CN111934846A - Multi-chaos Arnold image encryption method based on fractional Fourier transform - Google Patents

Multi-chaos Arnold image encryption method based on fractional Fourier transform Download PDF

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CN111934846A
CN111934846A CN202010686171.XA CN202010686171A CN111934846A CN 111934846 A CN111934846 A CN 111934846A CN 202010686171 A CN202010686171 A CN 202010686171A CN 111934846 A CN111934846 A CN 111934846A
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CN111934846B (en
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张志昌
浦绍质
于雪莲
马满满
王康维
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Harbin University of Science and Technology
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Abstract

The invention belongs to the technical field of information security and digital image processing, and mainly solves the problem that the correlation and scrambling degree of adjacent pixels in the traditional Arnold transform encryption technology are not high. The technical scheme adopted by the invention is multi-chaos Arnold image encryption based on fractional Fourier transform, and comprises the following steps: first, generating Arnold by chaos random sequenceParameters of the transformation: generating a random sequence by a hyper-chaotic Lorenz and one-dimensional chaotic Logistic system, and generating parameters of a transformation matrix of Arnold transformation; secondly, encrypting the image: first, the image is processed to have an order of p1The fractional order fourier transform of (a) and then scrambling the image by means of an improved Arnold transform; finally, the order of the image is p2Fractional order fourier transform of (a); thirdly, image decryption: performing inverse operation symmetrical to the image encryption process on the obtained encrypted image to obtain a decrypted image; the invention utilizes the chaotic system and fractional Fourier transform to enhance the key space and improve the security of encryption.

Description

Multi-chaos Arnold image encryption method based on fractional Fourier transform
Technical Field
The invention relates to the technical field of information security and image processing, in particular to a multi-chaos Arnold image encryption method based on fractional Fourier transform.
Background
The image information has the characteristics of image and intuition, and is widely applied to politics, military, medicine and daily life of people. Therefore, the security problem of how to ensure that the image information is not stolen or damaged by malicious attacks in the transmission process of the image is more worthy of attention.
Professor Refregior and Javidi applied double random phase coding technology to encrypt images in 1995, which opened the way of optical image encryption, and the optical encryption method can complete image encryption while completing image information transmission, and has efficient computing capability. And has the advantages of large capacity, high speed, multiple dimensionality and high safety. In 2000, Unnikrishnan et al introduced fractional Fourier transform into image encryption for the first time, because of the additivity of fractional order of fractional Fourier transform, the key space was well enhanced. Fractional fourier transform is gradually becoming an important method for optical image encryption because of its high degree of freedom in encryption, and is widely used. However, when the fractional order Fourier transform is applied to image encryption alone, the sensitivity of the fractional order is not very high, and potential safety hazards exist.
The image scrambling is an important approach in image encryption, and the idea of the image scrambling based on the spatial domain is to change the original pixel position of an image by mathematically transforming an image, and destroy the correlation of adjacent pixels of the original image, thereby achieving the effect of image encryption. The Arnold transformation, the knight tour transformation, the magic square transformation and other methods are common image scrambling methods, and the Arnold transformation is the most common method for image scrambling due to the simplicity and effectiveness of the Arnold transformation and the good chaotic dynamics characteristic in the image encryption process. And the chaos system is introduced in the encryption process, so that the image scrambling effect can be better enhanced. The chaos phenomenon is an uncertain phenomenon generated in a determined chaos system, a random sequence generated by a hyper-chaos Lorenz and one-dimensional chaos Logistic system has good randomness, and a good image scrambling effect can be achieved when the random sequence is combined with Arnold transformation. However, the Arnold transform only changes the pixel position of the image and does not change the pixel value of the image in the process of image encryption, and there is a high risk of being deciphered.
