CN110572252A - image encryption and decryption method based on fractional order translation chaotic system - Google Patents
image encryption and decryption method based on fractional order translation chaotic system Download PDFInfo
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Abstract
a method for encrypting and decrypting an image based on a fractional order translation chaotic system relates to the field of computer cryptography, image encryption processing, communication and network engineering and solves the problems that initial conditions of the existing chaotic system are not related to plaintext, and the existing chaotic system is not beneficial to resisting plaintext attack, so that the safety is poor and the like. Meanwhile, the fractional order multi-scroll chaotic system has higher dimensional space and stronger initial value sensitivity, the image encryption and decryption algorithm based on the system has strong randomness, more key parameters, and a multi-iteration mode is adopted in the encryption process, so that the encrypted ciphertext matrix becomes very uniform, thereby achieving the purpose of effectively hiding the original image information.
Description
Technical Field
The invention relates to the fields of computer cryptography, image encryption processing, communication and network engineering, in particular to an image encryption and decryption method based on a fractional order multi-scroll chaotic system.
Background
with the change of information technology, the life of people becomes more and more convenient. Digital images, one of the important forms of information, are easily stolen and tampered during transmission, and thus the security problem thereof has attracted a great deal of attention. The image encryption technology is the most economic and effective method for protecting the safety of image information, and is a very practical and urgent key technology to be rapidly developed. At present, there are many different image encryption schemes, such as spatial domain encryption scheme, frequency domain encryption scheme, adaptive encryption, and the like. Because the chaotic mapping has sensitivity to initial values and parameters, the chaotic mapping-based image encryption algorithm has higher efficiency and higher safety, and becomes a research hotspot in the field of image information safety. Meanwhile, the translational chaotic system is a high-dimensional chaotic system which is newly proposed in 2016 and generates a multi-directional scroll attractor, and the translational chaotic system can be equivalent to an existing classical scroll chaotic system such as a Chua system and a Jerk system by changing the parameter value of the system. Therefore, the system improves the theoretical system of the first-order chaotic system and has profound theoretical and practical significance for the exploration of the engineering application of the first-order chaotic system.
the prior art provides various chaos-based image encryption and decryption algorithms, which are divided into gray image encryption and color image encryption according to an encryption object and one-dimensional, two-dimensional and high-dimensional chaos mapping encryption according to a chaos system. The one-dimensional chaotic mapping and the two-dimensional chaotic mapping have the advantages of simple form, high operation efficiency and the like, but have the defects of small key space, low safety and the like; although the high-dimensional chaos has the advantages of higher complexity, better unpredictability and the like, pixel points are not changed, and the initial condition of the chaotic system is not related to the plaintext, so that the chaotic system is not beneficial to resisting plaintext attack.
disclosure of Invention
the invention provides an image encryption and decryption method based on a fractional order translation chaotic system, which aims to solve the problems that the initial condition of the conventional chaotic system is not associated with a plaintext, so that the conventional chaotic system is not beneficial to resisting plaintext attack, the safety is poor and the like.
