CN110598430B - Image encryption method based on improved logistic mapping - Google Patents
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Abstract
The invention relates to the technical field of chaotic systems and information security, which has effectiveness and feasibility in the aspects of shearing resistance, key sensitivity, histogram distribution, information entropy, correlation between adjacent coefficients and the like, and is applied to data encryption. 1) Encryption section: firstly, respectively scrambling pixel positions of an image to be encrypted by using a first pair of random sequences generated by using Logistic and improved Logistic mapping; then performing exclusive OR operation to obtain a final encrypted image; 2) Decryption section: firstly, performing exclusive OR operation on the second pair of random sequences and the encrypted image respectively, and recovering an initial pixel value; and then restoring the pixel positions of the original image according to the conversion sequence of the pixel positions in the first encryption process. The invention is mainly applied to information encryption occasions.
Description
Technical Field
The invention relates to the technical field of chaotic systems and information security, in particular to an improvement scheme aiming at the problem of uneven distribution of traditional Logistic mapping.
Background
With the development of computer and communication technologies, particularly the wide application of the Internet, the security and confidentiality of multimedia information has become more and more important. Among multimedia information, image information has become one of the important means for human expression of information because of its vivid image, and how to protect digital images from tampering, illegal copying and transmission has important practical significance.
The chaotic system provides a good path for image information encryption due to the characteristics of pseudo-randomness, uncertainty, unpredictability and extreme sensitivity to control parameters and initial conditions. Encryption systems that use a chaotic sequence to construct images appear sequentially. However, with the continuous perfection and innovation of encryption systems, it has been found that random sequences generated by conventional Logistic chaotic mapping are not uniformly distributed. Whether the simple image scrambling result based on Logistic mapping meets the uniform distribution required by the encryption system is not well verified.
Disclosure of Invention
To overcome the deficiencies of the prior art, the present invention is directed to an improved Logistic mapping scheme. The random sequence generated by the improved Logistic map not only satisfies the uniform distribution, but also has a larger key space than the traditional Logistic map. And then, combining a simple image pixel position scrambling and pixel value replacing image encryption algorithm, further verifying the effectiveness and feasibility of the improved Logistic mapping in the aspects of shearing resistance, key sensitivity, histogram distribution, information entropy, correlation between adjacent coefficients and the like. Therefore, the technical scheme adopted by the invention is that the image encryption method based on improved Logistic mapping comprises the following steps:
1) Encryption section: firstly, scrambling pixel positions of an image to be encrypted respectively by utilizing a first pair of random sequences generated by a Logistic map which does not meet uniform distribution and a Logistic map which meets uniform distribution, so as to realize first encryption; then, performing exclusive OR operation on the generated second pair of random sequences and the pixel values of the image after the first encryption respectively to realize the replacement of the pixel values and obtain a final encrypted image;
2) Decryption section: firstly, performing exclusive OR operation on the second pair of random sequences and the encrypted image respectively, and recovering an initial pixel value; and then restoring the pixel positions of the original image according to the conversion sequence of the pixel positions in the first encryption process, thus finishing the decryption process.
The iterative form of the Logistic mapping that does not satisfy the uniform distribution is as follows:
x n+1 ＝μx n (1-x n )
wherein x is n Is the chaos sequence value after n times of iteration, when the control parameter 3.5699 is more than mu and less than or equal to 4, the input initial value x 0 ∈[0,1]The Logistic map is in a chaotic state;
the Logistic mapping iteration form satisfying the uniform distribution is as follows:
x n+1 ＝(ux n+1 (1-x n ))mod1
y n+1 ＝rx n
wherein x is n+1 ，y n+1 The chaotic sequence values are respectively iterated n+1 times, and u and r are control parameters.
