CN113949782B - Complex chaotic system data encryption method based on cubic memristor - Google Patents
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- H—ELECTRICITY
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- H04N1/00—Scanning, transmission or reproduction of documents or the like, e.g. facsimile transmission; Details thereof
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- H04L9/00—Cryptographic mechanisms or cryptographic arrangements for secret or secure communications; Network security protocols
- H04L9/001—Cryptographic mechanisms or cryptographic arrangements for secret or secure communications; Network security protocols using chaotic signals
Abstract
The invention relates to the technical field of data processing, in particular to a data encryption method of a complex chaotic system based on a cubic non-linear memristor, wherein the complex chaotic system is derived from a 4D chaotic system with a cubic non-linear memristor, the expansion of variables 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, 0-1 inspection, complexity and the like. Then, on the basis of the 7D complex chaotic system with the cubic memristor, a complex chaotic system data encryption method based on the cubic memristor is provided in combination with Arnold transformation, the method is applied to encryption of smart grid data, verification experiments are carried out on images and data transmitted in the smart grid, and images are encrypted by utilizing histograms, correlation coefficients, information entropy, key sensitivity and reconstruction quality, so that the method has good robustness and enough capability of preventing data leakage and malicious attack.
Description
Technical Field
The invention relates to the technical field of data processing, in particular to a complex chaotic system data encryption method based on a cubic memristor.
Background
After the smart grid is built, the security of its 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 communication data between the different fields needs to be encrypted to prevent malicious attacks. With the rapid development of smart grid technology, the number and types of attacks on smart grids are increased remarkably, and huge losses and negative effects are brought to the grids. In sharp contrast to the importance of smart grids, concerns about their network and information security remain inadequate, which is also a cause of frequent occurrence of power system accidents. Therefore, the security encryption algorithm is a key for preventing malicious attacks and ensuring normal operation of the smart grid.
The chaotic system has a series of characteristics of initial value and extremely sensitive system parameters, ergodic property, track unpredictability, good pseudo-randomness and the like, 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, secret communication, and the like. Among them, the application of chaos in data encryption has received great attention in recent years and has become an important branch of chaos applications, but the following problems still remain: the low-dimensional chaotic system has the defects of short period, uneven chaotic sequence distribution, small key space and the like, and has poor data encryption effect. In addition, data in the smart grid is easy to attack and tamper, and misjudgment and economic loss of the system are caused.
Disclosure of Invention
The invention aims to overcome the defects in the prior art, and provides a complex chaotic system data encryption method based on a cubic memristor, which can effectively prevent the problem of data leakage.
The technical scheme adopted for solving the technical problems is as follows:
a complex chaotic system data encryption method based on a cubic memristor comprises the following steps:
step 1: separating the image channels and changing them to R, G and B channels;
step 2: randomly transforming the positions of the three primary color pixel values, called R1, G1 and B1, using a random function;
step 3: the positions of the three primary color pixel values are converted according to the Arnold conversion, and they are named R2, G2, and B2; arnold transformation is shown below:
wherein M and N are the rows and columns of the image matrix, respectively, and A and B are the sizes generated by the chaotic sequence of the 7D complex chaotic system with the cubic memristorIs a pseudo-random matrix of m×n; coordinates x of pixels in a digital image n ，y n ∈[0，255]；
Step 4: seven exclusive or is carried out on the seven-dimensional pseudo-random sequence with the R2, G2 and B2 images generated by the 7D complex chaotic system with the cubic memristor.
Further, in step 4, the exclusive-or sequence is random.
Steps 1,2 and 3 are scrambling portions and step 4 is a diffusion portion.
The complex chaotic system data encryption method based on the cubic memristor has the advantages of small image distortion, good diffusion and scrambling performance, better safety, better covering up of the distribution rule of an original image, 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, x i (i=1,..7.) is the argument and W is the memristor derivative.
Further, the establishment process of the 7D complex chaotic system with the cubic memristor comprises the following steps:
the mathematical expression of the cubic nonlinear memristor is:
wherein a, b is a constant,is an independent variable,/->Is a nonlinear memristor; definition of memristor derivative ∈>The method comprises the following steps:
the real chaotic system with the cubic nonlinear memristor is given by:
wherein α, β, r, d are constants and x, y, z are independent variables; the above formula is derived into the complex domain, where x=x 1 +jx 2 ，y＝x 3 +jx 4 ，z＝x 5 +jx 6 ，x i (i=1,.,. 7) is an argument and j is an imaginary number; the real part and the imaginary part are separated to obtain the following 7D complex chaotic system with the cubic memristor, and the expansion of the variable from the real domain to the complex domain is realized;
further, the complex chaotic system data encryption method based on the cubic memristor is applied to intelligent power 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 it as data 2;
step 3: seven exclusive-or is carried out on a seven-dimensional pseudo-random sequence with 2 data generated by a 7D complex chaotic system with a cubic memristor, and the exclusive-or sequence is random.
