CN113949782A - Chaotic system data encryption method based on cubic memristor - Google Patents
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Abstract
The invention relates to the technical field of data processing, in particular to a data encryption method for a complex chaotic system based on a three-dimensional memristor, wherein the complex chaotic system is derived from a 4D chaotic system with a cubic non-linear memristor, the expansion of a variable from a real domain to a complex domain is realized, and the chaotic performance of a new chaotic system is analyzed by utilizing a phase diagram, a bifurcation diagram, a 0-1 inspection, complexity and the like. Then, on the basis of the 7D complex chaotic system with the cubic memristor, a chaotic system data encryption method based on the cubic memristor is provided by combining the Arnold transformation, the chaotic system data encryption method is applied to encryption of data of the smart grid, verification experiments are carried out on images and data transmitted in the smart grid, the images are encrypted by using a histogram, a correlation coefficient, an information entropy, a key sensitivity and reconstruction quality, and the result shows that the method has good robustness and enough capacity of preventing data leakage and malicious attacks.
Description
Technical Field
The invention relates to the technical field of data processing, in particular to a chaos complex system data encryption method based on a three-dimensional memristor.
Background
After the smart grid is built, the security of the information system becomes more and more important. Smart grids are beginning to be involved in the application fields of smart factories, traffic networks, gas systems, etc. The data communicated between the different fields needs to be encrypted to prevent malicious attacks. With the rapid development of the smart grid technology, the number and types of attacks on the smart grid are remarkably increased, and huge loss and negative effects are brought to the power grid. In contrast to the importance of smart grids, concerns about their network and information security are still insufficient, which is also a frequent cause of power system accidents. Therefore, the security encryption algorithm is the key for preventing malicious attacks and ensuring the normal operation of the smart grid.
The chaotic system has a series of characteristics of extreme sensitivity, ergodicity, track unpredictability, good pseudo-randomness and the like of initial values and system parameters, and the characteristics just meet the encryption requirement. Therefore, chaos has been widely used in many fields such as chaos control, chaos spread spectrum communication, image data encryption, secure communication, etc. Among them, the application of chaos in data encryption has received great attention in recent years and has become an important branch of chaos application, but the following problems still exist: the low-dimensional chaotic system has the defects of short period, uneven chaotic sequence distribution, small key space and the like, and the data encryption effect is poor. In addition, data in the smart grid is easily attacked and tampered, so that system misjudgment and economic loss are caused.
Disclosure of Invention
The invention aims to overcome the defects in the prior art, and provides a chaotic system data encryption method based on a cubic memristor, which can effectively prevent the problem of data leakage.
The technical scheme adopted by the invention for solving the technical problems is as follows:
a chaotic system data encryption method based on a cubic memristor comprises the following steps:
step 1: separating the image channel and changing it into R, G and B channels;
step 2: randomly transforming the positions of the three primary color pixel values using a random function, which are referred to as R1, G1, and B1;
and step 3: transforming the positions of the three primary color pixel values according to the Arnold transform, named R2, G2, and B2; the Arnold transform is as follows:
m and N are rows and columns of an image matrix respectively, A and B are pseudo-random matrixes with the size of M multiplied by N generated by a chaotic sequence of a 7D complex chaotic system with a cubic memristor; coordinates x of pixels in digital imagesn，yn∈[0，255]；
And 4, step 4: and carrying out seven-dimensional exclusive OR on the seven-dimensional pseudo-random sequence with the images of R2, G2 and B2 generated by the 7D complex chaotic system with the cubic memristor.
Further, in step 4, the xor sequence is random.
The steps 1, 2 and 3 are scrambling portions and the step 4 is a diffusing portion.
The data encryption method of the chaotic system based on the cubic memristor has the advantages of small image distortion, good diffusion and scrambling performances, better safety, capability of covering the distribution rule of an original image better, enough anti-attack capability and capability of effectively preventing data leakage.