Disclosure of Invention
To overcome the deficiencies of the prior art, the present invention aims to reduce the correlation of adjacent pixels and increase the image key space, thereby improving the security of encrypted images. The technical scheme adopted by the invention is as follows: the multi-chaos Arnold image encryption method based on fractional Fourier transform comprises the following steps:
step 1: generating transform matrix parameters m and q of Arnold transform by using a random sequence generated by the hyper-chaos Lorenz and the Logistic of the one-dimensional chaotic system, improving the Arnold transform, scrambling and encrypting the image and enhancing the scrambling effect, wherein the process comprises the following steps:
1-1, performing discretization treatment on the hyperchaotic Lorenz system;
the hyper-chaotic Lorenz system formula is as follows:
Figure BDA0002587628210000021
wherein, a is 10, b is 8/3, c is 28, x is system parameteri,yi,zi,viRespectively representing the differentiation of the system variable with respect to time t;
step 1-2, solving a hyperchaotic Lorenz system equation by using a Runge Kutta method to obtain a result:
Figure BDA0002587628210000022
wherein x isi,yi,zi,viRepresenting the value of a system variable of the ith iteration, wherein h is a step length;
step 1-3, setting an input image matrix as I, wherein the image can be regarded as consisting of 8 bit planes, and the hyperchaotic Lorenz initial value x can be obtained by the formula0，y0，z0，w0The 4 th and 5 th bit planes, the 3 rd and 6 th bit planes, the 2 nd and 7 th bit planes, and the 1 st and 8 th bit planes of the graph respectively determine:
x0＝sum{sum[bitand(I,24)]}/(24×M×N)
y0＝sum{sum[bitand(I,36)]}/(36×M×N)
z0＝sum{sum[bitand(I,66)]}/(66×M×N)
w0＝sum{sum[bitand(I,129)]}/(129×M×N)
according to the formula, when the images I are different, the initial value x of the hyperchaotic Lorenz is obtained0，y0，z0，w0Also different;
step 1-4, adding x0，y0，z0，v0Substituting the initial value of the chaos into the hyper-chaos Lorenz for iteration to generate a chaos pseudorandom sequence { x ] with the length of 2MNi}；
Step 1-5, discard { xiLeading the hyper-chaos Lorenz system to fully enter a chaos state by the first 200 values of the hyper-chaos Lorenz system;
steps 1-6, to generate sufficiently long { x }iAfter 3000 iterations, the initial value x is checked0Plus small periodic perturbations, i.e. x0＝x0+hsiny0Then x is added0Substituting the hyper-chaos Lorenz for iteration;
step 1-7, mixing { xiAll conversion is of integer type Xi，XiE (1,2, …,10MN) and generate the following matrix of size M N:
U＝reshape(X(1:M×N),M,N)
step 1-8, performing initial value solving on the one-dimensional chaotic system Logistic, wherein a one-dimensional chaotic system Logistic expression is as follows:
xn＝μxn-1(1-xn-1)
wherein mu is a branch parameter and the initial value is x0；
Step 1 to 9, setting x0Expression:
Figure BDA0002587628210000031
wherein f (x, y) represents the gray value of a pixel at the position of the image (x, y), the image selection is different, and the initial value of the Logistic of the one-dimensional chaotic system is different;
step 1-10, adding x0Substituting mu into the Logistic expression of the one-dimensional chaotic system, and continuously iterating for 1000+ MxN times; step 1-11, in order to obtain a better Logistic chaotic sequence of the one-dimensional chaotic system, the first 1000 items are omitted to obtain a sequence { P }i}；
Step 1-12, sequence { P }iThe integer is transformed and a matrix of size M × N is generated according to:
V＝reshape(P,M,N)。
step 2: an image encryption step:
step 2-1, inputting a gray image to be encrypted with the size of M multiplied by N to obtain a two-dimensional image matrix h (x, y);
step 2-2, the order of the image line direction is p1Obtaining an image h after the primary transformation by the primary fractional Fourier transform1(x,y)；
And 2-3, scrambling and encrypting the image by applying the improved Arnold transformation, wherein the Arnold transformation expression is as follows:
Figure BDA0002587628210000032
wherein, (x ', y') is the coordinate of the pixel point (x, y) after transformation; a is a transformation matrix of the Arnold transformation; mod is the modulo operation; l is the image order; taking random sequences generated by the hyper-chaos Lorenz and the one-dimensional chaotic system Logistic as transformation matrix parameters m and q of Arnold transformation, and obtaining the improved Arnold transformation operation expression:
Figure BDA0002587628210000033
wherein the value of U (i, j) is m and the value of V (i, j) is q;
Figure BDA0002587628210000034
for the new coordinate vector calculated by transformation, the pixel at the coordinate vector of the image is then associated with h1Replacing the pixels at the (i, j) positions, and continuously and circularly replacing to obtain a scrambled image h2(x,y)；
Step 2-4: for image h2The order of progression in the (x, y) column direction is p2The final encrypted image f (x, y) is obtained by fractional fourier transform.