Constructing a fractional order translation chaotic system, constructing a fractional order form of the translation chaotic system according to fractional order differential definition, and performing translation transformation on the fractional order chaotic system through a piecewise function to obtain the fractional order translation chaotic system;
Step two, generating even and odd multi-scroll chaotic attractors according to the fractional order translation chaotic system constructed in the step one to obtain a fractional order multi-scroll chaotic system;
Thirdly, encrypting the image based on the fractional order multi-scroll chaotic system in the second step; the encryption process is as follows:
setting an original color image with the size of X multiplied by Y, and sequentially decomposing the original color image into a red layer matrix OA, a green layer matrix OG and a blue layer matrix OB according to red, green and blue components;
Performing scrambling transformation on the red layer matrix OA, the green layer matrix OG and the blue layer matrix OB generated in the step three through Logistic mapping to obtain a red layer matrix LA, a green layer matrix LG and a blue layer matrix LB after scrambling;
thirdly, setting initial values, orders and iteration times of parameters of the fractional order translation chaotic system to generate three key chaotic sequences C1, C2 and C3 with the size of X multiplied by Y;
Step three, performing modulus processing on the key chaotic sequences C1, C2 and C3 with the size of X multiplied by Y generated in the step three respectively to obtain 8-bit unsigned integer encryption key streams K1, K2 and K3 with the size of X multiplied by Y and the maximum element value of 255;
Performing exclusive-or operation on the scrambled red layer matrix LA, the scrambled green layer matrix LG and the scrambled blue layer matrix LB obtained in the third step and the key streams K1, K2 and K3 generated in the third step and the fourth step respectively to obtain an encrypted red layer matrix CA, an encrypted green layer matrix CG and an encrypted blue layer matrix CB;
step three, respectively combining the encrypted red layer matrix CA, the green layer matrix CG and the blue layer matrix CB obtained in the step three with the encryption key streams K1, K2 and K3 generated in the step three to generate three complex chaotic phase templates CCRPM1, CCRPM2 and CCRPM 3;
step three, multiplying the red layer matrix OA, the green layer matrix OG and the blue layer matrix OB in the step one with the composite chaotic phase templates CCRPM1, CCRPM2 and CCRPM3 of the three matrices obtained in the step three six respectively, and performing fractional Fourier transform on the multiplied results to obtain a transform result EF (EF 1, EF2 and EF 3);
And step III, compounding the transformation result EF in the step III with the red layer matrix CA, the green layer matrix CG and the blue layer matrix CB which are encrypted in the step III, so as to realize the encryption of the whole image.
The invention has the beneficial effects that: according to the image encryption and decryption method based on the fractional order multi-scroll chaotic system, the fractional order chaotic system is subjected to translation transformation by introducing the piecewise function, and parameters are set, so that 2(N +1) and 2N +1 scroll attractors with different numbers can be generated, and the fractional order translation chaotic system perfects the field of the fractional order chaotic system formed by three-dimensional one-time autonomous ordinary differentiation. Meanwhile, the fractional order multi-scroll chaotic system has higher dimensional space and stronger initial value sensitivity, the image encryption and decryption algorithm based on the system has strong randomness, more key parameters, and a multi-iteration mode is adopted in the encryption process, so that the encrypted ciphertext matrix becomes very uniform, thereby achieving the purpose of effectively hiding the original image information.
The chaotic phase template generated by the invention can effectively resist attack of decoding methods such as phase space reconstruction and the like, has strong confidentiality and large key space, and is more favorable for image encryption. Even if an illegal user masters the encrypted chaotic system, even the initial value of the system, the complete decryption of the image can not be realized on the premise of not knowing the fractional order and the phase template.
The fractional order translation chaotic system based on the invention can completely reset the image pixel matrix and form a key stream to be mixed and embedded into the pixel image, thereby realizing the design of the image encryption method. The unpredictability and the randomness of the image encryption scheme are improved, so that the image encryption scheme has a large key space and good initial value sensitivity, can effectively resist exhaustive attacks, and has high safety.
drawings
FIG. 1 is a flowchart of an image encryption and decryption method based on a fractional order translation chaotic system according to the present invention;
fig. 2 is an even number of multi-scroll chaotic attractors in the image encryption and decryption method based on the fractional order translational chaotic system according to the present invention, where fig. 2a is the multi-scroll chaotic attractor when n is 0; FIG. 2b is the multi-scroll chaotic attractor diagram with n equal to 3;
fig. 3 is an odd-numbered multi-scroll chaotic attractor diagram according to the present invention, wherein fig. 3a is the multi-scroll chaotic attractor diagram when n is 0; FIG. 3b is the multi-scroll chaotic attractor diagram when n is 3;
fig. 4 is an effect diagram of an encrypted image in the image encryption and decryption method based on the fractional order translation chaotic system according to the present invention, where fig. 4a is an original image, fig. 4b is an effect diagram of a graying process of the original image, fig. 4c is an encrypted image, and fig. 4d is a histogram of the encrypted image;
FIG. 5 is a diagram illustrating the effect of image decryption in the image encryption and decryption method based on the fractional order translation chaotic system according to the present invention; fig. 5a is a schematic diagram of wavelet reconstruction of the decrypted image, fig. 5b is a histogram of the decrypted image, fig. 5c is an effect diagram of the decrypted image, and fig. 5d is an effect diagram of decryption failure.