The specific steps are refined as follows:
(1) Encryption portion of image:
1) Firstly, scrambling pixel positions of an image I to be encrypted by utilizing a first pair of random sequences seq11 and seq12 generated by using a Logistic map which does not meet the uniform distribution and a Logistic map which meets the uniform distribution respectively;
wherein I (I, j) is the pixel value of the original image at (I, j), seq11, seq12 is a line number with the length of the original image, and the value range is an integer sequence of [0,255 ];
2) Then, the generated second pair of random sequences seq21 and seq22 are respectively exclusive-ored with the pixel values of the first encrypted image to realize the replacement of the pixel values, so as to obtain final encrypted images I1 and I2:
in the method, in the process of the invention,is an exclusive-or operation, seq21, seq22 is a size equal to the original image, and the value range is [0,255]]Is an integer matrix of (a);
(2) Decryption portion of image:
1) Firstly, performing exclusive OR operation on the generated second pair of random sequences seq21 and seq22 and the final encrypted images I1 and I2 respectively to restore initial pixel values;
2) Then, the pixel positions of the original image are restored according to the conversion order of the pixel positions in the first encryption process, and a decrypted image reI is obtained:
in the formula, reI is a decrypted image of two encryption methods.
The invention has the characteristics and beneficial effects that:
an improved Logistic chaotic mapping scheme is provided, so that the generated random sequence presents a uniform distribution state, and the randomness of the chaotic sequence is enhanced. Meanwhile, compared with the traditional Logistic mapping, the key space of the chaotic mapping in the scheme provided by the invention is greatly increased. Finally, through experiments, the effectiveness and feasibility of the improved Logistic mapping in the aspects of shearing resistance, key sensitivity, histogram distribution, information entropy, correlation between adjacent coefficients and the like are further verified by using a simple chaotic mapping-based image encryption algorithm.
Description of the drawings:
FIG. 1 is a schematic block diagram of the encryption scheme of the present invention.
Fig. 2 is a graph comparing the index profile.
FIG. 2 (a) is a Lyapunov exponent profile of a conventional Logistic map;
FIGS. 2 (b) - (d) are Lyapunov exponent profiles for improved Logistic mapping.
Fig. 3 is a distribution histogram contrast chart.
FIG. 3 (a) is a distribution histogram of a chaotic sequence generated by a conventional Logistic map;
fig. 3 (b) is a distribution histogram of the chaotic sequence generated by the improved Logistic map.
Fig. 4 is a diagram showing the comparison of the images to be encrypted, the conventional encryption and the encryption of the invention.
Fig. 4 (a) is an image to be encrypted;
FIG. 4 (b) is an encrypted image based on a conventional Logistic map (first scheme);
fig. 4 (c) is an encrypted image based on an improved Logistic mapping (second scheme).
Fig. 5 is a similarity comparison diagram of the decrypted image and the original image.
Fig. 5 (a) shows the value of the shear control coefficient k=d1/d;
fig. 5 (b) is a graph comparing the similarity of the decrypted image of the first scheme and the second scheme with the original image by the cut attack.
Fig. 6 is a gray histogram contrast chart.
Fig. 6 (a) is a gray level histogram of an original image;
fig. 6 (b) is a gray level histogram of the encrypted image of the first scheme;
fig. 6 (c) is a gray level histogram of the encrypted image of the second scheme.
Detailed Description
In order to overcome the defect that the random sequence generated by the traditional Logistic chaotic mapping does not meet the uniform distribution, the invention provides an improved Logistic mapping scheme. The random sequence generated by the improved Logistic map not only satisfies the uniform distribution, but also has a larger key space than the traditional Logistic map. And then, combining a simple image pixel position scrambling and pixel value replacing image encryption algorithm, further verifying the effectiveness and feasibility of the improved Logistic mapping in the aspects of shearing resistance, key sensitivity, histogram distribution, information entropy, correlation between adjacent coefficients and the like. The method comprises the following steps:
1) Encryption section: firstly, scrambling pixel positions of an image to be encrypted respectively by using a first pair of random sequences generated by the traditional Logistic mapping and the improved Logistic mapping, so as to realize first encryption; and then performing exclusive OR operation on the generated second pair of random sequences and the pixel values of the image after the first encryption respectively to realize the replacement of the pixel values and obtain a final encrypted image.
