Key stream generation method based on five-dimensional chaotic system

Abstract

The invention discloses a key stream generation method based on a five-dimensional chaotic system, solves the problem that an existing image encryption method based on the chaotic system is complex, and belongs to the field of image encryption. The method comprises the steps that a novel five-dimensional chaotic system is established, the size of a to-be-encrypted image is obtained, k is the length of the to-be-encrypted image, and w is the width of the to-be-encrypted image; discretizing the five-dimensional chaotic system, setting a secret key, taking the secret key as a system variable initial value and a system parameter of the five-dimensional chaotic system to obtain five groups of sequences of which the length and the width are k * w, and generating a secret key stream according to the five groups of sequences of which the length and the width are k * w; and performing image encryption on the to-be-encrypted image by using the key stream. The five-dimensional chaotic system provided by the invention has rich dynamic behaviors, and the key stream is generated by adopting a discretization processing method, so that the additional complexity of the algorithm caused by solving a differential equation in the traditional encryption algorithm is avoided.

Description

Technical Field

[0001] The invention relates to a key stream generation method based on a five-dimensional chaotic system, and belongs to the field of image encryption. Background Art

[0002] Chaotic system is an important component of nonlinear dynamic system, which is characterized by complex time dependence and sensitivity to initial conditions. In recent years, with the deepening of research, chaos theory has been widely used in many fields, such as secure communication, signal processing, image encryption, random number generation, etc. Therefore, designing and analyzing higher-dimensional chaotic systems in order to obtain more complex and more practical dynamic behaviors has become one of the hot topics of current research.

[0003] Traditional two-dimensional and three-dimensional chaotic systems have been widely studied and have achieved success in many applications. However, the complexity and uncertainty of such systems are limited, especially when dealing with high-dimensional data, and may not fully meet the needs. Therefore, the development of a new five-dimensional chaotic system has important theoretical and practical significance.

[0004] First, the five-dimensional chaotic system can provide higher complexity than the low-dimensional system and capture more dynamic information. This makes it have stronger application potential in fields such as information encryption, data hiding, and complex signal generation. Secondly, the establishment of the five-dimensional system also provides a new perspective for in-depth understanding of multidimensional chaotic phenomena and exploring issues such as chaotic synchronization and control. In addition, the potential application of the five-dimensional chaotic system in emerging fields such as deep learning, the Internet of Things, and smart cities is also worthy of in-depth study.

[0005] At present, chaotic systems have been widely used in the field of image encryption. Image encryption algorithms based on chaotic systems have many advantages over other types of encryption algorithms. First, chaotic systems exhibit richer dynamic behaviors, which means that they can generate more complex and unpredictable encryption key streams. Second, continuous chaotic systems have a wider range of chaotic parameters, which enables them to generate a larger range of key spaces when applied to encryption algorithms. However, although chaotic systems have many advantages in image encryption, they also have some potential disadvantages. Since the essence of continuous chaotic systems is in the form of differential equations, when they are applied to image encryption algorithms, they need to be solved to generate the key stream required for encryption, which will bring additional algorithmic complexity to the encryption algorithm. Summary of the invention

[0006] In view of the complexity of existing image encryption methods based on chaotic systems, the present invention provides a key stream generation method based on a five-dimensional chaotic system.

[0007] A key stream generation method based on a five-dimensional chaotic system of the present invention comprises:

[0008] S1. Establish a five-dimensional chaotic system:

[0009]

[0010] Among them, x, y, z, w, u represent system variables, and a, b, c, d represent system parameters;

[0011] S2. Obtain the size of the image to be encrypted, where k is the length of the image to be encrypted and w is the width of the image to be encrypted;

[0012] S3, discretizing the five-dimensional chaotic system, setting a key, using the key as the initial value of the system variable and the system parameter of the five-dimensional chaotic system, obtaining five groups of sequences x(n), y(n), z(n), w(n) and u(n) with a length and width of k*w, and generating a key stream according to the five groups of sequences with a length and width of k*w;

[0013] S4. Use the key stream to encrypt the image to be encrypted.

[0014] Preferably, in S3, the five-dimensional chaotic system is discretized:

[0015]

[0016] Where h represents the discrete step length;

[0017]

[0018]

[0019] Iterate according to Formula 1 to Formula 5 to obtain five groups of sequences with a length and width of k*w.

[0020] Preferably, iterate k*w+L times according to Formula 1 to Formula 5, take the five groups of sequences of the last k*w times, and use the first L times to eliminate the influence of transient processes.