Detailed Description
In a first embodiment, the first embodiment is described with reference to fig. 1 to 5, and a fractional order multi-scroll chaotic system-based image encryption and decryption method is provided, first, a fractional order translational chaotic system is constructed according to fractional order differential definition, and the fractional order translational chaotic system is obtained through translational transformation, which indicates that the fractional order system has more complex topological properties compared with an integer order;
Secondly, two translation rules for generating odd number and even number of scroll attractors are provided; finally, based on the system, the encryption of the overall reset of the image pixel matrix and the mixed embedding of a key stream formed by utilizing the translation chaotic subsystem into the pixel image are realized, and the design of the image encryption method is realized, and the method specifically comprises the following steps:
Step 1: in the three-dimensional ordinary differential equation, the translation chaotic system is
Constructing a fractional order form of the system, namely a fractional order translation chaotic system, according to the fractional order differential definition; wherein, a12＝a210, the expression is:
Then setting the state variable as x1＝x，x2＝y，x3z, parameter a11＝a，a12＝r，a23＝b，a31＝p，a32＝q，a33C; f (X) is a nonlinear part of the chaotic system and determines the generation method and the generation quantity of the multi-scroll chaotic attractors of the system.
The Jacobian matrix A of the fractional order chaotic system at the origin O (0,0,0) is described as follows:
At present, the definition of fractional order differential is not unified, but the inemann-Liouville definition is adopted in common academia, and the definition is mainly divided into an integral definition and a differential definition, which are briefly introduced as follows:
Let f ∈ C (0, + ∞),μ>0, w is the smallest positive integer of more than or equal to 1, namely v ═ w-alpha ≥ 0,0<α<1 then called
Riemann-Liouville differentiation of order alpha for the function f (t) and is noted
If it is notContinuing over interval J, we call f (t) a continuous differentiable function of order α over J, and we denote as:Obtaining the alpha-order Riemann-Liouville differential:
Where Γ (w- α) is a gamma function, α is the order of the fractional order, and 0< α <1 is a non-integer.
Taking the initial value of the above formula as zero, and performing Laplace conversion on the formula (5) to obtain
the fractional order chaotic system is subjected to translation transformation through a piecewise function, and the obtained fractional order chaotic translation system is as follows:
wherein f (x) is a nonlinear piecewise function, and is a translation transformation rule of the system, and alpha is greater than 0.88 after verification, the system has chaotic behavior. The system can be made to produce an even and odd number of scroll attractors.
step 2: according to the fractional order translation chaotic system, even number and odd number scroll attractors are generated. The method specifically comprises the following steps:
(1)f1(x) As a first translation transformation criterion, f (x), an even number of attractors can be generated, as follows:
When a is 1, a is 0.4, p is q is 1, r is 2, b is 1.2, c is 0.6, m is 0.2, and n is 3, the system generates 8 scroll attractors, as shown in fig. 2, where fig. 2a is an even number of multi-scroll chaotic attractors when n is 0, and fig. 2b is an even number of multi-scroll chaotic attractors when n is 3. Where N is an integer, and N is an integer whose value determines the scroll attractor position.
(2)f2(x) F (x) is the second translation transformation criterion, an odd number of attractors can be generated, and the formula is as follows:
When a is 1, a is 0.1, r is 2, p is q is 1, b is 1.2, c is 0.6, m is 0.2, and n is 3, the system generates 7 scroll attractors, as shown in fig. 3, where fig. 3a is an odd-numbered multi-scroll chaotic attractor when n is 0 and fig. 3b is an odd-numbered multi-scroll chaotic attractor when n is 3. Where n is an integer whose value determines the scroll attractor position.
because the fractional order differential operator can not be directly operated in time domain simulation, the fractional order operator can be effectively approximated by using a standard integer order operator under the condition that the approximation error is within the allowable range, and the dynamic characteristic analysis of the fractional order chaotic behavior is realized.
Further, in this embodiment, the step of finding the balance point of the fractional order translational chaotic system is as follows:
if even number of scroll attractors are generated, the positions of the balance points are respectively (+/-2 nA,0 and 0);
Odd number of vortex attractors are generated, and the positions of balance points are respectively (+/- [2n- (| n |/n) ] A,0 and 0).
