangent at point $P_{3} (x_{1}, f (x_{1}))$ of curve $f (x, y)$ , and $x_{2} = x_{1} + h$ , $y_{2} \approx y_{1} + hf (x_{1}, y_{1})$ , the rest can be done in the same manner, and we obtain the approximate solution of $y_{n} (n = 0, 1, 2, \dots, N)$ , $hf (x_{i}, y_{i}) \approx hy' (x_{i})$ is the increment of the numerical solution of the function, and the brokenline of the $P_{0}, P_{1}, \dots, P_{N}$ can be regarded as the approximate curve of the solution curve $f (x, y)$ .

From the definition of the derivative, we know that for the sufficiently small $(y (x_{n + 1}) - y (x_{n})) / h \approx y' (x_{n})$ $\approx f (x_{n}, y_{n})$ , then we get $y (x_{n + 1}) \approx y (x_{n}) + hf (x_{n}, y_{n})$ , we can use the Euler method to discretize the Lorenz equation as

{\begin{matrix} x_{n + 1} = x_{n} - 10 (x_{n} - y_{n}) h \\ y_{n + 1} = y_{n} + (28 x_{n} - y_{n} - x_{n} z_{n}) h \\ z_{n + 1} = z_{n} + (x_{n} y_{n} - \frac{8}{3} z_{n}) h \end{matrix}

(7)

Euler method needs to set the step size, the selection of the step size in the numerical solution is very important, if the step is too big, after each step of the calculation, there will be a large local truncation error. If the step size is too small, although the value of the truncation error calculated for each step is smaller but when the scope of the calculations has been determined, a small step size results in the need to complete more calculation steps, which not only increases the amount of calculation but also the accumulation of calculation errors. Digital systems should use as few calculations as possible to meet specific error requirements, while also minimizing the amount of calculation and the accumulation of unnecessary errors.

A new 3D discrete chaotic system

In this article, a new three-dimensional discrete chaotic system (3DDCS) is proposed. The left end of the equation is simpler than the 3D Lorenz chaotic differential equation. If the characteristic meets the requirements, it can be used instead of the traditional Lorenz system in digital secure communication system.^15–18 Through a large number of experimental tests and data analysis, a new 3DDCS is found. A new 3DDCS is shown in the following equation

{\begin{matrix} x_{1} (k + 1) = x_{1} (k) x_{2} (k) - x_{3} (k) \\ x_{2} (k + 1) = x_{1} (k) + 0.001 x_{2} (k) \\ x_{3} (k + 1) = x_{2} (k) \end{matrix}

(8)

The Lyapunov exponents of 3DDCS are as follows: $L E_{1} = 0.00016$ , $L E_{2} = 0.00016$ , and $L E_{3} = - 0.00032$ . Because there are two positive Lyapunov exponents, it is a hyper-chaotic system.

The 3DDCS has two equilibrium points, and the points are as follows $Q_{1} = (0, 0, 0)$ , $Q_{2} = (1.999, 1.999 /$ $0.999, 1.999 / 0.999)$ .

According to the eigenvalues of the Jacobian matrix of the equilibrium point $Q_{1}$ , it can be determined that $Q_{1}$ is an unstable equilibrium point. The eigenvalues are as follows

λ_{1} = - 1.0003, λ_{2} = 0.49967 + 0.86603 i, λ_{3} = 0.49967 - 0.86603 i

According to the eigenvalues of the Jacobian matrix of the equilibrium point $Q_{2}$ , it can be determined that $Q_{2}$ is an unstable equilibrium point. The eigenvalues are as follows

λ_{1} = - 2.6173, λ_{2} = - 1.001, λ_{3} = 0.38169

When the initial condition is $X (0) = (x_{1} (0), x_{2} (0),$ $x_{3} (0)) = (0.7, 0.7, - 1.4)$ , the dynamic orbit is shown in Figure 5.

Figure 5.