[0021] Preferably, the key streams K1(i), K2(i), K3(i), K4(i), and K5(i) are generated according to the sequences of five groups of k*w:

[0022]

[0023] Among them, mod represents the modulo operation.

[0024] Preferably, S4 includes:

[0025] Extract the RGB color components of the image to be encrypted, and use the key streams K1(i), K2(i), and K3(i) to perform diffusion encryption on each extracted color component;

[0026] Each component after diffusion encryption is scrambled using key streams K4(i) and K5(i) to generate an index matrix, and the pixel positions of the image to be encrypted are rearranged using the index matrix to obtain a scrambled encrypted image.

[0027] Preferably, S3 further includes:

[0028] Taking the system parameter b as the control variable, a sequence with expanding or shrinking dynamic behavior is obtained.

[0029] Preferably, S3 further includes:

[0030] The system parameters a, b, c, d are used as control variables to obtain a sequence with symmetrical behavior.

[0031] Preferably, S3 further includes:

[0032] Using the initial value of the system variable as the control variable, a sequence with offset behavior is obtained.

[0033] Preferably, S3 further includes:

[0034] Taking the system parameter a as the control variable, a sequence with rotation and contraction behavior is obtained.

[0035] The beneficial effects of the present invention are as follows: the present invention provides a new five-dimensional chaotic system and analyzes its dynamic behavior under different parameters, proving that it has rich dynamic behavior. At the same time, the five-dimensional chaotic system is discretized and applied to the field of image encryption, thus avoiding the additional complexity of the traditional encryption algorithm caused by solving differential equations. BRIEF DESCRIPTION OF THE DRAWINGS

[0036] Figure 1 This is a schematic diagram of the principle of image encryption in this implementation mode;

[0037] Figure 2 Schematic diagram of the scrambling algorithm;

[0038] Figure 3 The xy-plane phase trajectory diagram of the chaotic system as b changes, (a) is the shrinking behavior, and (b) is the expanding behavior;

[0039] Figure 4 It is the time series diagram of the chaotic system changing with b, (a) is the shrinking behavior, (b) is the expanding behavior;

[0040] Figure 5Symmetrical behavior of phase trajectories exhibited by chaotic systems, (a) a = -30, -25, -20, 36, 24.5, 16; (b) b = -50, -30, -10, 35, 75, 105; (c) c = -30, -20, -10, 10, 20, 30; (d) d = -30, -20, -10, 20.5, 12, 9;

[0041] Figure 6 The shift behavior of the chaotic system with the change of u0, where (a) xy plane phase trajectory diagram, (b) average value diagram;

[0042] Figure 7 Phase trajectory deviation behavior of the chaotic system, (a) x(1) = 60, 65, 70, 75, 80; (b) y(1) = -98, -96, -94, -92, -90; (c) z(1) = 80, 85, 90, 95, 100; (d) w(1) = 20, 25, 30, 35, 40;

[0043] Figure 8 This is the xu plane phase diagram generated when the chaotic system changes with a. DETAILED DESCRIPTION

[0044] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0045] It should be noted that, in the absence of conflict, the embodiments of the present invention and the features in the embodiments may be combined with each other.

[0046] The present invention will be further described below in conjunction with the accompanying drawings and specific embodiments, but they are not intended to limit the present invention.

[0047] The key stream generation method based on the five-dimensional chaotic system of this embodiment includes:

[0048] Step 1: Establish a new five-dimensional chaotic system:

[0049]

[0050] Where x, y, z, w, and u represent system variables, and a, b, c, and d represent system parameters. Respectively represent the time derivatives of x, y, z, w, and u;

[0051] Step 2: Using the five-dimensional chaotic system to obtain the key stream required for encryption, and using the key stream to encrypt the image to be encrypted;

[0052] First, read the matrix of the image to be encrypted to obtain the size of the image to be encrypted; k*w, k is the length of the image to be encrypted, w is the width of the image to be encrypted;

[0053] Secondly, the five-dimensional chaotic system is discretized through formula (2)-formula (6):

[0054]

[0055] In the formula, H 01 Indicates the first dimension parameter corresponding to the first recursion, H 02 Indicates the second dimension parameter corresponding to the first recursion, H 03 Indicates the third dimension parameter corresponding to the first recursion, H 04 Indicates the fourth dimension parameter corresponding to the first recursion, H 05 Indicates the fifth dimension parameters corresponding to the first recursion, all of which are intermediate variables;

[0056] H 11 Indicates the first dimension parameter corresponding to the second recursion, H 12 Indicates the second dimension parameter corresponding to the second recursion, H 13 Indicates the third dimension parameter corresponding to the second recursion, H 14 Indicates the fourth dimension parameter corresponding to the second recursion, H 15 Indicates the fifth dimension parameters corresponding to the second recursion, all of which are intermediate variables;