And step 3: the image encryption method based on the fractional order multi-scroll chaotic system comprises the following specific steps:
Step 3.1: considering a color image of X Y size, the original pixel point is Pdsthis is denoted by d being the row vector of the matrix and s being the column vector of the matrix (d 0.., X-1, s 0., Y-1). Decomposing an original image into a red layer matrix OA, a green layer matrix OG and a blue layer matrix OB in sequence according to red, green and blue color components;
Step 3.2: scrambling and transforming the matrices OA, OG, OB generated in step 3.1 through Logistic mapping to obtain new red layer matrix LA, green layer matrix LG, blue layer matrix LB, that is, pixel scrambling and encrypting are performed on the image, so as to achieve the purpose of destroying the correlation between adjacent pixels of the original image, and the specific expression is as follows:
first, an X × Y color image is formed with P ═ P for the original pixel pointds]this is denoted by d being the row vector of the matrix and s being the column vector of the matrix (d 0.., X-1, s 0., Y-1). By Logistic mapping, the initial condition is x0：
Xg+1＝4xg(1–xg) (10)
after a series of g iterations, the final state x is obtainedg. Further, let state xgis new x0And (4) iterating once until X groups of different data (0-X-1) are obtained. Then, rearranging the number of rows of pixel points P according to the row vector, and recording as Phh represents a row vector of the matrix after reset; then setting the initial condition as y0：
Yg+1＝4yg(1–yg) (11)
after a series of g iterations, the final state y is obtainedg. Further, let state ygIs new y0And (4) iterating once until X groups of different data (0-Y-1) are obtained. Subsequently, each column in P is rearranged in the same way, denoted as PlAnd l represents the column vector of the matrix after reset, thereby obtaining the overall reset of the image pixel matrix, and each pixel point is marked as Phl. The following equation is used to calculate the pixel location for each row of the blended column
hd＝mod(x0×1014,X)
ls＝mod(y0×1014,Y) (12)
step 3.3: setting initial values, orders and iteration times of system parameters from the fractional order translation chaotic systems (7) to (9), generating three variables, and then converting the variables into integers:
xi＝mod((|x0|-Floor(|x0|))×1014,256) (13)
wherein x1＝x，x2＝y，x3＝z。
Next, it is generated as follows
According toFrom three variables (x) according to equation (13)1，x2，x3) The three variables (C1, C2, C3) are selected to perform the encryption operation as follows
step 3.4: the key chaotic sequences C1, C2 and C3 with the size of X × Y generated in step 3.3 are respectively subjected to modulus processing, and become 8-bit unsigned integer encryption key streams K1, K2 and K3 with the size of X × Y and the maximum element value of 255.
step 3.5: performing exclusive or operation on the scrambled matrices LA, LG and LB obtained in the step 3.2 and the key streams K1, K2 and K3 generated in the step 3.3 respectively to obtain an encrypted red layer matrix CA, an encrypted green layer matrix CG and an encrypted blue layer matrix CB, which is specifically implemented as follows:
wherein the content of the first and second substances,Respectively representing position coordinates of each pixel point in the matrixes LA, LG and LB after scrambling; bitxor (·) denotes bitxor operations performed bitwise.
step 3.6: the encrypted red layer matrix CA, green layer matrix CG and blue layer matrix CB obtained in step 3.5 are respectively operated together with the 3 key chaotic sequences K1, K2 and K3 generated in step 3.4, that is: the red layer matrix CA plus the key stream K1 generated in the third and fourth steps, the green layer matrix CG plus the key stream K2 generated in the third and fourth steps and the blue layer matrix CB plus the key stream K3 generated in the third and fourth steps respectively obtain complex chaotic phase templates CCRPM1, CCRPM2 and CCRPM3 of three matrices, and the expressions are as follows:
in the formula (u)d,us) And expressing the position coordinates of the corresponding domain after the pixel points in each matrix are subjected to Fourier transform, wherein j is an imaginary part. In the known original image matrix, the original pixel point is Pdsexpressed, i.e. the position coordinate is (p)d,ps)，d＝0,...,X-1，s＝0,...,Y-1；
step 3.7: multiplying the red layer matrix OA, the green layer matrix OG and the blue layer matrix OB in the step 3.1 by the composite chaotic phase templates CCRPM1, CCRPM2 and CCRPM3 of the three matrices obtained in the step 3.6 respectively, and performing fractional fourier transform on the multiplied results to obtain results EF ═ EF1, EF2 and EF 3.