Further, the 7D complex chaotic system with the cubic memristor is:
wherein α, β, r, d are constants, xi(i 1.., 7) is an argument and W is the memristor derivative.
Further, the establishment process of the 7D complex chaotic system with the cubic memristor is as follows:
the mathematical expression of the cubic nonlinear memristor is as follows:
wherein, a and b are constants,
is an independent variable, and the number of the independent variables,
a non-linear memristor; definition of memristor derivatives
The method comprises the following steps:
the real chaotic system with cubic nonlinear memristors is given by:
wherein α, β, r, d are constants, x, y, z are independent variables; the above equation is derived into a complex field, where x ═ x1+jx2，y＝x3+jx4，z＝x5+jx6，
xi(i) 1, 7) is an argument, j is an imaginary number; separating a real part and an imaginary part of the formula to obtain a 7D complex chaotic system with a cubic memristor, so that the expansion of a variable from a real domain to a complex domain is realized;
further, the complex chaos system data encryption method based on the three-power memristor is applied to smart grid data encryption and comprises the following steps:
step 1: randomly changing the position of the temperature data by using a random function, and naming the position as data 1;
step 2: changing the position of data 1 according to Arnold transformation, and naming the changed position as data 2;
and step 3: and carrying out seven-time XOR on a seven-dimensional pseudo-random sequence with 2 data generated by the 7D complex chaotic system with the cubic memristor, wherein the XOR sequence is random.
The invention has the technical effects that:
compared with the prior art, the chaotic system data encryption method based on the three-power memristor is provided by combining the Arnold transformation on the basis of the 7D complex chaotic system with the three-power memristor, realizes the expansion of variables from a real domain to a complex domain, has small image distortion, good diffusion and scrambling performances, better safety, better covering of the distribution rule of an original image, capability of effectively preventing data leakage and malicious attack, and wide application range, and can be applied to the encryption of data of an intelligent power grid and the like.
Drawings
FIG. 1 is a phase diagram of a 7D complex chaotic system attractor with a cubic memristor according to the present invention;
FIG. 2 is a Lyapunov exponent spectrum of a 7D complex chaotic system with a cubic memristor according to the present invention;
FIG. 3 is a bifurcation diagram of the variables of the present invention as a function of the parameters;
FIG. 4 is a 0-1 test schematic diagram of a 7D complex chaotic system with a cubic memristor according to the present invention;
FIG. 5 is a chromatogram of the variation of a parameter of the present invention with the parameter;
FIG. 6 is a flow chart of an image encryption algorithm of the present invention;
FIG. 7 is a schematic diagram of the Lena image encryption process of the present invention;
FIG. 8 is a histogram of original Lena and encrypted Lena of the present invention;
FIG. 9 is a schematic diagram of the encryption process of "image 1" according to the present invention;
FIG. 10 is a schematic diagram of the encryption process for "image 2" according to the present invention;
FIG. 11 is an isometric distribution histogram of the invention after "image 1" encryption;
FIG. 12 is an isometric distribution histogram of the invention after "image 2" encryption;
FIG. 13 is a diagram of the decryption result of the wrong key according to the present invention;
FIG. 14 is a flow chart of a data encryption algorithm of the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention are clearly and completely described below with reference to the drawings of the specification.
Example 1:
image encryption is generally divided into two parts: scrambling and diffusing; the scrambling operation may shift the position of the image and the diffusion operation may change the pixel values of the image.