[0057] In the formula, H 21 Indicates the first dimension parameter corresponding to the third recursion, H 22 Indicates the second dimension parameter corresponding to the third recursion, H 23 Indicates the third dimension parameter corresponding to the third recursion, H 24 Indicates the fourth dimension parameter corresponding to the third recursion, H 25 Indicates the fifth dimension parameters corresponding to the third recursion, all of which are intermediate variables;

[0058] h represents the discrete step length, and in this embodiment, h can be 0.0001;

[0059]

[0060] In the formula, H 31 Indicates the first dimension parameter corresponding to the fourth recursion, H 32 Indicates the second dimension parameter corresponding to the fourth recursion, H 33 Indicates the third dimension parameter corresponding to the fourth recursion, H34 Indicates the fourth dimension parameter corresponding to the fourth recursion, H 35 Indicates the fifth dimension parameters corresponding to the fourth recursion, all of which are intermediate variables;

[0061]

[0062] Set the key and use the key as the initial value of the system variable and system parameter of the five-dimensional chaotic system. The initial values ​​of the system variables are x(1), y(1), z(1), w(1), and u(1). Substitute the set key into formula (2)-formula (6) and iterate k*w+500 times. After each iteration, n=n+1. The first 500 iterations of k*w+500 are to eliminate the influence of transient processes. After the iteration, the first 500 values ​​need to be discarded. In this way, five groups of k*w sequences x(n), y(n), z(n), w(n), and u(n) are generated. At this time, n∈[501,500+k×w]. In the encryption algorithm, in order to obtain the key stream required for encryption, the sequence obtained after the iteration needs to be further preprocessed. According to formula (7), the key streams K1(i), K2(i), K3(i), K4(i), and K5(i) are generated:

[0063]

[0064] Extract the RGB three color components of the image to be encrypted;

[0065] Perform a diffusion algorithm on each color component:

[0066] Using three sets of k*w sequences, each extracted color component is diffused and encrypted using formula (8):

[0067]

[0068] Where i represents the pixel number, the value range is 1 to k*w, P R (i) P G (i) P B (i) represents the extracted RGB three color components, C R (i) C G (i) C B (i) RGB color components after diffusion algorithm;

[0069] Use another two sets of k*w sequences to scramble each component after diffusion encryption:

[0070] The three components C after diffusion encryption R (i) C G (i) C B(i) Perform scrambling. First, process K4(i) and K5(i) into matrix form. Assume that the encrypted image is 4*4 in size. Multiply K4(i) and K5(i) to generate a new matrix s and arrange the matrix in ascending order to obtain a new matrix s1. Comparing matrix s and s1, it is found that 0.0500, which is the first position, is arranged in a new position, so the index matrix at this position is 1. Repeat this process until an index matrix is ​​generated. The encrypted image after scrambling can be obtained by rearranging the pixel positions of the encrypted image through the index matrix. The schematic diagram of this process is shown in the figure. Figure 2 shown.

[0071] The five-dimensional chaotic system of this embodiment has rich dynamic behaviors, which can make the sequence it generates more unpredictable. After changing the value, the range of the generated sequence also changes accordingly. This feature can be applied in image encryption to make the encryption algorithm more secure.

[0072] 1. The expansion and contraction phenomenon with the change of system parameter b

[0073] By analyzing the dynamic behavior of formula (1), it is found that when the system parameter b is used as the control variable, the other system parameters are set to a = 1, c = 1, d = 1, and the initial values ​​are all set to (x0, y0, z0, w0, u0) = (1, 1, 1, 1, 1), the phase trajectory of formula (1) expands and shrinks with the change of parameter b. The specific transformation is as follows: Figure 3 As shown, Figure 3 The xy plane phase trajectory diagram generated by formula (1) is shown in the figure. When b is greater than 0, with an interval of 10, as b increases, the overall outline of the phase trajectory generated by the chaotic system continues to expand; when b is less than 0, with an interval of 10, as b increases, the overall outline of the sequence phase trajectory continues to shrink. That is, the phase trajectory expands as the absolute value of b increases. At the same time, in order to further clearly demonstrate the transformation behavior of this image trajectory, Figure 4 A time sequence diagram is drawn to further demonstrate this changing behavior. Therefore, this embodiment uses the system parameter b as the control variable to obtain a sequence with an expanding or shrinking dynamic behavior.