Orbit evolution graph of new 3D discrete chaotic mapping: (a) $x_{1} (k) - x_{2} (k)$ coordinate projection, (b) $x_{1} (k) - x_{3} (k)$ coordinate projection, (c) $x_{2} (k) - x_{3} (k)$ coordinate projection, and (d) $x_{1} (k) - x_{2} (k) - x_{3} (k)$ coordinate projection.

According to the Lyapunov exponent, the system is hyper-chaotic system. According to the chaotic map orbit evolution graph, it can be seen that the system has chaotic deterministic regional distribution characteristics and can be applied to chaotic secure communication system.

Based on the field-programmable gate array/complex programmable logic device (FPGA/CPLD) development tool Quartus II 64-Bit, the 64-bit 3DDCS system programmed with programmable hardware description language verilog is shown in Figure 6, which uses 20 9-bit multipliers and 356 logical elements. For the actual comparison, the 64-bit Lorenz system uses 40 9-bit multipliers and 1272 logical elements. The comparison between these two resources is shown in Table 1. The complexity of the hyper-chaotic system avoids the procedure of the selection of step size, while the resource occupancy rate is low. It can be applied to the application of encryption system or secure communication synchronization system.

Figure 6.

Circuit diagram of 3DDCS.

Table 1.

Resource occupation comparison table of Lorenz and 3DDCS.

	Lorenz	3DDCS
Family	Cyclone IV E	Cyclone IV E
Device	EP4CE15F23C8	EP4CE15F23C8
Timing models	Final	Final
Total logicelements	1272/15,408 (8%)	356/15,408 (2%)
Total combinationalfunctions	1265/15,408 (8%)	286/15,408 (2%)
Dedicatedlogic registers	192/15,408 (1%)	192/15,408 (1%)
Total registers	192	192
Total pins	194/344 (56%)	194/344 (56%)
Embeddedmultiplier9-bit elements	40/112 (36%)	20/112 (18%)

3DDCS: three-dimensional discrete chaotic system.

From the Lorenz and the 3DDCS resource occupation comparison table, 3DDCS has advantages in some aspects, the circuit is simple to implement, and further application designs can be carried out.

The three outputs X, Y, and Z of 3DDCS are quantized by interval quantization, and the mathematical expression is as follows

Q_{0 - 1} [x (n)] = \begin{matrix} {\begin{matrix} \begin{matrix} 1, & x (n) \in \cup_{k = 0}^{2^{m} - 1} I_{2 k}^{m} \end{matrix} \\ \begin{matrix} 0, & x (n) \in \cup_{k = 0}^{2^{m} - 1} I_{2 k + 1}^{m} \end{matrix} \end{matrix}; & k = 0, 1, 2, \dots \end{matrix}

(9)

where m is arbitrary integer that is greater than 0, and $I_{0}^{m}$ , $I_{1}^{m}$ , $I_{2}^{m}, \dots$ are $2^{m}$ consecutive equal intervals on the range of the real value sequence. If the real value falls on the odd interval, the result of quantization is 0, and if the real value falls on the even interval, the result is 1. By the selection of quantization method, the discrete chaotic sequence value has been quantized as 0-1 chaotic pseudo-random sequence, which can be used as chaotic pseudo-random sequence of sequence cryptography, and the security of the encryption results is completely dependent on the randomness and unpredictability of the pseudo-random sequence. Based on the above quantization method, we make a permutation entropy analysis of the quantized sequence with the purpose of testing the complexity of chaotic binary sequence. The test results are shown in Table 2. The experimental results show that the chaotic binary sequence has good complexity.

Table 2.

Resource occupation comparison table of Lorenz and 3DDCS.