As shown in fig. 6, the present embodiment relates to a method for encrypting data of a chaotic system based on a three-power memristor, which includes the following steps:
step 1: separating the image channel and changing it into R, G and B channels;
step 2: randomly transforming the positions of the three primary color pixel values using a random function, which are referred to as R1, G1, and B1;
and step 3: transforming the positions of the three primary color pixel values according to the Arnold transform, named R2, G2, and B2; the Arnold transform is as follows:
where M and N are the moments of the image, respectivelyThe method comprises the following steps that (1) rows and columns of an array, A and B are pseudo-random matrixes with the size of M multiplied by N generated by a chaotic sequence of a 7D complex chaotic system with a cubic memristor; coordinates x of pixels in digital imagesn，yn，∈[0，255]；
And 4, step 4: seven-dimensional pseudo-random sequences with R2, G2 and B2 images generated by the 7D complex chaotic system with the cubic memristor are subjected to seven-time exclusive OR, and the exclusive OR sequence is random.
The steps 1, 2 and 3 are scrambling portions and the step 4 is a diffusing portion.
Is provided with
The initial conditions were (0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1).
The above encryption algorithm uses the "Lena" image, and the encryption process is shown in fig. 7, where fig. 7(a) is the original image, fig. 7(b) is the scrambled image, fig. 7(c) is the encrypted image, fig. 7(d) is the decrypted image, and fig. 7(c) hides the features of the original image.
The reconstruction quality, the correlation, the histogram and the information entropy of the complex chaotic system data encryption method are analyzed.
A. Reconstructed mass analysis
A peak signal-to-noise ratio (PSNR) is introduced to measure the distortion between the original image and the decrypted image. When the peak signal-to-noise ratio is greater than 30dB and less than 40dB, image distortion is small. The peak signal-to-noise ratio method is as follows:
step 1: calculating Mean Square Error (MSE):
step 2: calculating peak signal-to-noise ratio:
PSNR(f，g)＝20log10(255/MSE(f，g))。
the peak signal-to-noise ratio between fig. 7(a) and fig. 7(d) was calculated to be about 30dB, and the distortion of fig. 7(a) and fig. 7(d) was small.
Structural Similarity (SSIM) is another measure of the similarity of two images and consists of three independent components, brightness, contrast and structure. SSIM can be expressed as follows:
the SSIM between fig. 7(a) and fig. 7(d) is calculated to be 1, and fig. 7(a) and fig. 7(d) are the same in SSIM.
B. Correlation analysis
In order to prevent the original information from being broken by the similarity between pixels, it is necessary to remove the correlation between adjacent pixels. The N pairs of marked pixels are randomly selected in the original image. The correlation coefficient between the two is shown as follows:
the correlation coefficients of neighboring pixels range from-1 to 1. The closer the value is to 1, the higher the correlation. Accordingly, if the value is close to-1, the neighboring pixels are substantially uncorrelated. Table 1 the phase-adjacent pixel correlation,
TABLE 1 neighboring Pixel correlation
Table 1 shows that the complex chaotic system data encryption method has good diffusion and scrambling performance, and thus, the algorithm has better security for statistical analysis.
C. Histogram of the data
Fig. 8 shows the pixel distribution at each pixel level of the three primary color matrices. Fig. 8(a), 8(b) and 8(c) show a large fluctuation and a large difference between the peak value and the bottom value. Some pixel values have a large frequency and others have a small frequency. After encryption, the pixel distribution of each pixel level of the three-primary-color matrix is relatively uniform, the value frequencies of each pixel value are basically the same, and the distribution rule of the original image is better covered.
D. Entropy of information
The information entropy is used to reflect the average uncertainty of all pixel values, and the formula is as follows:
l is the maximum gray value 255, p (x)i) Is the grey value probability; the larger the entropy of the image information is (the maximum value is 8), the more uniform the distribution of image pixels is, and the non-random distribution of the image pixels shows that the encryption effect is better. The entropy of the image information is shown in table 2,
TABLE 2 entropy of image information
Table 2 shows that the complex chaotic system data encryption method has a large image information entropy, which indicates that the method has sufficient anti-attack capability. The encryption algorithm provided by the invention is proved to be capable of effectively preventing data leakage from the aspects of correlation coefficient, histogram, information entropy and the like.