And 3, an image decryption process, which comprises the following specific steps:
step 3-1, carrying out the order of-p in the f (x, y) row direction of the image2To obtain a primary inverse transform image f1(x,y)；
Step 3-2, scanning the image f from right to left and from bottom to top by means of the Arnold matrix used in the encryption process1(x, y) is decrypted to obtain an image f after reverse scrambling2(x,y)；
Step 3-3, to the image f2The order of progression in the (x, y) row direction is-p1The second fractional order inverse Fourier transform to obtain the final decrypted image f3(x,y)。
The invention has the following advantages and beneficial effects:
(1) the method utilizes the Lorenz system and the Logistic system of the one-dimensional chaotic system to generate random sequences, then introduces Arnold transformation to generate transformation matrix parameters m and q, and utilizes the improved Arnold transformation to scramble the image, thereby well enhancing the scrambling degree of the image.
(2) The invention applies a multi-chaos Arnold image encryption method based on fractional Fourier transform to change the pixel position in a space domain and change the pixel value in a frequency domain at the same time, thereby improving the anti-attack capability of the encrypted image.
(3) The invention applies fractional Fourier transform to carry out graph encryption, thereby enhancing the sensitivity of encryption keys and improving the flexibility of encryption combination.
(4) The invention applies an encryption method of mixing a plurality of methods and has good safety.
Description of the drawings:
FIG. 1 is a flow chart of a multi-chaos Arnold image encryption method based on fractional Fourier transform according to the present invention;
FIG. 2 is a graph showing the results of scrambling an image using a conventional Arnold transform and the improved Arnold transform, respectively, in accordance with the present invention; wherein (a) is 256 × 256 pixel 256 gray level plaintext image; (b) is the result of scrambling through one traditional Arnold transformation; (c) the result of scrambling by one improved Arnold transform; (d) the result of decryption encrypted by one time of the improved Arnold transformation;
FIG. 3 is a graph of neighboring pixel correlation plots for the improved Arnold transform described in this invention; wherein, (a) a plaintext image horizontal adjacent pixel correlation point diagram, (b) a ciphertext image horizontal adjacent pixel correlation point diagram, (c) a plaintext image vertical adjacent pixel correlation point diagram, (d) a ciphertext image vertical adjacent pixel correlation point diagram, (e) a plaintext image diagonal adjacent pixel correlation point diagram, (f) a ciphertext image diagonal adjacent pixel correlation point diagram
FIG. 4 is a graph of the multi-chaos Arnold image encryption result based on fractional Fourier transform; the image encryption method comprises the following steps that (a) a ciphertext image encrypted by a multi-chaos Arnold image based on fractional Fourier transform; (b) the image is decrypted when the fractional Fourier order has 0.1 error and other key parameters are correct; (c) the image is decrypted when the encryption key parameters are all correct.
Detailed Description
The invention is implemented by encrypting and decrypting the image, and the encrypting steps are as follows:
in the specific implementation process of the invention, a gray image lena with 256 gray levels and 256 × 256 sizes is used as an image to be encrypted.
