Wavelet transform image encryption method based on henon mapping

Abstract

This invention discloses an image encryption method based on Henon mapping and wavelet transform, belonging to the field of information security technology. Addressing the problem that traditional scrambling-diffusion encryption structures require a large number of pseudo-random numbers and suffer from low security, this invention proposes an image encryption method based on Henon mapping and wavelet transform. The method employs a Henon chaotic system to generate a pseudo-random sequence, which is then used to generate a keystream through S-box operations. Simultaneously, the original image is transformed into a one-dimensional sequence, further divided into odd-indexed and even-indexed sequences. The encrypted image is then generated using prediction and update algorithms within the wavelet transform. Theoretical analysis and experimental results demonstrate that the algorithm of this invention has high security, resists statistical attacks, and possesses broad application prospects and practical value.

Description

Technical Field

[0001] This invention belongs to the field of information security technology, specifically relating to a wavelet transform image encryption method based on Henon mapping. Background Technology

[0002] In this era, with the continuous development and widespread acceptance of computer technology and digital networks, digitized information has become an indispensable core issue in many fields, playing an increasingly irreplaceable role. Therefore, security has become a key focus for experts. Due to the limitations of computer science and technology and computer networks, the transmission of information over the network carries inherent risks, threatening information security in various fields. [1] Because information is expressed in diverse forms, and image information, with its strong intuitiveness and comprehensiveness, has become an important method for the public to express information. Digital images, due to their inherent characteristics such as large data volume, strong correlation between adjacent pixels, high redundancy, and blockiness, are subject to traditional encryption methods such as DES and AES. [2] This approach is no longer suitable for digital image encryption. Furthermore, the security of digital images has begun to attract widespread attention, and with the advancement of technology, many new encryption methods have gradually emerged.

[0003] Chaos is not a linear phenomenon, but rather a revelation of an internal randomization process within a deterministic system. Because the pseudo-randomness and sensitivity to initial states of chaotic systems meet the requirements of cryptography, chaotic cryptography has attracted considerable attention from experts and scholars. In chaotic image encryption schemes, the initial value of the chaotic system is often used as the key for transmission; therefore, the sensitivity of chaos to initial values ​​corresponds to the key sensitivity in the encryption scheme. In image encryption, a commonly used structure is the scrambling-diffusion structure. The scrambling process mainly changes the original positions of image pixels to reduce the correlation between adjacent pixels. Common scrambling algorithms include row and column sorting scrambling. [3] Matrix transformation based on cat mapping [4] Diffusion, or diffusion, alters the pixel values ​​of an original image. Using a key, it distributes information from each pixel across the entire image to other pixels, resulting in a flat and uniform histogram of pixel values ​​in the encrypted image. This achieves the purpose of information hiding. Common diffusion algorithms include those based on cyclic shifting. [5] XOR operation between adjacent pixels [6]Wait a minute. It's clear that chaos is the core of the entire encryption method; therefore, when choosing an encryption scheme, a chaotic system should be prioritized. There are many classic chaotic systems, but some suffer from small key spaces due to their limited chaos range, making them vulnerable to attacks. Furthermore, many scholars, both domestically and internationally, have proposed numerous methods to crack classic scrambling-diffusion encryption structures, such as chosen-plaintext attacks. Dou et al. employed a chaotic image encryption algorithm based on DNA encoding. [7] However, its use of a single DNA encoding rule makes it difficult to resist chosen-plaintext attacks. Although Chen et al. used dynamic DNA encoding to address the above problem, it is still vulnerable to chosen-plaintext attacks. [8] Ping et al. proposed an image pixel position scrambling algorithm using Henon chaotic mapping. [9] However, it does not include a diffusion operation that changes pixel values, thus reducing system complexity. The algorithm proposed by Wu et al. is a typical scramble-diffusion algorithm.

[10] However, its key is independent of the plaintext, making it vulnerable to plaintext attacks. Although Belazi et al. improved the algorithm's plaintext sensitivity by using a key generated by a hash function.