The 7D complex chaotic system with the cubic memristor is established as follows:
the mathematical expression of the cubic nonlinear memristor is as follows:
wherein, a and b are constants,
is an independent variable, and the number of the independent variables,
a non-linear memristor; definition of memristor derivatives
The method comprises the following steps:
the real chaotic system with cubic nonlinear memristors is given by:
wherein α, β, r, d are constants, x, y, z are independent variables; the above equation is derived into a complex field, where x ═ x1+jx2，y＝x3+jx4，z＝x5+jx6，
xi(i 1.., 7) is an argument, j is an imaginary number; separating a real part and an imaginary part of the formula to obtain a 7D complex chaotic system with a cubic memristor, so that the expansion of a variable from a real domain to a complex domain is realized;
attractors, bifurcation diagrams, complexity and 0-1 tests of the 7D complex chaotic system with the cubic memristor are analyzed below.
A. Chaotic attractor
Is provided with
(x1，x2，x3，x4，x5，x6，x7) The initial value of (1), (0.1), and the phase diagram of the attractor is shown in FIG. 1.
The lyapunov exponent represents a numerical characteristic of the average exponential divergence rate of adjacent trajectories in a phase space, which is one of numerical characteristics for identifying chaotic motion, a negative value indicates that motion in the direction is stable, and a positive value indicates that motion in the direction is unstable. When the Lyapunov exponent is positive, negative, and zero, the system is chaotic. The Lyapunov exponent of the 7D complex chaotic system with the cubic memristor is as follows: LE1 ═ 2.041, LE2 ═ 0.425, LE3 ═ 0.189, LE4 ═ 0, LE5 ═ -0.041, LE6 ═ -3.355, and LE7 ═ -4.205, so the system is chaotic and the corresponding lyapunov indices are shown in fig. 2.
B. Bifurcation diagram
The bifurcation diagram can clearly reflect the whole process of the system entering chaos. When a large number of unpredictable distribution points appear on the bifurcation diagram, the system is indicated to be chaotic. The bifurcation of the variables with the parameters is shown in fig. 3. As can be seen from fig. 3, the system continuously branches between different states as time goes by, and finally the system enters a chaotic state.
C. 0-1 test
The 0-1 test method is as follows:
step 1: setting x (j) (j ═ 1, 2.., N) as a chaotic sequence;
step 2: calculating the transformation variables p (n) and q (n) of the sequence:
wherein c is (0, pi); if x (j) is not chaotic, p (n) -q (n) tracks are stable; if the sequence is in a chaotic state, the trace plot of p (n) -q (n) will be in a Brownian state. The '0-1 test' schematic diagram of the 7D complex chaotic system with the cubic memristor is shown in FIG. 4, and it can be seen that the 7D complex chaotic system with the cubic memristor shows Brownian motion, and therefore, the system is chaotic.
D. Complexity analysis
SE algorithm and C0 algorithm based on Fourier transform and wavelet transform are both spectral entropy algorithms; in order to verify and analyze the complexity of the two parameters when they vary, chromatograms were introduced; the parameter-dependent chromatogram is shown in fig. 5, the lighter the color, the less complex.
Example 2:
in this embodiment, the encryption method proposed in embodiment 1 is verified by using data and images in the smart grid, and security analysis is performed.
A. Image encryption
The "image 1" and "image 2" transmitted in the smart grid will be encrypted using the algorithm described above. The size of "image 1" is 660 × 783. The size of "image 2" is 456 × 639. The encryption process for "image 1" is shown in fig. 9, where fig. 9(a) is the original "image 1", fig. 9(b) is the scrambled "image 1", fig. 9(c) is the encrypted "image 1", the characters in the original image cannot be directly recognized from the image, and fig. 9(d) is the decrypted "image 1". As shown in fig. 10, "image 2" of the same encryption process as "image 1" is shown.
The decrypted "image 1" and decrypted "image 2" are then analyzed in terms of histogram, correlation coefficient, information entropy, key sensitivity, and reconstruction quality.