Step 1: generating transform matrix parameters m and q of Arnold transform by using a random sequence generated by the hyper-chaos Lorenz and the Logistic of the one-dimensional chaotic system, improving the Arnold transform, scrambling and encrypting images and enhancing scrambling effect, wherein the process is as follows:
1-1, performing discretization treatment on the hyperchaotic Lorenz system;
the hyper-chaotic Lorenz system formula is as follows:
Figure BDA0002587628210000051
wherein, a is 10, b is 8/3, c is 28, x is system parameteri,yi,zi,viRespectively representing the differentiation of the system variable with respect to time t;
step 1-2, solving the hyperchaotic Lorenz system equation by using a Runge Kutta method to obtain a result:
Figure BDA0002587628210000052
wherein x isi,yi,zi,viThe value of the system variable representing the ith iteration is h, the step length is h, and the value is set to be 0.002;
wherein the content of the first and second substances,
Figure BDA0002587628210000061
Figure BDA0002587628210000062
Figure BDA0002587628210000063
Figure BDA0002587628210000064
step 1-3, setting an input image matrix as I, wherein each pixel value range is 0-255, so that the image can be regarded as consisting of 8 bit planes, and an initial value x of the chaotic system can be obtained by the following formula0，y0，z0，w0The 4 th and 5 th bit planes, the 3 rd and 6 th bit planes, the 2 nd and 7 th bit planes, and the 1 st and 8 th bit planes of the graph respectively determine:
x0＝sum{sum[bitand(I,24)]}/(24×M×N)
y0＝sum{sum[bitand(I,36)]}/(36×M×N)
z0＝sum{sum[bitand(I,66)]}/(66×M×N)
w0＝sum{sum[bitand(I,129)]}/(129×M×N)
according to the formula, when the images I are different, the initial value x of the hyper-chaos Lorenz system is obtained0，y0，z0，w0Also different;
step 1-4, the chaotic initial value x is processed0，y0，z0，v0Substituting the hyper-chaos Lorenz system for iteration to generate a chaos pseudorandom sequence { x with the length of 2MNi}；
Step 1-5, discard { xiLeading the hyper-chaos Lorenz system to fully enter a chaos state by the first 200 values of the hyper-chaos Lorenz system;
steps 1-6, to generate sufficiently long { x }i3000 times after each iteration, for the initial value x0Plus small periodic perturbations, i.e. x0＝x0+hsiny0Then x is added0Substituting the hyper-chaos Lorenz for iteration;
step 1-7, mixing { xiAll conversion is of integer type Xi，XiE (1,2, …,10MN) and generate a matrix of size each M N as follows:
U＝reshape(X(1:M×N),M,N)；
step 1-8, performing initial value solving on the one-dimensional Logistic chaotic system, wherein a one-dimensional Logistic chaotic system expression is as follows: x is the number ofn＝μxn-1(1-xn-1)；
Wherein mu is a branch parameter and the initial value is x0；
Step 1 to 9, setting x0Expression:
Figure BDA0002587628210000071
wherein f (x, y) represents the gray value of a pixel located at a position of an image (x, y), the image selection is different, and the initial values of the one-dimensional Logistic chaotic system are also different.
Step 1-10, adding x0Substituting the mu into the one-dimensional Logistic chaotic system expression, and continuously iterating for 1000+ MxN times;
step 1-11, in order to obtain a better one-dimensional Logistic chaotic sequence, the first 1000 items are omitted to obtain a sequence { P }i}；
Step 1-12, sequence { P }iThe integer is transformed and a matrix of size M × N is generated according to:
V＝reshape(P,M,N)。
step 2: an image encryption step:
step 2-1, inputting a lena gray image to be encrypted with the size of M multiplied by N to obtain a two-dimensional image matrix h (x, y);
step 2-2, carrying out order p on the image h (x, y) row direction1By a first fractional Fourier transform of p1Set to 0.2, obtain the image h after the first transformation1(x,y)；
The fractional order fourier transform expression is:
Figure BDA0002587628210000072
wherein the content of the first and second substances,
Figure BDA0002587628210000073
a transform kernel, which is a fractional fourier transform, is defined as:
Figure BDA0002587628210000081
in the formula, p1，p2In order to transform the order of the order,
Figure BDA0002587628210000082
is the angle of rotation.
Step 2-3, implementing image h by applying the improved Arnold transformation1(x, y) wherein the Arnold transform expression is:
Figure BDA0002587628210000083
wherein, (x ', y') is the coordinate of the pixel point (x, y) after transformation; a is a transformation matrix of the Arnold transformation; mod is the modulo operation; l is the image order; taking random sequences generated by the hyper-chaos Lorenz and the one-dimensional chaotic system Logistic as transformation matrix parameters m and q of Arnold transformation, and obtaining the improved Arnold transformation operation expression:
Figure BDA0002587628210000084
wherein the value of U (i, j) is m and the value of V (i, j) is q;
Figure BDA0002587628210000085
for the new coordinate vector calculated by transformation, the image h is then1(x, y) the pixel at the coordinate vector and h1Replacing the pixels at the (i, j) positions, and continuously and circularly replacing to obtain a scrambled image h2(x,y)；
Step 2-4, to the image h2The order of progression in the (x, y) column direction is p2Fractional Fourier transform of p2Set to 0.6, resulting in the final encrypted image f (x, y).