2) Decryption section: firstly, performing exclusive OR operation on the second pair of random sequences and the encrypted image respectively, and recovering an initial pixel value; and then restoring the pixel positions of the original image according to the conversion sequence of the pixel positions in the first encryption process, thus finishing the decryption process.
In one example, the specific steps are as follows:
(1) Encryption portion of image:
1) Firstly, scrambling pixel positions of an image I to be encrypted by using a first pair of random sequences seq11 and seq12 generated by a traditional Logistic map and an improved Logistic map respectively;
in the formula, seq11, seq12 is an integer sequence with the length of the line number of the original image and the value range of [0,255 ].
2) Then, the generated second pair of random sequences seq21 and seq22 are respectively exclusive-ored with the pixel values of the first encrypted image, so that the pixel values are replaced, and the final encrypted images I1 and I2 are obtained.
In the method, in the process of the invention,is an exclusive-or operation, seq21, seq22 is a size and original graphThe value range is 0,255]Is a matrix of integers.
(2) Decryption portion of image:
1) Firstly, performing exclusive OR operation on the generated second pair of random sequences seq21 and seq22 and the final encrypted images I1 and I2 respectively to restore initial pixel values;
2) Then, the pixel positions of the original image are restored according to the conversion order of the pixel positions in the first encryption process, and the decrypted image reI can be obtained.
Aiming at the defect that the random sequence generated by the traditional Logistic mapping does not meet the uniform distribution, the invention provides an improved Logistic mapping scheme. The random sequence generated by the improved Logistic map not only satisfies the uniform distribution, but also has a larger key space than the traditional Logistic map. And then, a simple image scrambling algorithm is designed by combining the scheme provided by the invention with the traditional Logistic mapping, and the effectiveness and feasibility of the scheme are further verified. The details are described below:
lyapunov index
Lyapunov exponent describes an important index of the chaos phenomenon, and the calculation formula is as follows:
wherein x is k Is a chaotic sequence. When the number of the pins is small,
λ < 0, the mapping converging at a certain point;
λ=0, the mapping performs a periodic motion;
lambda > 0, the mapping is in a chaotic state.
Logistic mapping
Logistic mapping is a simple and widely used nonlinear dynamics system. The iterative form is as follows:
x n+1 ＝μx n (1-x n )
wherein x is n Is the chaos sequence value after n times of iteration. When the control parameter 3.5699 is smaller than mu and smaller than or equal to 4, the input initial value x 0 ∈[0,1]The Logistic map is in a chaotic state. Fig. 3 (a) is a distribution histogram of a random sequence generated for a Logistic map.
Improved Logistic mapping
x n+1 ＝(ux n+1 (1-x n ))mod1
y n+1 ＝rx n
Wherein x is n+1 ，y n+1 Respectively, the chaotic sequence values of n+1 iterations. The improved Logistic map is in a chaotic state when the control parameters u and r satisfy the relationship in fig. 2 (b) calculated according to equation (1). Fig. 3 (b) is a distribution histogram of a random sequence generated for the improved Logistic map.
In order to make the purpose and the technical advantages of the present invention clearer, a simple image scrambling algorithm is designed by combining the scheme provided by the present invention with the traditional Logistic mapping, and is further described in detail.
Example 1
A simple image pixel position scrambling and pixel value replacing image encryption method has the encryption principle shown in figure 1. The method consists of an encrypted portion and a decrypted portion of the image.
(1) Encryption portion of image:
in the encryption process, firstly, a first pair of random sequences generated by the traditional Logistic mapping and the improved Logistic mapping are used for respectively scrambling the pixel positions of an image to be encrypted, so that first encryption is realized; and then performing exclusive OR operation on the generated second pair of random sequences and the pixel values of the image after the first encryption respectively to realize the replacement of the pixel values and obtain a final encrypted image.