1) Histogram of the data
As in fig. 11 and 12, the encrypted equi-distribution histogram successfully conceals the features.
2) Coefficient of correlation
TABLE 3 Adjacent Pixel correlation comparison
According to table 3, comparing the encrypted image with the original image, there is little or no correlation, even negative, between adjacent pixels.
3) Entropy of information
TABLE 4 information entropy of images
As can be seen from table 4, the entropy of the information of the encrypted image is close to 8 (the maximum value is 8), and it is difficult for the encrypted image to reveal the information.
4) Key sensitivity
For initial conditions (0.10001, 0.10001, 0.10001, 0.10001, 0.10001, 0.10001, 0.10001), values of other parameters in the 7D complex chaotic system with cubic memristors remain unchanged. The generated key is used to decrypt the cryptographic images 1 and 2. According to fig. 13, even a slight change in the key, to which the encryption algorithm is very sensitive, cannot successfully decrypt images 1 and 2.
5) Reconstruction mass analysis
Table 5 reconstructed mass analysis
As can be seen from table 5, the reconstruction quality of "image 1" and "image 2" is very good; from the aspects of correlation coefficient, histogram, information entropy, key sensitivity, reconstruction quality and the like, the encryption method provided by the invention plays a good role in the security image transmission of the smart grid.
B. Data encryption
The data set used for testing the security of the encryption method is temperature data in the smart grid.
The Modbus protocol is used for transmitting temperature data and comprises the following components:
0x13 0x04 0x00 0x00 0x00 0x01 0x32 0xB8
the data encryption algorithm is as follows:
step 1: the location of the temperature data was randomly changed using a random function, which was named data 1.
Step 2: the position of data 1 is modified according to the Arnold transformation, which is named data 2.
And step 3: and carrying out seven-time XOR on a seven-dimensional pseudo-random sequence with 2 data generated by the 7D complex chaotic system with the cubic memristor, wherein the XOR sequence is random.
A flow chart of the data encryption algorithm is shown in fig. 14.
The ciphertext data is composed as follows:
0xFC 0xCF 0xFC 0xFC 0xFC 0xFC 0xF8 0xF8
after decryption, the data composition is the same as the initial data; the security of the ciphertext mainly depends on whether the key is random or not, and if the key is random and variable, the security of the ciphertext can be ensured. Next, the randomness of the key is analyzed from the NIST test.
TABLE 6 NIST test
| Species of | Numerical value | Whether or not to pass |
| Approximate entropy | 0.437274 | By passing |
| Intra block frequency check | 0.122325 | By passing |
| Sum check | 0.834308 | By passing |
| Discrete Fourier transform inspection | 0.437274 | By passing |
| Frequency checking | 0.739918 | By passing |
| Linear complexity inspection | 0.213309 | By passing |
| Intra-block longest run check | 0.122325 | By passing |
| Non-overlapping module matching verification | 0.991468 | By passing |
| Overlay module match check | 0.213309 | By passing |
| Random walk test | 0.964295 | By passing |
| Random walk state frequency check | 0.710216 | By passing |
| Binary matrix rank test | 0.122325 | By passing |
| Intra-block longest run check | 0.911413 | By passing |
| Sequence testing | 0.834308 | By passing |
| Run length check | 0.350485 | By passing |
The NIST statistical test suite consists of 15 statistical tests and can detect the randomness of the sequences generated by the new chaotic system. In general, statistical tests are successful when the test results are between 0.01 and 1. Meanwhile, the larger the value, the more random the sequence. For convenience, 20000 ten thousand real numbers generated by the 7D complex chaotic system with the cubic memristor are normalized to a binary sequence and then directly used as experimental data of a NIST test suite. As shown in table 6, the NIST test results were between 0.01 and 1, which means that the statistical test was successful. The key is verified to have good randomness, the security of the method is improved, the key has good randomness, and the security of the algorithm in the data encryption process can be improved.