And 3, an image decryption process, which comprises the following specific steps:
step 3-1, the image f (x, y) row direction is steppedA number of-p2To obtain a primary inverse transform image f1(x,y)；
The two-dimensional fractional order inverse fourier transform equation is as follows:
Figure BDA0002587628210000086
wherein the content of the first and second substances,
Figure BDA0002587628210000087
the kernel of the two-dimensional fractional order inverse Fourier transform is expressed as follows:
Figure BDA0002587628210000088
in the formula, p1，p2In order to transform the order of the order,
Figure BDA0002587628210000089
is a rotation angle;
step 3-2, scanning the image f according to the sequence from right to left and from bottom to top by the improved Arnold transformation matrix in the encryption process1(x, y) is decrypted to obtain an image f after reverse scrambling2(x,y)；
Step 3-3, to the image f2The order of progression in the (x, y) row direction is-p1The second fractional order inverse Fourier transform to obtain the final decrypted image f3(x,y)。
The image h (x, y) of fig. 2(a) is scrambled once using the conventional Arnold transform and the improved Arnold transform, respectively, and by comparing the effect of the scrambling degree of the images 2(b) and (c), it can be found that the improved Arnold transform can achieve a good scrambling effect only once for the image.
The image has the great characteristic of high correlation between adjacent pixels, the gray values of a plurality of pixels in a limited area are almost the same, the horizontal, vertical and diagonal adjacent pixels of the plaintext image have high correlation, and the ciphertext is opposite to the horizontal, vertical and diagonal adjacent pixels. The image neighboring pixel correlation is analyzed as follows:
and (3) randomly taking N pairs of adjacent pixel points, and recording the gray values of the N pairs of adjacent pixel points as:
(ui,vi),i＝1,2,3……,N
the correlation coefficient expression is:
Figure BDA0002587628210000091
cov (u, v) in the correlation coefficient expression is:
Figure BDA0002587628210000092
in the correlation coefficient expression, D (u) is:
Figure BDA0002587628210000093
d (u) wherein E (u) is:
Figure BDA0002587628210000094
from the simulation results, the correlation coefficient of the neighboring pixels of the image obtained by scrambling the conventional Arnold transform 8 times and scrambling the image obtained by scrambling the conventional Arnold transform 1 time can be calculated, as shown in Table 1:
table 1: scrambling of image neighboring pixel correlation coefficient tables by conventional Arnold and improved Arnold transforms
Figure BDA0002587628210000095
As shown in table 1, the correlation coefficient of the plaintext image is close to 1, and the correlation coefficient of the ciphertext image is close to 0. The calculation result shows that the improved Arnold transform encryption method has better adjacent pixel correlation effect when scrambling the image for only one time compared with the traditional Arnold transform scrambling for multiple times, and meanwhile, a simulation experiment can be used for making a point diagram of the adjacent pixel correlation of the improved Arnold transform plaintext and the ciphertext image, and the result is shown in FIG. 3, and the improved Arnold transform can be found to effectively destroy the adjacent pixel correlation of the plaintext image.
The improved Arnold transform image scrambling method only changes the image pixel position but not the pixel value, and the introduction of fractional Fourier transform can well improve the problem.
The simulation result of the multi-chaos Arnold image encryption method based on fractional Fourier transform is shown in FIG. 4:
when the order of the fractional Fourier has 0.1 error, the simulation result figure 4(b) shows that the decrypted image can not be restored, the fractional Fourier order in the encryption method has higher key sensitivity, the size of a key space is improved, and the encryption method has good safety.