step 3.8: and (4) compounding the result EF in the step (3.7) with the encrypted red layer matrix CA, green layer matrix CG and blue layer matrix CB obtained in the step (3.5) to realize the complete encryption of the whole image.
and 4, step 4: the image is decrypted, the decryption algorithm is the inverse process of the encryption algorithm, namely, under the condition that the same parameters and initial values are selected and encrypted, the plate is backwashed, and the original image can be obtained, and the method comprises the following specific steps:
Step 4.1: performing layer decomposition on the completely encrypted image obtained in the step 3.8 to sequentially obtain a red layer matrix AC, a green layer matrix GC and a blue layer matrix BC with ciphertexts, wherein the red layer matrix AC, the green layer matrix GC and the blue layer matrix BC have the size of X multiplied by Y;
step 4.2: performing inverse fourier transform on the AC, GC and BC obtained in step 4.1, respectively, to obtain a result IEF ═ IEF1, IEF2, IEF 3;
step 4.3: generating corresponding decryption matrixes IC1, IC2 and IC3 by using key information such as an initial value, an order and an iteration number of a known fractional order translation chaotic system;
step 4.4: respectively carrying out modulus extraction on the secret matrixes IC1, IC2 and IC3 generated by 4.3 to obtain decryption chaotic streams IK1, IK2 and IK3 with the size of X multiplied by Y;
step 4.5: the red layer matrix AC with the ciphertext, the green layer matrix GC and the blue layer matrix BC which are decomposed in the step 4.1 and the decrypted chaotic streams IK1, IK2 and IK3 which are obtained in the step 4.4 are operated together to respectively generate a conjugate chaotic phase template CCRPM1 of the three matrices*、CCRPM2*、CCRPM3*the expression is as follows:
step 4.6: multiplying the inverse Fourier transform result IEF obtained in the step 4.2 by the conjugate chaotic phase template in the step 4.5 respectively to obtain results DR, DG and DB;
Step 4.7: performing inverse diffusion transformation on the red layer matrix AC with the ciphertext, the green layer matrix GC and the blue layer matrix BC decomposed in the step 4.1 to respectively obtain results NAC, NGC and NBC;
Step 4.8: combining initial value, order and iteration number information of the translation chaotic system in decryption matrixes IC1, IC2 and IC3 in the step 4.3 to obtain inverse mapping results AO, GO and BO;
Step 4.9: and (4) compounding the DR, DG and DB obtained in the step (4.6) with the AO, GO and BO obtained in the step (4.8) to realize the complete decryption of the whole image.
It can be seen from fig. 4 and 5 that the fractional order multi-scroll chaotic system has a higher dimensional space and a stronger initial value sensitivity, and the image encryption and decryption algorithm based on the system has strong randomness and a plurality of key parameters, and adopts a multi-iteration mode in the encryption process to make the encrypted ciphertext matrix become very uniform, thereby achieving the purpose of effectively hiding the original image information.
the fractional order multi-scroll chaotic attraction subsystem constructed in the embodiment has more complex chaotic dynamic characteristics. The complexity is that the value or the phase track of the state variable of the chaotic attractor can randomly jump among a plurality of different scrolls, so that the larger the number of the scrolls is, the larger the randomness of the jump is, meanwhile, the multi-scroll chaotic system has a plurality of key parameters, can effectively resist the attack of decoding methods such as phase space reconstruction and the like, has strong confidentiality and large key space, and is more favorable for image encryption.
It will be apparent to those skilled in the art that the foregoing specific embodiments are merely preferred embodiments of the invention, and thus, modifications and variations may be made in the invention to which the invention pertains, which will still embody the principles of the invention and which will still achieve the objects of the invention, within the scope of the invention as defined by the appended claims.