[0074] 2. Symmetrical phenomena with changes in system parameters:

[0075] Through further study of the phase trajectory diagram of formula (1), it is found that the system also has symmetrical behavior with the change of system parameters. Figure 5 The symmetry phenomenon of formula (1) under different system parameters is plotted. Figure 5In (a), when all other system parameters are set to 1, six sets of values ​​are selected and assigned to the system parameter a, which are -30, -25, -20, 36, 24.5, and 16. By observing the evolution of the yw phase trajectory under the six system parameters, the symmetry of the system can be found. Similarly, with the system parameter b as the variable, the initial values ​​of the other system parameters are all set to 1, and six sets of values ​​are selected and assigned to the system parameter b, which are -50, -30, -10, 35, 75, and 105. The evolution of the xy phase trajectory under the six system parameters is shown in Figure 5 As shown in (b), the system parameter c is used as a variable, and the initial values ​​of the other system parameters are all set to 1. Six sets of values ​​are selected and assigned to the system parameter c, namely -30, -20, -10, 10, 20, and 30. The evolution diagram of the zw phase trajectory under the six parameters is shown in Figure 5 As shown in (c), the system parameter d is used as a variable, and the initial values ​​of the other system parameters are all set to 1. Six sets of values ​​are selected and assigned to the system parameter d, namely -30, -20, -10, 20.5, 12, and 9. The evolution diagram of the Xu phase trajectory under the six system parameters is shown in Figure 5 As shown in (d). The above phenomena all indicate that formula (1) has symmetry under different parameter modulations. Therefore, in this embodiment, the system parameters a, b, c, d can be used as control variables to obtain a sequence with symmetric behavior;

[0076] 3. Offset phenomenon with changes in the initial value of system variables:

[0077] By changing the initial value of the system variable u in formula (1), the phase trajectory can be offset in position. Figure 6 (a) shows the xu phase trajectory diagram when the initial values ​​of the other parameters are 1 and only the initial value u0 is changed (u0=53,54,55,56,57). It can be seen that the positions of these conservative flows move along the u axis with the change of the initial value. To supplement the explanation and verify its correctness, Figure 6 In (b), the average value graph with the change of u0 is drawn. The average value graph can be used to explain whether the phase trajectory is offset. When the curve is monotonically increasing or monotonically decreasing, it means that the phase trajectory has offset behavior. It can be seen that only the average value of u is a monotonically increasing oblique line, which means that the conservative flow will offset along the u axis with the change of the initial value u0. This result further explains that formula (1) produces the offset behavior of the phase trajectory. Similarly, this offset behavior exists in each initial value of formula (1). Figure 7 The phase trajectory diagrams under different dimensions are drawn with different initial values ​​as control variables. Figure 7 (a) shows the zw phase trajectory diagram when the initial values ​​of other parameters are 1 and only the initial value x0 is changed (x0=60,65,70,75,80). Figure 7(b) shows the zw phase trajectory diagram when the initial values ​​of the other parameters are 1 and only the initial value y0 is changed (y0=-98,-96,-94,-92,-90). Figure 7 (c) shows the zw phase trajectory diagram when the initial values ​​of the other parameters are 1 and only the initial value z0 is changed (z0=80,85,90,95,100). Figure 7 (d) shows the zw phase trajectory diagram when the initial values ​​of the other parameters are 1 and only the initial value w0 is changed (w0 = 20, 25, 30, 35, 40). The above phenomena all show that there is an offset phenomenon in formula (1) under different initial value modulation. Therefore, in this embodiment, the initial value of the system variable can be used as the control variable to obtain a sequence with offset behavior.

[0078] 4. Rotation and contraction behavior with changes in a

[0079] When studying the effect of the system parameter a in formula (1), the experimental results show that with the change of the value of the system parameter a, the phase trajectory of the system has rotation and contraction behavior in the phase space. Figure 8 As shown. First, the system parameter a is fixed as the control variable, and the initial values ​​of the other parameters are all 1. As the system parameter a increases, the shape of the phase trajectory continues to rotate counterclockwise, and at the same time, the phase trajectory continues to shrink, showing a rotational contraction dynamic behavior that changes with the system parameter a. Therefore, in this embodiment, the system parameter a can be used as the control variable to obtain a sequence with rotation and contraction behavior.

[0080] Although the present invention is described herein with reference to specific embodiments, it should be understood that these embodiments are merely examples of the principles and applications of the present invention. It should therefore be understood that many modifications may be made to the exemplary embodiments and that other arrangements may be devised without departing from the spirit and scope of the present invention as defined by the appended claims. It should be understood that the various dependent claims and features described herein may be combined in a manner different from that described in the original claims. It should also be understood that features described in conjunction with individual embodiments may be used in other described embodiments.