Chaotic binary sequence	Permutation entropy
$Q_{0 - 1} [x_{1} (n)]$	0.81084
$Q_{0 - 1} [x_{2} (n)]$	0.81052
$Q_{0 - 1} [x_{3} (n)]$	0.81052

Here we take color image encryption as an example to illustrate the encryption process. The color image is composed of tricolor matrices, which are divided into three color components: R, G, and B. Each color component matrix is composed of pixel matrix values, which are denoted as three integer matrices of $R_{n \times m} (x, y)$ , $G_{n \times m} (x, y)$ , and $B_{n \times m} (x, y)$ , where n is the number of rows of the matrix, m is the number of columns of the matrix, and $n \times m$ is the number of pixels of the image, and the range of values is between 0 and 255. Color image is first converted into three matrices of three color components. Since the sequence encryption algorithm is used, we convert the matrix of the image into a sequence of integer values line by line, and the three matrices are represented by $R (i)$ , $G (i)$ , and $B (i)$ , where $i \in (0, n \times m)$ . Since the pixel values are in the range of 0–255, each pixel value can be converted into an 8-bit binary number. At this time, the image-based binary sequences $R (j)$ , $G (j)$ , and $B (j)$ are obtained, where $j \in (0, n \times m \times 8)$ , then the three binary sequences $R (j)$ , $G (j)$ , and $B (j)$ are respectively XORed with the chaotic binary sequence X which is obtained by the previous discretization and quantization bit by bit. Finally, we get the sequence value of three color components and make a decimal number by 8 bits, restore it to image sequence value and image pixel matrix and finally get the encrypted color image, where Lena of size $256 * 256$ is encrypted by the above chaotic binary sequence and encryption method. The encrypted image is shown in Figure 7. In addition, in this section, the adjacent pixel point analysis experiment is carried out for plain-image and cipher-image of Lena. The mathematical equations can be given as follows

ρ_{xy} = \frac{cov (x, y)}{\sqrt{D (x) D (y)}}

(10)

where $cov (x, y) = (1 / N) \sum_{i = 1}^{N} (x_{i} - E (x)) (y_{i} - E (y))$ , $D (x) = (1 / N) \sum_{i = 1}^{N} {(x_{i} - E (x))}^{2}$ , $E (x) = (1 / N) \sum_{i = 1}^{N} x_{i}$ , $x_{i}$ and $y_{i}$ represent the gray values of two adjacent pixels. Correlation analysis results of adjacent pixels are listed in Table 3. It can be seen from the table that the correlation of adjacent pixels of ciphertext images tends to 0.

Figure 7.

Experimental results of encryption Lena image: (a) the original image and (b) the encrypted image.

Table 3.

Correlation analysis of adjacent pixels.

Direction	Plain-image	Cipher-image
Horizontal	0.9871	0.0532
Vertical	0.9312	0.0043
Diagonal	0.9364	0.0234

Furthermore, the process of image decryption is the inverse of the encryption process, that is, the encrypted image and the chaotic sequence to do a bit-wise XOR operation, then you can restore the original image. The formula for implementing encryption is as follows

{\begin{matrix} R_{n \times m} (x, y) \to R (i) \to R (j) \oplus L (j) = E_{R} (j) \\ G_{n \times m} (x, y) \to G (i) \to G (j) \oplus L (j) = E_{G} (j) \\ B_{n \times m} (x, y) \to B (i) \to B (j) \oplus L (j) = E_{B} (j) \end{matrix}

(11)

where $i \in (0, n \times m)$ , $j \in (0, n \times m \times 8)$ , $E (j)$ is the binary sequence of the encrypted image, ⊕ is the bit-wise XOR operation, and $E (j)$ is converted into a decimal integer sequence for every 8 bits, and then converted into a $n \times m$ -order matrix, and the three color component matrices are combined into a colored encrypted image. Decryption is the inverse of the encryption process, the process is as follows

{\begin{matrix} E_{R} (j) \oplus L (j) = R (j) \\ E_{G} (j) \oplus L (j) = G (j) \\ E_{B} (j) \oplus L (j) = B (j) \end{matrix}

(12)

The outstanding advantage of this algorithm is that it is easy to operate, and the encryption and decryption process is simple and easy to implement. In the case of large amounts of information such as image information, this encryption algorithm has obvious advantages and is suitable for VSNs applications.