(2) Decryption portion of image:
in the decryption process, firstly, performing exclusive OR operation on the second pair of random sequences and the encrypted image respectively, and recovering an initial pixel value; and then restoring the pixel positions of the original image according to the conversion sequence of the pixel positions in the first encryption process, thus finishing the decryption process.
In summary, the present invention combines the improved Logistic mapping scheme with the traditional Logistic mapping scheme to design a simple image scrambling algorithm. The method not only solves the defect that the random sequence generated by the traditional Logistic mapping does not meet the uniform distribution, but also has larger key space compared with the traditional Logistic mapping. Meanwhile, the validity and feasibility of the scheme are verified in detail in aspects of key sensitivity, histogram distribution of the encrypted image, information entropy, correlation between adjacent pixels of the encrypted image and the like.
Example 2
The scheme of embodiment 1 is described in detail below in conjunction with the design principle of fig. 1, and is described in detail below:
in order to verify the validity and feasibility of the improved Logistic mapping scheme in detail, a simple image scrambling algorithm is designed by combining the improved Logistic mapping scheme with the traditional Logistic mapping scheme. The encryption principle block diagram is shown in fig. 1, and the encryption method consists of an encryption part of an image and a decryption part of the image. The following detailed description of the two embodiments is provided.
(1) Encryption portion of image:
1) Firstly, scrambling pixel positions of an image I to be encrypted by using a first pair of random sequences seq11 and seq12 generated by a traditional Logistic map and an improved Logistic map respectively;
where seq11, seq12 is a number of lines of length of the original image, and the range of values is an integer sequence of [0,255 ].
2) Then, the generated second pair of random sequences seq21 and seq22 are respectively exclusive-ored with the pixel values of the first encrypted image, so that the pixel values are replaced, and the final encrypted images I1 and I2 are obtained.
In the method, in the process of the invention,is an exclusive-or operation, seq21, seq22 is a size equal to the original image, and the value range is [0,255]]Is a matrix of integers.
(2) Decryption portion of image:
1) Firstly, performing exclusive OR operation on the generated second pair of random sequences seq21 and seq22 and the final encrypted images I1 and I2 respectively to restore initial pixel values;
2) Then, the pixel positions of the original image are restored according to the conversion order of the pixel positions in the first encryption process, and the decrypted image reI can be obtained.
Example 3
The feasibility of the solutions of examples 1 and 2 is verified in the following description, in conjunction with the specific figures, given in detail below:
the Lyapunov index profiles of the conventional Logistic map and the improved Logistic map proposed in the present invention are shown in FIGS. 2 (a) and 2 (b) - (d).
Fig. 3 (a) and 3 (b) are distribution histograms of random sequences generated for a conventional Logistic map and a modified Logistic map, respectively. The random sequences generated by the improved Logistic map satisfy a uniform distribution compared to the traditional Logistic map.
After the original image (as shown in fig. 4 (a)) is encrypted by adopting the first scheme and the second scheme in the present invention, the obtained encrypted image is shown in fig. 4 (b) and fig. 4 (c).
As can be seen from fig. 5 (b), the second solution is significantly stronger in terms of shear attack resistance than the first solution.
Fig. 6 (a) - (c) are gray-scale histogram distributions of the original image, the first-scheme encrypted image, and the second-scheme encrypted image, respectively. It follows that the second scheme is more resistant to statistical attacks than the first scheme.
Table 1 shows the key space of the first scheme and the second scheme and the sensitivity of the two schemes to keys was quantitatively analyzed. The second scheme is illustrated to have a larger key space and higher key sensitivity than the first scheme.
Table 2 is the information entropy of the original image and the encrypted images of the two encryption schemes. It can be seen that the information entropy of the encrypted image of the second scheme is closer to 8 than that of the first scheme.
Table 3 shows the correlation between adjacent coefficients of the encrypted image in three directions in two schemes. It can be seen that the second scheme has a lower correlation between adjacent coefficients in three directions than the encrypted image of the first scheme.