[11] However, the encryption process only relies on XOR operations and cannot defend against chosen-plaintext attacks.

[0004] A new method for computing wavelet transforms was proposed by Sweldens in 1996.

[12] Inspired by wavelet transform, this invention does not employ the traditional scrambling diffusion structure. Instead, it designs an image encryption method based on Henon chaotic mapping combined with wavelet transform, which offers high security and is applicable to most current image data. Summary of the Invention

[0005] The purpose of this invention is to address the low security and high requirement for a large number of pseudo-random numbers in scrambled diffusion structures. It proposes a novel image encryption method based on Henon mapping combined with wavelet transform to improve image encryption security. The wavelet transform enhancement process mainly consists of three steps: splitting, prediction, and updating. Based on the odd and even index sequences obtained during the splitting process, the low-frequency approximate components and high-frequency detail components of the image are obtained through the prediction and update operations in the proposed wavelet transform. Each round of encryption in this system consists of wavelet transform, and the number of encryption rounds can be increased according to security requirements and encryption / decryption speed. Here, the number of rounds is set to two, with a different key for each round. A pseudo-random sequence with the key generated by the chaotic system is used to scramble the detail components of the image, thereby scrambling the approximate components. The two scrambled components are flipped, and the prediction and update operations are performed again to achieve the encryption effect. [13,14] .

[0006] The technical solution adopted by the present invention to solve the above-mentioned technical problems is: to design an image encryption method based on Henon mapping and wavelet transform.

[0007] This invention mainly involves two main modules: a pseudo-random sequence generation module and a wavelet transform encryption module.

[0008] 1. Henon chaotic systems and the generation of pseudo-random sequences

[0009] This invention uses the Henon map, a typical two-dimensional discrete chaotic map, whose equation is expressed as follows:

[0010]

[0011] Prasad et al. studied the influence of parameters a and b on the mapping. When a = 1.4 and b = 0.3, the system belongs to a hyperchaotic state.

[15] The bifurcation graph of the Henon map, with the Lyaponuv exponent as shown. Figure 1 and Figure 2 As shown, the Henon mapping determines the iterative equations simultaneously by two variables, x and y, which is more complex than simply solving two independent one-dimensional chaotic equations. Furthermore, compared to three-dimensional chaotic systems, the Henon mapping is discrete and can be viewed as consisting of three mappings. Additionally, the sequence does not encounter fixed points during the Henon mapping iteration process. Therefore, this paper utilizes the ease of implementation and complexity of the Henon mapping as a pseudo-random sequence generator for image encryption.

[0012] Image encryption systems typically require a large number of pseudo-random numbers, and the quality of these pseudo-random numbers often determines the quality of the encryption system. A pseudo-random sequence, approximately half the size of a normal image, is generated by a Henon chaotic system. Sequence J is generated using the S-box in the AES algorithm and serves as the keystream for the image encryption system. For a given key K, sequence J is uniquely determined.

[0013] 2. Wavelet Transform Design

[0014] Sweldens proposed a new method for calculating wavelet transforms. The main steps of this method are as follows (assuming that n represents the length of the sequence and is even):

[0015] Step 1: Splitting. In this process, sequence f is divided into odd-numbered sequences and even-numbered sequences, as shown below:

[0016]

[0017] Even-numbered sequences are used to store approximate information about the discrete signal f, while odd-numbered sequences are used to store detailed information about the discrete signal f.

[0018] Step 2: Prediction. The prediction process uses approximate and detailed information to calculate the detailed components of the signal. In this process, a new sequence is obtained as the detailed information for f. The specific operations are as follows:

[0019]

[0020] Step 3: Update. The update process updates the approximate components with detailed components, as follows:

[0021]

[0022] The obtained new sequence is an approximation of f.