Claims (7)
1. an image encryption and decryption method based on a fractional order translation chaotic system is characterized in that: the method is realized by the following steps:
step one, constructing a fractional order translation chaotic system; according to the fractional order differential definition, constructing a fractional order form of a translation chaotic system, and performing translation transformation on the fractional order chaotic system through a piecewise function to obtain the fractional order translation chaotic system;
Step two, generating even and odd multi-scroll chaotic attractors according to the fractional order translation chaotic system constructed in the step one to obtain a fractional order multi-scroll chaotic system;
thirdly, encrypting the image based on the fractional order multi-scroll chaotic system in the second step; the encryption process is as follows:
setting an original color image with the size of X multiplied by Y, and sequentially decomposing the original color image into a red layer matrix OA, a green layer matrix OG and a blue layer matrix OB according to red, green and blue components;
performing scrambling transformation on the red layer matrix OA, the green layer matrix OG and the blue layer matrix OB generated in the step three through Logistic mapping to obtain a red layer matrix LA, a green layer matrix LG and a blue layer matrix LB after scrambling;
thirdly, setting initial values, orders and iteration times of parameters of the fractional order translation chaotic system to generate three key chaotic sequences C1, C2 and C3 with the size of X multiplied by Y;
step three, performing modulus processing on the key chaotic sequences C1, C2 and C3 with the size of X multiplied by Y generated in the step three respectively to obtain 8-bit unsigned integer encryption key streams K1, K2 and K3 with the size of X multiplied by Y and the maximum element value of 255;
performing exclusive-or operation on the scrambled red layer matrix LA, the scrambled green layer matrix LG and the scrambled blue layer matrix LB obtained in the third step and the key streams K1, K2 and K3 generated in the third step and the fourth step respectively to obtain an encrypted red layer matrix CA, an encrypted green layer matrix CG and an encrypted blue layer matrix CB;
Step three, respectively combining the encrypted red layer matrix CA, the green layer matrix CG and the blue layer matrix CB obtained in the step three with the encryption key streams K1, K2 and K3 generated in the step three to generate three complex chaotic phase templates CCRPM1, CCRPM2 and CCRPM 3;
step three, multiplying the red layer matrix OA, the green layer matrix OG and the blue layer matrix OB in the step one with the composite chaotic phase templates CCRPM1, CCRPM2 and CCRPM3 of the three matrices obtained in the step three six respectively, and performing fractional Fourier transform on the multiplied results to obtain a transform result EF (EF 1, EF2 and EF 3);
And step III, compounding the transformation result EF in the step III with the red layer matrix CA, the green layer matrix CG and the blue layer matrix CB which are encrypted in the step III, so as to realize the encryption of the whole image.
2. the image encryption and decryption method based on the fractional order translation chaotic system according to claim 1, characterized in that: the decryption process is as follows:
And step A, decrypting the encrypted image obtained in the step III-VIII, wherein the specific process is as follows:
A1, carrying out layer decomposition on the encrypted image, and sequentially obtaining a red layer matrix AC, a green layer matrix GC and a blue layer matrix BC with ciphertexts, wherein the red layer matrix AC, the green layer matrix GC and the blue layer matrix BC have the size of X multiplied by Y;
Step a2, performing fourier inverse transformation on the red layer matrix AC with the ciphertext, the green layer matrix GC and the blue layer matrix BC obtained in step a1, respectively, to obtain an inverse transformation result IEF, where IEF is [ IEF1, IEF2, IEF3 ];
Step A3, generating corresponding decryption matrixes IC1, IC2 and IC3 according to the initial value, the order, the iteration times and the key chaotic sequence of the fractional order translation chaotic system in the step three;
Step A4, respectively carrying out modulus extraction on the decryption matrixes IC1, IC2 and IC3 generated in the step A3 to obtain decryption chaotic streams IK1, IK2 and IK3 with the size of X multiplied by Y;
step A5, adding the red layer matrix AC with the ciphertext, the green layer matrix GC and the blue layer matrix BC decomposed in the step A1 with the decrypted chaotic streams IK1, IK2 and IK3 obtained in the step A4 respectively to obtain a chaotic complex phase conjugate template CCRPM1*、CCRPM2*and CCRPM3*；
Step A6, multiplying the inverse transformation result IEF obtained in step A2 by the conjugate chaotic phase template in step A5 respectively to obtain multiplied results DR, DG and DB;
step A7: performing inverse diffusion transformation on the red layer matrix AC with the ciphertext, the green layer matrix GC and the blue layer matrix BC decomposed in the step A1 to respectively obtain inverse diffusion results NAC, NGC and NBC;
Step A8, performing chaos inverse mapping on the inverse transformation result IEF obtained in the step A2 according to the initial value, the order and the iteration number of the translation chaotic system in the decryption matrixes IC1, IC2 and IC3 in the step A3 to obtain inverse mapping results AO, GO and BO;
And step A9, compounding the multiplied results DR, DG and DB obtained in the step A6 with the inverse mapping results AO, GO and BO obtained in the step A8, and realizing decryption of the whole image.