Claims

1. A key stream generation method based on a five-dimensional chaotic system, characterized in that: The method comprises: S1. Establish a five-dimensional chaotic system: Among them, x, y, z, w, u represent system variables, and a, b, c, d represent system parameters; S2. Obtain the size of the image to be encrypted, where k is the length of the image to be encrypted and w is the width of the image to be encrypted; S3, discretizing the five-dimensional chaotic system, setting a key, using the key as the initial value of the system variable and the system parameter of the five-dimensional chaotic system, obtaining five groups of sequences x(n), y(n), z(n), w(n) and u(n) with a length and width of k*w, and generating a key stream according to the five groups of sequences with a length and width of k*w; S4. Use the key stream to encrypt the image to be encrypted.

2. The key stream generation method based on the five-dimensional chaotic system according to claim 1 is characterized in that: In S3, the five-dimensional chaotic system is discretized: Where h represents the discrete step length; In the formula, H 01 Indicates the first dimension parameter corresponding to the first recursion, H 02 Indicates the second dimension parameter corresponding to the first recursion, H 03 Indicates the third dimension parameter corresponding to the first recursion, H 04 Indicates the fourth dimension parameter corresponding to the first recursion, H 05 Indicates the fifth dimension parameter corresponding to the first recursion; H 11 Indicates the first dimension parameter corresponding to the second recursion, H 12 Indicates the second dimension parameter corresponding to the second recursion, H 13 Indicates the third dimension parameter corresponding to the second recursion, H 14 Indicates the fourth dimension parameter corresponding to the second recursion, H 15 Indicates the fifth dimension parameter corresponding to the second recursion; H 21 Indicates the first dimension parameter corresponding to the third recursion, H 22 Indicates the second dimension parameter corresponding to the third recursion, H 23 Indicates the third dimension parameter corresponding to the third recursion, H 24 Indicates the fourth dimension parameter corresponding to the third recursion, H 25 Indicates the fifth dimension parameter corresponding to the third recursion; H 31 Indicates the first dimension parameter corresponding to the fourth recursion, H 32 Indicates the second dimension parameter corresponding to the fourth recursion, H 33 Indicates the third dimension parameter corresponding to the fourth recursion, H 34 Indicates the fourth dimension parameter corresponding to the fourth recursion, H 35 Indicates the fifth dimension parameter corresponding to the fourth recursion h represents the discrete step length; Iterate according to Formula 1 to Formula 5 to obtain five groups of sequences with a length and width of k*w.

3. The key stream generation method based on the five-dimensional chaotic system according to claim 2 is characterized in that: Iterate k*w+L times according to Formula 1 to Formula 5, take the five groups of sequences after k*w times, and the first L times are used to eliminate the influence of transient processes.

4. The key stream generation method based on the five-dimensional chaotic system according to claim 3 is characterized in that: Generate key streams K1(i), K2(i), K3(i), K4(i), and K5(i) based on five sequences with length and width k*w: Among them, mod represents the modulo operation.

5. The key stream generation method based on the five-dimensional chaotic system according to claim 4 is characterized in that S4 include: Extract the RGB color components of the image to be encrypted, and use the key streams K1(i), K2(i), and K3(i) to perform diffusion encryption on each extracted color component; Each component after diffusion encryption is scrambled using key streams K4(i) and K5(i) to generate an index matrix, and the pixel positions of the image to be encrypted are rearranged using the index matrix to obtain a scrambled encrypted image.

6. The key stream generation method based on the five-dimensional chaotic system according to claim 1 is characterized in that: The S3 further includes: Taking the system parameter b as the control variable, a sequence with expanding or shrinking dynamic behavior is obtained.

7. The key stream generation method based on the five-dimensional chaotic system according to claim 1 is characterized in that: The S3 further includes: The system parameters a, b, c, d are used as control variables to obtain a sequence with symmetrical behavior.

8. The key stream generation method based on the five-dimensional chaotic system according to claim 1 is characterized in that: The S3 further includes: Using the initial value of the system variable as the control variable, a sequence with offset behavior is obtained.

9. The key stream generation method based on a five-dimensional chaotic system according to claim 1 is characterized in that: The S3 further includes: Taking the system parameter a as the control variable, a sequence with rotation and contraction behavior is obtained.

10. An image encryption device based on a five-dimensional chaotic system, comprising a storage device, a processor, and a computer program stored in the storage device and executable on the processor, characterized in that: The processor executes the computer program to implement the steps of the key stream generation method based on a five-dimensional chaotic system as described in any one of claims 1 to 9.

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