The invention has the technical effects that:
compared with the prior art, the data encryption method of the complex chaotic system based on the three-dimensional memristor is provided by combining Arnold transformation on the basis of the 7D complex chaotic system with the three-dimensional memristor, and has the advantages of realizing the expansion of variables from a real domain to a complex domain, having smaller image distortion, good diffusion and scrambling performance, better safety, better covering up the distribution rule of an original image, effectively preventing data leakage and malicious attack, being applicable to encryption of data of smart power grids and the like, and having wide application range.
Drawings
FIG. 1 is a phase diagram of a 7D complex chaotic system attractor with a cubic memristor of the present disclosure;
FIG. 2 is a Lyapunov exponent spectra of a 7D complex chaotic system with a cubic memristor of the present disclosure;
FIG. 3 is a bifurcation diagram of the variable according to the parameter of the present invention;
FIG. 4 is a 0-1 test schematic diagram of a 7D complex chaotic system with a cubic memristor of the present disclosure;
FIG. 5 is a chromatogram of the variation of parameters with parameters according to the present invention;
FIG. 6 is a flow chart of an image encryption algorithm according to the present invention;
FIG. 7 is a schematic diagram of the encryption process of the Lena image of the present invention;
FIG. 8 is a histogram of an original Lena and an encrypted Lena of the present invention;
FIG. 9 is a schematic diagram of the encryption process of "image 1" of the present invention;
FIG. 10 is a schematic diagram of the encryption process of "image 2" of the present invention;
FIG. 11 is an isodistribution histogram of an encrypted "image 1" according to the present invention;
FIG. 12 is an isodistribution histogram of an encrypted "image 2" according to the present invention;
FIG. 13 is a diagram showing the decryption result of the wrong key according to the present invention;
fig. 14 is a flow chart of the 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 more clear, the technical solutions in the embodiments of the present invention are clearly and completely described below with reference to the accompanying drawings of the specification.
Example 1:
image encryption is typically divided into two parts: scrambling and spreading; 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 embodiment relates to a complex chaotic system data encryption method based on a cubic memristor, which comprises the following steps:
step 1: separating the image channels and changing them to R, G and B channels;
step 2: randomly transforming the positions of the three primary color pixel values, called R1, G1 and B1, using a random function;
step 3: the positions of the three primary color pixel values are converted according to the Arnold conversion, and they are named R2, G2, and B2; arnold transformation is shown below:
m and N are the rows and columns of the image matrix respectively, and 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 a digital image n ，y n ∈[0，255]；
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 exclusive OR for seven times, and the exclusive OR sequence is random.
Steps 1,2 and 3 are scrambling portions and step 4 is a diffusion portion.
Setting up
α＝10，r＝0.1，/>
The initial condition was (0.1,0.1,0.1,0.1,0.1,0.1,0.1).
The encryption algorithm adopts a 'Lena' image, and the encryption process is shown in fig. 7, wherein fig. 7 (a) is an original image, fig. 7 (b) is a scrambled image, fig. 7 (c) is an encrypted image, fig. 7 (d) is a decrypted image, and fig. 7 (c) hides the characteristics of the original image.
The reconstruction quality, correlation, histogram and information entropy of the complex chaotic system data encryption method are analyzed.
A. Reconstruction mass analysis
A peak signal-to-noise ratio (PSNR) is introduced to measure distortion between the original image and the decrypted image. When the peak signal-to-noise ratio is greater than 30dB and less than 40dB, the image distortion is small. The peak signal-to-noise ratio method is as follows:
step 1: mean square deviation (MSE) is calculated:
step 2: calculating peak signal-to-noise ratio:
PSNR(f，g)＝20log 10 (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 indicator of the similarity of two images, consisting of three independent components of brightness, contrast, and structure. SSIM can be expressed as follows:
the SSIM between fig. 7 (a) and fig. 7 (d) was calculated, the number was 1, and fig. 7 (a) and fig. 7 (d) were identical in the 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. N pairs of labeled pixels are randomly selected in the original image. The correlation coefficient formula between the two is as follows:
the correlation coefficients of adjacent 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, then the neighboring pixels are substantially uncorrelated. Table 1 the phase is of pixel relevance,
TABLE 1 Adjacent Pixel correlation
Table 1 shows that the complex chaotic system data encryption method has good diffusion and scrambling performance, so that the algorithm has better safety for statistical analysis.