[0023] In the enhancement scheme, the three steps of splitting, predicting, and updating are reversible. Image encryption can be achieved by replacing the methods in the prediction and updating steps.

[0024] 3. Experimental Results and Safety Analysis

[0025] The experimental results and safety analysis in this section serve as a demonstration of the actual effects of this invention. Through actual data analysis, the beneficial effects of this invention can be seen intuitively.

[0026] 3.1 The effect of encryption and decryption

[0027] Extensive testing was conducted on the image encryption algorithm proposed in this paper, yielding satisfactory results. We used plaintext images such as Lena, Female, and House with dimensions of [missing information]. Test results are as follows Figure 3 As shown. Figure 3 The `ac` directive displays the plaintext image, `df` represents the encrypted image, and `gi` displays the image after decryption of `df`. From... Figure 3 We can see that all encrypted images visually present a noise-like pattern, and all decrypted images are identical to the plaintext images, which indicates that the image encryption system can function normally.

[0028] 3.2 Security Analysis

[0029] To determine whether an encryption system is excellent, in addition to achieving good encryption results, it must also have the ability to resist various attacks. The following analysis will evaluate the security of the encryption algorithm proposed in this paper.

[0030] 3.2.1 Key Space

[0031] In Advanced Encryption Standard (AES), the key length is 256 bits, so the maximum key space for AES is 2^356 bits. 256In this invention, the key length is 512 bits, therefore the key space size is 2. 512 Therefore, the key space of the proposed encryption method is sufficient to resist attacks.

[0032] 3.2.2 Histogram Analysis

[0033] To resist statistical attacks, encrypted images need to maintain a certain degree of diffusion and perturbation, ensuring that attackers cannot deduce the key from the ciphertext image. Histogram analysis can be used to measure the perturbation of encryption algorithms. A histogram reflects the distribution of pixel values ​​in an image. Figure 4 a, c, and e represent Figure 4 Histograms of the plaintext image, b, d, and f show the histograms of the encrypted image, from... Figure 4 As can be seen, the histogram of the plaintext image has obvious fluctuations, while the histogram of the encrypted image is relatively flat and uniformly distributed, without retaining any plaintext information. This indicates that the encrypted image can resist attacks.

[0034] 3.2.3 Correlation Analysis

[0035] Correlation analysis uses statistical methods to analyze the diffusion of an algorithm. Generally, two adjacent pixels in a plaintext image have a high correlation, with a normal correlation coefficient usually above 0.9. However, when an image is encrypted, the redundancy of the plaintext image diffuses into the ciphertext, and the correlation coefficient of the ciphertext image should be very small. Correlation reflects the degree of correlation between the gray values ​​of adjacent pixels in an image, and is generally measured by the correlation coefficient r. The specific calculation formula is as follows:

[0036]

[0037] Where K represents the number of pairs of adjacent pixels randomly selected from an image, and uiui and vivi represent the gray values ​​of two adjacent pixels. When r = 1, it indicates that there is a strong correlation between adjacent pixels, and when r = 0, it indicates that there is no correlation between adjacent pixels. Statistical calculations were performed on 10,000 pairs of adjacent pixels randomly selected from the horizontal, vertical, and diagonal directions, respectively. The results are shown in the table below:

[0038] Table 1. Correlation coefficients of adjacent pixels between plaintext and encrypted images.

[0039]

[0040] The correlation coefficient between adjacent pixels in the encrypted image is much smaller than that before encryption, and is close to 0, indicating that the encryption algorithm proposed in this paper has good diffusion performance. To more intuitively analyze the correlation characteristics between the plaintext and ciphertext images, Figure 5 A phase map showing the correlation between adjacent pixels is presented. Figure 5 Taking the phase maps of horizontally adjacent pixels a and b as an example, the horizontal phase map of a corresponds to the Boat with a correlation coefficient r = 0.9711 in Table 1, and the horizontal phase map of b is equivalent to the encrypted Boat with a correlation coefficient r = 0.0651 in Table 1. Similarly, it can be observed that... Figure 5 In the graph, regardless of whether the pixels a, c, or e are horizontal, vertical, or diagonal, each pair of adjacent pixels is close to y=x, indicating a strong correlation between adjacent pixels. Figure 5 In the image, adjacent pixel pairs of pixels b, d, and f are evenly distributed throughout the phase space, regardless of whether they are horizontal, vertical, or diagonal, indicating that there is almost no correlation between adjacent pixels in the encrypted image.