3. The image encryption and decryption method based on the fractional order translation chaotic system according to claim 1, characterized in that: the specific process of constructing the fractional order translation chaotic system in the first step is as follows:
step one, constructing a fractional order form of the translation chaotic system according to the fractional order differential definition, and expressing the form as follows:
Setting a state variable value to x1＝x，x2＝y，x3Z, the parameter is a11＝a，a12＝r，a23＝b，a31＝p，a32＝q，a33c; wherein F (X) is chaosa system non-linear portion;
The Jacobian matrix of the fractional order chaotic system at the origin (0,0,0) is A and is expressed by the following formula:
performing translation transformation on the fractional order chaotic system through a piecewise function to obtain the fractional order translation chaotic system, wherein the fractional order translation chaotic system is expressed by the following formula:
Wherein f (x) is a nonlinear piecewise function, and is a system translation transformation criterion, alpha is the order of fractional order and is a non-integer, and 0< alpha < 1.
4. the image encryption and decryption method based on the fractional order translation chaotic system according to claim 1, characterized in that: in the second step, according to the fractional order translation chaotic system constructed in the first step, even and odd multiple scroll chaotic attractors are generated, and a fractional order multiple scroll chaotic system is obtained; the method comprises the following specific steps:
Step two, setting f1(x) If f (x) is the first shift transformation rule, an even number of multi-scroll chaotic attractors are generated, and the formula is as follows:
if the parameters A, a, p, q, r, b, c, m and n are set to be different values, generating even number of scroll attractors with different numbers; n is an integer;
Step two, setting f2(x) F (x) is the second translation transformation criterion, then an odd number of attractors is generated, and the formula is as follows:
Generating a different number of odd scroll attractors;
If even number of scroll attractors are generated, the positions of the balance points are respectively (+/-2 nA,0 and 0);
if odd number of scroll attractors are generated, the balance point positions are respectively (+/- [2n- (| n |/n) ] A,0 and 0).
5. The image encryption and decryption method based on the fractional order translation chaotic system according to claim 1, characterized in that: the third step five is to obtain the encrypted red layer matrix CA, the encrypted green layer matrix CG and the encrypted blue layer matrix CB, and the third step five is specifically realized by the following formula:
in the formula (I), the compound is shown in the specification,Respectively representing the position coordinates of each pixel point in the red layer matrix LA, the green layer matrix LG and the blue layer matrix LB after scrambling; bitxor (·) is a bitwise XOR operation, Phrearranging the original pixel point matrix P according to the row vector, wherein h is the row vector of the matrix after resetting; plTo rearrange the original pixel point matrix P according to the column vector, l is the column vector of the matrix after reset, Phlfor each pixel in the matrix after reset.
6. the image encryption and decryption method based on the fractional order translation chaotic system according to claim 1, characterized in that: complex chaotic phase templates CCRPM1, CCRPM2 and CCRPM3 of the three matrices generated in step three and six are expressed by the following equations:
CCRPM1＝exp[jπ(CA(ud,us))+K1(ud,us)]
CCRPM2＝exp[jπ(CG(ud,us))+K2(ud,us)]
CCRPM3＝exp[jπ(CB(ud,us))+K3(ud,us)]
in the formula (u)d,us) And j is an imaginary part of the position coordinate of the corresponding domain after the pixel points in each matrix are subjected to Fourier transform.
7. The image encryption and decryption method based on the fractional order translation chaotic system according to claim 1, characterized in that: in step A5, conjugate chaotic phase templates CCRPM1 of three matrices are generated respectively*、CCRPM2*、CCRPM3*Expressed by the following formula:
CCRPM1*＝exp[-jπ(AC(ud,us))+IK1(ud,us)]
CCRPM2*＝exp[-jπ(GC(ud,us))+IK2(ud,us)]
CCRPM3*＝exp[-jπ(BC(ud,us))+IK3(ud,us)]。
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