C. Histogram
Fig. 8 shows a pixel distribution at each pixel level of three primary color matrices. Fig. 8 (a), 8 (b) and 8 (c) show that the fluctuation is large, and the peak and the valley are greatly different. Some pixel values are more frequent, while others are less frequent. After encryption, the pixel distribution of each pixel level of the three-primary-color matrix is relatively uniform, the value frequency of each pixel value is basically the same, and the distribution rule of an original image is better covered.
D. Information entropy
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 a gray value probability; the larger the entropy of the image information (maximum value is 8), the more uniform the distribution of the image pixels, and the non-random distribution of the image pixels indicates 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 larger image information entropy, which indicates that the method has enough anti-attack capability. The encryption algorithm provided by the invention can effectively prevent 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:
wherein, alpha, b are constants,is an independent variable,/->Is a nonlinear memristor; definition of memristor derivative ∈>The method comprises the following steps:
the real chaotic system with the cubic nonlinear memristor is given by:
wherein α, β, r, d are constants and x, y, z are independent variables; the above formula is derived into the complex domain, where x=x 1 +jx 2 ，y＝x 3 +jx 4 ，z＝x 5 +jx 6 ，x i (i=1,.,. 7) is an argument and j is an imaginary number; the real part and the imaginary part are separated to obtain the following 7D complex chaotic system with the cubic memristor, and the expansion of the variable from the real domain to the complex domain is realized;
the attractors, bifurcation diagrams, complexity and 0-1 test of the 7D complex chaotic system with three-dimensional memristors are analyzed below.
A. Chaotic attractor
Setting up
α＝10，r＝0.1，/>(x 1 ，x 2 ，x 3 ，x 4 ，x 5 ，x 6 ，x 7 ) The initial value of (0.1,0.1,0.1,0.1,0.1,0.1,0.1) and the phase diagram of the attractor is shown in FIG. 1.
The lyapunov exponent represents a numerical feature of the average exponential divergence rate of adjacent tracks in phase space, which is one of the numerical features used to identify chaotic motion, a negative value indicates that motion in that direction is stable, and a positive value indicates that motion in that direction is unstable. The system is chaotic when the Lyapunov index is positive, negative and zero. 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, le7= -4.205, and therefore the system is chaotic, with the corresponding lyapunov index as 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 shown to be chaotic. A bifurcation diagram of the variable with parameter is shown in fig. 3. As can be seen from fig. 3, the system is continuously branched between different states with time variation, 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) to a chaotic sequence;
step 2: the transformation variables p (n) and q (n) of the sequence are calculated:
wherein c ε (0, pi); if x (j) is not chaotic, the p (n) -q (n) track is stable; if the sequence is in a chaotic state, the trace of p (n) -q (n) will be in a Brownian state. The schematic diagram of the "0-1 test" 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, so that the system is chaotic.
D. Complexity analysis
The SE algorithm and the C0 algorithm based on Fourier transformation and wavelet transformation are both spectral entropy algorithms; to verify and analyze the complexity of these two parameters as they change, chromatograms were introduced; the color of the chromatogram with the parameter variation is shown in fig. 5, and the lighter the color is, the lower the complexity is.
Example 2:
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 recognized directly from the image, and fig. 9 (d) is the decrypted "image 1". As shown in fig. 10, "image 2" which has the same encryption process as "image 1" is shown.
Then, the decrypted "image 1" and the decrypted "image 2" are analyzed in terms of histogram, correlation coefficient, information entropy, key sensitivity, reconstruction quality, and the like.
1) Histogram
As in fig. 11 and 12, the encrypted isodistribution histogram successfully conceals the features.
2) Correlation coefficient
TABLE 3 correlation of adjacent pixels
According to table 3, the encrypted image is compared with the original image, and the adjacent pixels have little correlation, even negative correlation.
3) Information entropy
TABLE 4 information entropy of images
As can be seen from table 4, the information entropy of the encrypted image is close to 8 (maximum value is 8), and the encrypted image hardly reveals 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 the cubic memristor remain unchanged. The generated key is used to decrypt the cryptographic images 1 and 2. According to fig. 13, even a small change of the key does not successfully decrypt images 1 and 2, and the encryption algorithm is very sensitive to the key.