[0041] 3.2.4 Information Entropy

[0042] Information entropy reflects the uniformity of the grayscale value distribution in an image. The more uniform the grayscale value distribution, the greater the information entropy. Theoretically, it is 8. The closer it is to 8, the better the encryption effect. The definition of information entropy is as follows:

[0043]

[0044] Where P(i) represents the probability of grayscale value i appearing.

[0045] This paper calculates the information entropy of the experimental images, and the specific results are shown in Table 2:

[0046] Table 2 Information Entropy of Plaintext and Encrypted Images

[0047]

[0048] As can be seen from the table, the encrypted image is closer to 8 than the plaintext image, indicating a good encryption effect.

[0049] To address the shortcomings of traditional scrambling-diffusion encryption methods, this invention proposes an image encryption algorithm based on Henon mapping and wavelet transform. It primarily utilizes Henon mapping to generate chaotic sequences, and then generates keys through operations such as cyclic shifting and S-boxing of the sequences. The algorithm focuses on three operations in the wavelet transform: splitting, prediction, and updating the sequence, achieving image encryption with good security. In practical applications, this algorithm can be extended to encrypt digital files such as color images and videos. Furthermore, the prediction and update operations can be iterated multiple times, and different keys can be added in each iteration to increase the algorithm's complexity. Further research will focus on developing more complex prediction and update functions to reduce the system size; this is also a future research objective. Attached image description:

[0050] Figure 1This is a bifurcation diagram of the Henon chaotic system used in this invention;

[0051] Figure 2 This is the Lyaponuv exponent diagram of the Henon chaotic system used in this invention;

[0052] Figure 3 These are the original image, encrypted image, and decrypted image of this invention; where (a)-(c) are plaintext images, (d)-(f) are encrypted images, and (g)-(i) are decrypted images;

[0053] Figure 4 These are histograms of the original image and the encrypted image of this invention; where (a), (c), and (e) are histograms of the plaintext image, and (b), (d), and (f) are histograms of the encrypted image.

[0054] Figure 5 This is a phase map showing the correlation between adjacent pixels of the original image and the encrypted image in this invention; where (a), (c), and (e) represent the correlation of the plaintext image in three directions, and (b), (d), and (f) represent the correlation of the encrypted image in three directions.

[0055] Figure 6 This is a flowchart of the encryption method of the present invention;

[0056] Figure 7 This is a flowchart of the decryption method of the present invention. Detailed Implementation

[0057] The invention will be further illustrated below with reference to examples. Figure 6 Figure 7 The flowchart of the image encryption and decryption process of the present invention is shown in the figure. As can be seen from the figure, in order to encrypt the image data, a pseudo-random sequence is used to scramble the details of the image data. Let the input be a plaintext image and the encrypted image be C.

[0058] A wavelet transform-based image encryption method based on Henon mapping is implemented in the following ten steps:

[0059] first step, The plaintext image is unfolded into a one-dimensional vector P of length MN and serialized. The key stream J is obtained from the chaotic system through S-box operations.