Conclusion

In order to facilitate the application of VSNs, this article explores the realization method of multidimensional chaotic discrete system. In the method of discretization of multidimensional chaotic continuous system, the Runge–Kutta method and the Euler method are analyzed. The former calculation complexity is high, which is difficult for the application of digital system design. The latter is dependent on the discrete step size, the need to select the step size will increase the calculation steps. In this article, we propose a 3DDCS method and find its Lyapunov exponents, which prove that there exist two positive Lyapunov exponents, so 3DDCS is a hyper-chaotic system. By the distribution map simulation of chaotic region, we can identify that the distribution is uniform. On the basis of this, the hardware design of the two chaotic systems in the same environment is realized, and the Lorenz and 3DDCS chips are realized, and the resources occupied are compared. Through the multi-comparison analysis, it is superior to the Lorenz system in terms of resource utilization, and at the same time, we designed a scheme based on 3DDCS sequence image encryption, which can be applied to VSNs. Since there is no more systematic method for constructing chaotic systems, the construction of chaotic systems is often stochastic. Therefore, this article will further explore the systematic method of constructing the discrete multidimensional chaotic system in order to avoid the step length selection problem and apply it to the lightweight digital encryption field.

Footnotes

Handling Editor: Jeng-Shyang Pan

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is partially supported by the Natural Science Foundation of China (grant no.: 61471158)

ORCID iD

Xin Huang

References

Lorenz

EN.

Deterministic nonperiodic flow. J Atmos Sci 1963; 20: 130–141.

Lorenz

EN.

The essence of chaos. Washington: University of Washington Press, 1993, pp.7–11.

Yorke

JA.

Period three implies chaos. Am Math Month 1975; 8210: 985–992.

Ruelle

Takens

On the nature of turbulence. Commun Math Phys 1971; 20: 167–192.

May

RM.

Simple mathematical models with very complicated dynamics. Nature 1976; 261: 459–467.

Feigenbaum

MJ.

Quantitative universality for a class of non-linear transformations. Stat Phys 1978; 19: 125–152.

Yang

Chua

LO.

Cryptography based on chaotic systems. IEEE T Circuit Syst 1997; 445: 469–472.

Fridrich

Symmetric ciphers based on two-dimensional chaotic maps. Int J Bifurc Chaos 1998; 86: 1259–1284.

Frey

DR.

Chaotic digital encoding: an approach to secure communication. IEEE T Circuit Syst 1993; 4010: 660–666.

10.

Zhao

Fang

JQ.

Modern information safety and advances in application research of chaos-based security communication. Prog Phys 2003; 23(2): 212–255.

11.

Ding

Fang

JQ.

A circuit design method of chaotic system output sequence based on FPGA technology and its possible application in secure communication network. Ann Report China Inst Atomic Energ 2005; 1: 174.

12.

Ding

Pang

Fang

et al . Designing of chaotic system output sequence circuit based on FPGA and its possible applications in network encryption card. Int J Innov Comput Inf Control 2007; 3(2): 449–456.

13.

Ding

Zang

et al . Discrete chaotic circuit and the property analysis of output sequence. In: Proceedings of the international symposium on communications and information technologies (IEEE ISCIT 2005), Beijing, China, 12–14 October 2005, pp.1009–1012. New York: IEEE.

14.

Cheng

Ding

et al . Study of SMS4 based on of FPGA realization. Chin J Sci Instrum 2011; 32(12): 58–63.

15.

Pan

Xue

et al . Field programmable gate array-based chaotic encryption system design and hardware realization of cell phone short massage. Acta Phys Sin 2012; 61(28): 180504.

16.

Zheng

Song

et al . A novel detection of periodic phenomena of binary chaotic sequences. Acta Phys Sin 2012; 61(23): 230501.

17.

Wang

Song

Pan

et al . FPGA design and applicable analysis of discrete chaotic maps. Int J Buficat Chaos 2014; 24: 1450054.

18.

Ding

Yang

ZH.

Secure communication system based on hardware logic encryption. Beijing: Posts & Telecom Press, 2015.

A multidimensional discrete digital chaotic encryption system

Abstract

Keywords

Introduction

The characteristics of Lorenz chaotic system

Lorenz attractor

Lyapunov exponents of Lorenz system

Construction of discrete chaoticsystem by Runge–Kutta method

Construction of Lorenz system by Euler method

A new 3D discrete chaotic system

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References