The invention provides a novel 7D complex chaotic system with a memristor, which is derived from a 4D chaotic system with a cubic nonlinear memristor, realizes the expansion of variables from a real domain to a complex domain, and analyzes the chaotic performance of the novel chaotic system by utilizing Lyapunov indexes, phase diagrams, bifurcation diagrams, 0-1 test and complexity. Then, on the basis of the 7D complex chaotic system with the cubic memristor, a chaotic system data encryption method based on the cubic memristor is provided by combining the Arnold transformation, the chaotic system data encryption method is applied to encryption of data of the smart grid, verification experiments are carried out on images and data transmitted in the smart grid, and the image encryption is analyzed by utilizing a histogram, a correlation coefficient, an information entropy, a key sensitivity and a reconstruction quality. The NIST test is used for analyzing data encryption, and test results show that the method has good robustness and enough capability of preventing data leakage and malicious attacks.
The above embodiments are only specific examples of the present invention, and the protection scope of the present invention includes but is not limited to the product forms and styles of the above embodiments, and any suitable changes or modifications made by those skilled in the art according to the claims of the present invention shall fall within the protection scope of the present invention.
Claims (5)
1. A chaotic system data encryption method based on a cubic memristor is characterized by comprising the following steps of: the method comprises the following steps:
step 1: separating the image channel and changing it into R, G and B channels;
step 2: randomly transforming the positions of the three primary color pixel values using a random function, which are referred to as R1, G1, and B1;
and step 3: transforming the positions of the three primary color pixel values according to the Arnold transform, named R2, G2, and B2; the Arnold transform is as follows:
wherein M andn are rows and columns of an image matrix respectively, A and B are pseudo-random matrixes with the size of M multiplied by N generated by a chaotic sequence of a 7D complex chaotic system with a cubic memristor; coordinates x of pixels in digital imagesn，yn∈[0，255]；
And 4, step 4: and carrying out seven-dimensional exclusive OR on the seven-dimensional pseudo-random sequence with the images of R2, G2 and B2 generated by the 7D complex chaotic system with the cubic memristor.
2. The method of claim 1, wherein said method comprises: in step 4, the exclusive or sequence is random.
3. The method of claim 1, wherein said method comprises: the 7D complex chaotic system with the cubic memristor is as follows:
wherein α, β, r, d are constants, xi(i 1.., 7) is an argument and W is the memristor derivative.
4. The method of claim 3, wherein the three-dimensional memristor-based complex chaotic system data encryption method comprises: the establishment process of the 7D complex chaotic system with the cubic memristor is as follows:
the mathematical expression of the cubic nonlinear memristor is as follows:
wherein, a and b are constants,
is an independent variable, and the number of the independent variables,
a non-linear memristor; definition of memristor derivatives
The method comprises the following steps:
the real chaotic system with cubic nonlinear memristors is given by:
wherein α, β, r, d are constants, x, y, z are independent variables; the above equation is derived into a complex field, where x ═ x1+jx2，y＝x3+jx4，z＝x5+jx6，
xi(i 1.., 7) is an argument, j is an imaginary number; separating a real part and an imaginary part of the formula to obtain a 7D complex chaotic system with a cubic memristor, so that the expansion of a variable from a real domain to a complex domain is realized;
5. the method of any of claims 1-4, wherein said method comprises: the chaotic system data encryption method based on the cubic memristor is applied to data encryption of the intelligent power grid, and comprises the following steps of:
step 1: randomly changing the position of the temperature data by using a random function, and naming the position as data 1;
step 2: changing the position of data 1 according to Arnold transformation, and naming the changed position as data 2;
and step 3: and carrying out seven-time XOR on a seven-dimensional pseudo-random sequence with 2 data generated by the 7D complex chaotic system with the cubic memristor, wherein the XOR sequence is random.
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