5) Reconstruction quality analysis
Table 5 reconstruction quality analysis
As can be seen from table 5, the reconstruction quality of "image 1" and "image 2" is very good; the encryption method provided by the invention plays a good role in the safe image transmission of the smart grid in terms of correlation coefficient, histogram, information entropy, key sensitivity, reconstruction quality and the like.
B. Data encryption
The data set used to test the security of the encryption method described above 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 changed randomly using a random function, which was designated as data 1.
Step 2: the position of data 1 was changed according to the Arnold transformation and was designated as data 2.
Step 3: seven exclusive-or is carried out on a seven-dimensional pseudo-random sequence with 2 data generated by a 7D complex chaotic system with a cubic memristor, and the exclusive-or sequence is random.
A flow chart of the data encryption algorithm is shown in fig. 14.
Ciphertext data consisted of:
0xFC 0xCF 0xFC 0xFC 0xFC 0xFC 0xF80xF8
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
The NIST statistical test suite consists of 15 statistical tests, and can detect the randomness of a sequence 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 thousands of real numbers generated by the 7D complex chaotic system with the cubic memristor are standardized into binary sequences and then directly used as experimental data of a NIST test suite. As shown in Table 6, 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 using Lyapunov indexes, phase diagrams, bifurcation diagrams, 0-1 inspection and complexity. Then, based on the 7D complex chaotic system with the cubic memristor, the complex chaotic system data encryption method based on the cubic memristor is provided in combination with Arnold transformation, the method is applied to encryption of smart grid data, verification experiments are carried out on images and data transmitted in the smart grid, and images are encrypted and analyzed by means of histograms, correlation coefficients, information entropy, key sensitivity and reconstruction quality. 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 attack.
The foregoing embodiments are merely examples of the present invention, and the scope of the present invention includes, but is not limited to, the forms and styles of the foregoing embodiments, and any suitable changes or modifications made by those skilled in the art, which are consistent with the claims of the present invention, shall fall within the scope of the present invention.
Claims (4)
1. A complex chaotic system data encryption method based on a cubic memristor is characterized in that: the method comprises the following steps:
step 1: separating the image channels and changing them to R, G and B channels;
step 2: randomly transforming the positions of the three primary color pixel values, called R1, G1 and B1, using a random function;
step 3: the positions of the three primary color pixel values are converted according to the Arnold conversion, and they are named R2, G2, and B2; arnold transformation is shown below:
m and N are the rows and columns of the image matrix respectively, and 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 a digital image n ，y n ∈[0，255]；
Step 4: performing seven exclusive OR on the seven-dimensional pseudo-random sequence with the R2, G2 and B2 images generated by the 7D complex chaotic system with the cubic memristor;
the 7D complex chaotic system with the cubic memristor comprises the following components:
wherein α, β, r, d are constants, x i (i=1,..7.) is the argument and W is the memristor derivative.
2. The complex chaotic system data encryption method based on the cubic memristor, according to claim 1, is characterized in that: in step 4, the exclusive-or sequence is random.
3. The complex chaotic system data encryption method based on the cubic memristor, according to claim 1, is characterized in that: the establishment process of the 7D complex chaotic system with the cubic memristor comprises the following steps:
the mathematical expression of the cubic nonlinear memristor is:
wherein a, b is a constant,is an independent variable,/->Is a nonlinear memristor; definition of memristor derivative ∈>The method comprises the following steps:
the real chaotic system with the cubic nonlinear memristor is given by:
wherein α, β, r, d are constants and x, y, z are independent variables; the above formula is derived into the complex domain, where x=x 1 +jx 2 ，y＝x 3 +jx 4 ，x i (i=1,.,. 7) is an argument and j is an imaginary number; the real part and the imaginary part are separated to obtain the following 7D complex chaotic system with the cubic memristor, and the expansion of the variable from the real domain to the complex domain is realized;
4. the complex chaotic system data encryption method based on the cubic memristor according to any one of claims 1 to 3, wherein the complex chaotic system data encryption method is characterized in that: the complex chaotic system data encryption method based on the cubic memristor is applied to intelligent power 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 it as data 2;
step 3: seven exclusive-or is carried out on a seven-dimensional pseudo-random sequence with 2 data generated by a 7D complex chaotic system with a cubic memristor, and the exclusive-or sequence is random.
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