Claims (4)

1. The multi-chaos Arnold image encryption method based on fractional Fourier transform is characterized by comprising the following steps of:
step 1, generating a random sequence by a hyper-chaos Lorenz and a one-dimensional chaotic system Logistic to generate parameters of a transformation matrix of Arnold transformation for improving the Arnold transformation;
step 2, encrypting the image:
step 2-1, inputting a gray image to be encrypted with the size of M multiplied by N to obtain a two-dimensional image matrix h (x, y);
step 2-2, carrying out order p on the image h (x, y) row direction1Obtaining an image h after the primary transformation by the primary fractional Fourier transform1(x,y)；
Step 2-3, implementing image h by applying the improved Arnold transformation1Scrambling and encrypting (x, y) to obtain a scrambled image h2(x,y)；
Step 2-4, to the image h2The order of progression in the (x, y) column direction is p2Obtaining a final encrypted image f (x, y) by fractional Fourier transform;
and 3, image decryption, namely obtaining a decrypted image by the inverse process of the image encryption process.
2. The multi-chaos Arnold image encryption method based on fractional Fourier transform as claimed in claim 1, wherein a transformation matrix parameter m of the Arnold transform is generated by the hyper-chaos Lorenz system,
discretizing the hyperchaotic Lorenz system to obtain an expression:
Figure FDA0002587628200000011
wherein, a is 10, b is 8/3, c is 28, x is system parameteri,yi,zi,viRespectively expressing the differential of system variables to time t, and solving a hyperchaotic Lorenz system equation by using a Runge Kutta method to obtain a result:
Figure FDA0002587628200000012
Figure FDA0002587628200000021
Figure FDA0002587628200000022
Figure FDA0002587628200000023
Figure FDA0002587628200000024
wherein x isi,yi,zi,viRepresenting the value of a system variable of the ith iteration, wherein h is a step length;
setting an input image matrix as I, wherein the image can be regarded as 8 bit planes, and the initial value x of the hyperchaotic Lorenz system can be obtained by the formula0，y0，z0，w0The 4 th and 5 th bit planes, the 3 rd and 6 th bit planes, the 2 nd and 7 th bit planes, and the 1 st and 8 th bit planes of the graph respectively determine:
x0＝sum{sum[bitand(I,24)]}/(24×M×N)
y0＝sum{sum[bitand(I,36)]}/(36×M×N)
z0＝sum{sum[bitand(I,66)]}/(66×M×N)
w0＝sum{sum[bitand(I,129)]}/(129×M×N)
wherein sum represents a summation operation, and bitand represents a bit and operation; according to the formula, when the images I are different, the initial value x is chaotic0，y0，z0，w0Also different, x0，y0，z0，w0Substituting the chaos initial value into the hyper-chaos Lorenz system for iteration to generate a chaos random sequence { x with the length of 2MNi}; discard { xiLeading the hyper-chaotic Lorenz system to fully enter a chaotic state by the first 200 values of the random sequence; in order to generate a long enough chaotic sequence by the hyper-chaotic Lorenz system, after 3000 times of iteration, an initial value x is calculated0Plus small periodic perturbations, i.e. x0＝x0+hsiny0Then x is added0Substituting the hyper-chaos Lorenz for iteration; will { xiAll conversion is of integer type Xi，XiE (1,2, …,10MN) and generate a matrix of size M N:
U＝reshape(X(1:M×N),M,N)
where m is the value at coordinate (i, j) of matrix U.
3. The multi-chaos Arnold image encryption method based on fractional order Fourier transform as claimed in claim 1, wherein the transformation matrix parameter q of Arnold transform is generated by the one-dimensional chaotic system Logistic, and the initial value solution is performed, wherein the one-dimensional Logistic chaotic system expression is:
xn＝μxn-1(1-xn-1)
wherein mu is a branch parameter, and an initial value is set as x0The value is given by the following relationship:
Figure FDA0002587628200000031
wherein f (x, y) represents the gray value of the pixel at the position of the image (x, y), and the initial value of the one-dimensional chaotic system Logistic obtained by different images is different according to the formula; x is to be0Substituting mu into the Logistic expression of the one-dimensional chaotic system, and continuously iterating for 1000+ MxN times; in order to obtain a better Logistic chaotic sequence of the one-dimensional chaotic system, the first 1000 items of the sequence generated by the Logistic of the one-dimensional chaotic system are omitted to obtain a sequence { PiH, will sequence { P }iThe integer is transformed and a matrix of size M × N is generated according to:
V＝reshape(P,M,N)
where q is the value at coordinate (i, j) of matrix V.