TABLE 1
TABLE 2
TABLE 3 Table 3
It can be seen that the random sequence generated by the improved Logistic mapping proposed in the present invention not only satisfies the uniform distribution, but also has a larger key space than the conventional Logistic mapping. Moreover, the effectiveness and feasibility of the improved Logistic mapping in the aspects of shearing resistance, key sensitivity, histogram distribution, information entropy, correlation between adjacent coefficients and the like are verified by combining a simple image pixel position scrambling and pixel value substitution image encryption algorithm. Meets various requirements in practical application.
Those skilled in the art will appreciate that the drawings are schematic representations of only one preferred embodiment, and that the above-described embodiment numbers are merely for illustration purposes and do not represent advantages or disadvantages of the embodiments.
The foregoing is only illustrative of the present invention and is not to be construed as limiting thereof, but rather, any modifications, equivalent arrangements, improvements, etc. that fall within the spirit and principles of the present invention are intended to be included within the scope of the present invention.
Claims (2)
1. An image encryption method based on improved Logistic mapping is characterized by comprising the following steps:
1) Encryption section: firstly, scrambling pixel positions of an image to be encrypted respectively by utilizing a first pair of random sequences generated by a Logistic map which does not meet uniform distribution and a Logistic map which meets uniform distribution, so as to realize first encryption; then, performing exclusive OR operation on the generated second pair of random sequences and the pixel values of the image after the first encryption respectively to realize the replacement of the pixel values and obtain a final encrypted image;
2) Decryption section: firstly, performing exclusive OR operation on the second pair of random sequences and the encrypted image respectively, and recovering an initial pixel value; then, according to the conversion sequence of the pixel positions in the first encryption process, the pixel positions of the original image are restored, and the decryption process can be completed;
the iterative form of the Logistic mapping that does not satisfy the uniform distribution is as follows:
x n+1 ＝μx n (1-x n )
wherein x is n Is the chaos sequence value after n times of iteration, when the control parameter 3.5699 is more than mu and less than or equal to 4, the input initial value x 0 ∈[0,1]The Logistic map is in a chaotic state;
the Logistic mapping iteration form satisfying the uniform distribution is as follows:
x n+1 ＝(ux n+1 (1-x n ))mod1
y n+1 ＝rx n
wherein x is n+1 ，y n+1 The chaotic sequence values are respectively iterated n+1 times, and u and r are control parameters.
2. The improved Logistic mapping-based image encryption method according to claim 1, wherein the specific steps are refined as follows:
(1) Encryption portion of image:
1) Firstly, scrambling pixel positions of an image I to be encrypted by utilizing a first pair of random sequences seq11 and seq12 generated by using a Logistic map which does not meet the uniform distribution and a Logistic map which meets the uniform distribution respectively;
wherein I (I, j) is the pixel value of the original image at (I, j), seq11, seq12 is a line number with the length of the original image, and the value range is an integer sequence of [0,255 ];
2) Then, the generated second pair of random sequences seq21 and seq22 are respectively exclusive-ored with the pixel values of the first encrypted image to realize the replacement of the pixel values, so as to obtain final encrypted images I1 and I2:
in the method, in the process of the invention,is an exclusive-or operation, seq21, seq22 is a size equal to the original image, and the value range is [0,255]]Is an integer matrix of (a);
(2) Decryption portion of image:
1) Firstly, performing exclusive OR operation on the generated second pair of random sequences seq21 and seq22 and the final encrypted images I1 and I2 respectively to restore initial pixel values;
2) Then, the pixel positions of the original image are restored according to the conversion order of the pixel positions in the first encryption process, and a decrypted image reI is obtained:
in the formula, reI is a decrypted image of two encryption methods.
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| US11734435B2 (en) * | 2020-10-16 | 2023-08-22 | Qilu University Of Technology | Image encryption and decryption communication algorithm based on two-dimensional lag complex logistic map |
| CN112258520B (en) * | 2020-10-16 | 2022-03-29 | 齐鲁工业大学 | Image encryption and decryption communication method based on two-dimensional hysteresis complex Logistic mapping |
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