[0060] The second step is to divide the sequence P into an even-indexed sequence G and an odd-indexed sequence H. Here, we assume that MN is even, and the size and specific expression of L are as follows:

[0061] L = floor(MN+1) / 2 (7)

[0062] G i =P 2i Hi =P 2i+1 ,i=1,2,3,...,L (8)

[0063] The third step is the prediction process. Based on G and H, the first prediction is performed to obtain H_yc1. The specific generation method is as follows:

[0064] H_yc1 i =mod((H(i)+floor(0.5*(G(i)+G(i+1)))),256),i=1,2,3,...,L,G L+1 =G1 (9)

[0065] Fourth step: Perform an XOR operation between the key stream J and H_yc1 to obtain a new sequence V, calculated as follows:

[0066]

[0067] The fifth step is the update process. Based on V and G, the first update process is performed to obtain U_gx1, which is expressed as follows:

[0068] U_gx1 i ={G i +floor(0.001*(V i-1 +V i +2))}mod 256,i=1,2,3,...,L,V0=0 (11)

[0069] Step 6: Reverse the sequences U and V obtained above to obtain two new sequences G1 and H1 respectively. Now G1 is considered the detailed component, while H1 is the approximate component.

[0070] Step 7, the second prediction process, obtains G_yc2 based on G1 and H1. The specific generation method is as follows:

[0071]

[0072] Step 8: Perform an XOR operation between the key stream J and G_yc2 to obtain a new sequence U, calculated as follows:

[0073]

[0074] Step 9, the second update process, obtains V_gx2 based on H1 and U, specifically expressed as follows:

[0075] V_gx2 i ={H1 i +floor(0.001*(U i-1 +U i+2))}mod 256,i=1,2,3,...,L,U0=0 (14)

[0076] Step 10: Merge the generated U and V_gx2 into a single sequence, where U is the even-indexed sequence of sequence W, and V_gx2 is the odd-indexed sequence of sequence W. Then transform sequence W into a sequence of size... The matrix C is the final encrypted image.

[0077] In lifting wavelet transform, the three steps of splitting, prediction, and updating are reversible. Therefore, the decryption process is the inverse of the encryption process. The prediction process in the decryption process is exactly the same as the encryption process, and the updating process is the opposite of the encryption process.

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Claims

1. A wavelet transform-based image encryption method based on Henon mapping, which is implemented in the following ten steps: The first step is to unfold the M×N plaintext image into a one-dimensional vector P of length MN and serialize it. The key stream J is obtained from the chaotic system through S-box operations. The second step is to divide the sequence P into an even-indexed sequence G and an odd-indexed sequence H. Here, we assume that MN is even, and the size and specific expression of L are as follows: L = floor(MN+1) / 2 (1) G i =P 2i ,H i =P 2i+1 ,i=1,2,3,...,L (2) The third step is the prediction process. Based on G and H, the first prediction is performed to obtain H_yc1. The specific generation method is as follows: H_yc1 i =mod((H(i)+floor(0.5*(G(i)+G(i+1)))),256),i=1,2,3,...,L,G L+1 =G1 (3) Fourth step: Perform an XOR operation between the key stream J and H_yc1 to obtain a new sequence V, calculated as follows: The fifth step is the update process. Based on V and G, the first update process is performed to obtain U_gx1, which is expressed as follows: U_gx1 i ={G i +floor(0.001*(V i-1 +V i +2))}mod256,i=1,2,3,...,L,V0=0 (4) Step 6: Reverse the sequences U and V obtained above to obtain two new sequences G1 and H1 respectively. Now G1 is considered the detailed component, while H1 is the approximate component. Step 7, the second prediction process, obtains G_yc2 based on G1 and H1. The specific generation method is as follows: Step 8: Perform an XOR operation between the key stream J and G_yc2 to obtain a new sequence U, calculated as follows: Step 9, the second update process, obtains V_gx2 based on H1 and U, specifically expressed as follows: V_gx2 i ={H1 i +floor(0.001*(U i-1 +U i +2))}mod256,i=1,2,3,...,L,U0=0 (7) The tenth step is to merge the generated U and V_gx2 into a single sequence, where U is the even-indexed sequence of sequence W and V_gx2 is the odd-indexed sequence of sequence W. Then, sequence W is transformed into a matrix C of size M×N, where C is the final encrypted image.

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