4. The multi-chaos Arnold image encryption method based on fractional Fourier transform as claimed in claim 1, wherein the random sequence generated by the hyper-chaos Lorenz and the one-dimensional chaos system Logistic is used to improve Arnold transform, and the Arnold transform expression is as follows:
Figure FDA0002587628200000032
wherein, (x ', y') is the coordinate of the pixel point (x, y) after transformation; a is a transformation matrix of the Arnold transformation; mod is the modulo operation; l is the image order; taking random sequences generated by the hyper-chaos Lorenz and the Logistic of a one-dimensional chaotic system as transformation matrix parameters m and q of Arnold transformation, and obtaining the improved Arnold transformation operation expression:
Figure FDA0002587628200000041
wherein the value of U (i, j) is m and the value of V (i, j) is q;
Figure FDA0002587628200000042
for the new coordinate vector calculated by transformation, the image h to be scrambled is then1(x, y) pixel value at new coordinate vector and h1Replacing the pixels at the (i, j) positions, and continuously circularly replacing to obtain a scrambled encrypted image h2(x,y)。
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Cited By (4)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN112837203A (en) * 2021-04-12 2021-05-25 中国矿业大学 Chaos and multi-azimuth based multi-image encryption method
CN113726975A (en) * 2021-08-11 2021-11-30 国网河北省电力有限公司保定供电分公司 Image encryption method, device, medium and electronic equipment based on chaotic system
CN114679250A (en) * 2021-12-17 2022-06-28 中国人民解放军海军工程大学 Image encryption algorithm based on mixed chaos and Arnold transformation
CN116094690A (en) * 2023-04-07 2023-05-09 中铁工程设计咨询集团有限公司 Geometric figure digital asset encryption method, device, equipment and readable storage medium

Citations (4)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US20110311042A1 (en) * 2008-10-23 2011-12-22 University Of Ulster Encryption method
CN103716502A (en) * 2013-12-25 2014-04-09 西安理工大学 Encryption method based on iteration FrFT and chaotic single-channel color images
CN105447396A (en) * 2015-12-15 2016-03-30 西安电子科技大学 Fractional domain image encryption method based on Arnold transformation and compound chaos
CN108270944A (en) * 2018-01-02 2018-07-10 北京邮电大学 A kind of method and device of the digital image encryption based on fractional order transformation

Patent Citations (4)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US20110311042A1 (en) * 2008-10-23 2011-12-22 University Of Ulster Encryption method
CN103716502A (en) * 2013-12-25 2014-04-09 西安理工大学 Encryption method based on iteration FrFT and chaotic single-channel color images
CN105447396A (en) * 2015-12-15 2016-03-30 西安电子科技大学 Fractional domain image encryption method based on Arnold transformation and compound chaos
CN108270944A (en) * 2018-01-02 2018-07-10 北京邮电大学 A kind of method and device of the digital image encryption based on fractional order transformation

Cited By (8)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN112837203A (en) * 2021-04-12 2021-05-25 中国矿业大学 Chaos and multi-azimuth based multi-image encryption method
CN112837203B (en) * 2021-04-12 2023-11-24 中国矿业大学 Multi-image encryption method based on chaos and multi-azimuth
CN113726975A (en) * 2021-08-11 2021-11-30 国网河北省电力有限公司保定供电分公司 Image encryption method, device, medium and electronic equipment based on chaotic system
CN113726975B (en) * 2021-08-11 2024-02-09 国网河北省电力有限公司保定供电分公司 Image encryption method and device based on chaotic system, medium and electronic equipment
CN114679250A (en) * 2021-12-17 2022-06-28 中国人民解放军海军工程大学 Image encryption algorithm based on mixed chaos and Arnold transformation
CN114679250B (en) * 2021-12-17 2024-04-16 中国人民解放军海军工程大学 Image encryption algorithm based on mixed chaos and Arnold transformation
CN116094690A (en) * 2023-04-07 2023-05-09 中铁工程设计咨询集团有限公司 Geometric figure digital asset encryption method, device, equipment and readable storage medium
CN116094690B (en) * 2023-04-07 2023-06-06 中铁工程设计咨询集团有限公司 Geometric figure digital asset encryption method, device, equipment and readable storage medium

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