Abstract
Everybody wants to maintain solitariness to some extent or entirely in his dealings with other people during different modes of communication. To retain privacy, researchers materialized distinct image encryption algorithms using chaotic maps. Due to their extraordinary features, most researchers employed multidimensional chaotic maps to barricade clandestine information or digital images from potential invaders. Still, multidimensional chaotic maps have many impediments conferred in the literature review. In this paper, we developed a cryptosystem utilizing multiple chaotic maps to mitigate the shortcoming of multidimensional chaotic maps. A distinctive approach is adopted to sire a key stream using a combination of chaotic maps and create a sequence of random integers linked with the pixels of the plain image to shatter the association between neighboring pixels of a plain image. Finally, diffusion is accomplished using the previously diffused pixels at a decimal level. Security and statistical analysis demonstrate that the presented encryption algorithm is robust against well-known attacks. An ample key space indicates that it is best suited for secure communication.
1. Introduction
The importance of image encryption is increased to transmit confidential data (images, videos, audios, and agreements) via the internet because digital images are frequently used to communicate and share sensitive information, whether in the military, banking, or personal moments. Many image encryption techniques [1] have been constructed based on a variety of chaotic [2, 3] to protect the plain images from prying eyes or unlawful access. Traditional security encryption schemes like AES [4], DES [5], and IDEA [6] are inadequate to maintain acceptable standards for the safety of digital images because of the strong association among neighboring pixels, enormous storage capacities, and significant redundancy.
Chaotic systems are deemed to be appropriate for image encryption schemes due to their features like ergodicity, pseudorandomness, ambiguity, high sensitivity to initial conditions, and controlled parameters. Most researchers designed cryptosystems [7, 8] either by modifying existing chaotic systems [9, 10] or developing new chaotic systems [11, 12], keeping in mind the usefulness of chaotic systems due to their characteristics, as mentioned earlier. The authors of [13] proposed a 2D-LSM chaotic system and ascertained that it is more chaotic than some lately developed two-dimensional chaotic maps using a bifurcation diagram, LE, phase plane trajectory, and sample entropy. Furthermore, it materializes a color image encryption algorithm (CIEA) using orthogonal Latin square and 2D-LSM. Hua et al. [14] conferred a new 2D-PPCS (two-dimensional parametric polynomial chaotic system) to mitigate the flaws of existing chaotic systems, and theoretical analysis exhibits the chaotic behavior of the 2D PPCS. The study of [15] modifies the chaotic behavior of a 1-dimensional chaotic map, further analyzed the chaotic behavior through phase diagram and Lyapunov exponent spectrum, and constructs a novel image encryption scheme. There are two types of chaotic systems: (i) one or dimensional chaotic maps and (ii) multidimensional chaotic maps. Most recent image encryption techniques are based on multidimensional chaotic maps [16, 17], which are more sensitive than 1-dimensional chaotic maps. It contains many initial values and controlled parameters that help enlarge the length of key space to resist brute force attacks. In [18], Malik et al. developed a cryptosystem for a color image using two multidimensional chaotic maps and used the idea of histogram equalization to equalize histogram of chaotic sequences of Lorenz system. Hua et al. [19] suggested an image encryption scheme using S-box, yielded using a complete Latin square. Experimental consequences divulge that the presented cryptosystem can defy all well-known security and statistical seizures from potential invaders. The authors of [20] designed an image encryption scheme utilizing parallel compressing sensing with adaptive thresholding sparsification. In [21], to get the random sequences of a 5-dimensional hyperchaotic map, first generate the initial values that depend on pixels of a plain image. A unique key is generated after rearranging the random sequences and constructed an image encryption algorithm. Multidimensional chaotic maps take more time during the execution process due to their intricate structure and method to find the solutions of a multidimensional chaotic map. Alternatively, one or dimensional chaotic maps are simple in structure and execute efficiently in the available MATLAB versions, not taking more time during the computational process. Therefore, we use a combination of chaotic maps rather than the multidimensional chaotic system in this paper. The logistic map, piecewise linear chaotic map, tent map, and Henon map are used to propose an encryption algorithm. Two unique keys are generated using the aforementioned chaotic maps to diffuse the pixels of the plain image at the decimal level. To break the connection among nearby pixels in a plain image, generate a random sequence of integers using any chaotic map, relate it with the diffused pixels, and further use it to scramble the diffused pixels. Finally, the presented encryption algorithm is evaluated on various images, including Lena, Baboon, Pepper, Cameraman, and House. The experimental results demonstrate the effectiveness of our presented image encryption algorithm.
The remaining sections of the paper are organized as follows. The basic features of distinct chaotic maps are described in Section 2. Each step of the proposed encryption method is discussed in detail in Section 3. Various security and statistical tests are performed in Section 4 to assess the performance of the constructed cryptosystem. Conclusions are discussed in Section 5.
2. Chaotic Map
In this section, we will discuss some chaotic maps which are further utilized in an image encryption scheme.
2.1. Logistic Map
Logistic map is a 1-dimensional chaotic map with excellent chaotic features and its numerical expression is described as follows:where is a controlled parameter and its range is [0, 4], but the bifurcation diagram of a logistic map shows chaotic behavior only when can be seen in Figure 1(a). This map highly sensitive to initial condition and the range of initial condition is (0, 1).
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2.2. Piecewise Linear Chaotic Map
Piecewise linear chaotic map [22] is a one dimensional map that generates the random sequence necessary for image encryption scheme during the process of confusion and diffusion. Mathematical formula of piecewise linear chaotic map is given as follows:where is a parameter and its initial value range is . The bifurcation diagram of piecewise linear chaotic is in Figure 1(b) that shows the chaotic behavior of piecewise linear chaotic map.
2.3. Tent Map
Numerical formula of tent map is described as follows:where is a parameter and its initial value range is . Chaotic behavior of tent map can be seen in Figure 1(c).
2.4. Hénon Map
Hénon map [23] is a 2-dimensional chaotic map and its mathematical equation is described as follows:
The Hénon map depends on two control parameters and , and highly sensitive to initial conditions which belongs (0, 1). The chaotic behavior of the Hénon map can be seen in bifurcation diagram against the parameter in Figure 1(d).
3. Proposed Image Encryption Scheme
This section will discuss all the necessary things of an encryption process in detail, like key generation, block scrambling, and pixels scrambling process, and decimal level diffusion process. Step 1. Transformation Let be a grayscale image of size , where and indicate rows and columns, respectively. Now, we transform image into a dimensional array of size . Step 2. Key Stream Generation(i)Let , , , , , , , , , and be the initial conditions and control parameters that are used to iterate the equations from (1) to (5).(ii)After iteration the pseudorandom generated sequences , , , , and .(iii)To remove the transient effect, discard the first iterations from each sequence , , , , and and obtained new sequences , , , , and , respectively, of length .(iv)Choose the one sequence from the sequences , , , , and and after selecting the sequence, store them in , in ascending order.(v)Now generate sequence of random integers by storing the position of each member of in the selected sequence.(vi)Divide the sequence into four subsequences , , , and , the length of each subsequence is ;(vii)Now, select the state of variables from the remaining four sequence named as , , , and and get the subsequences , , , and , respectively of length using the expression described as follows: where means number of encryption rounds and for each , and . if then .(viii)The subsequences , , , and are used to form a sequence of length .(ix)We have ways to form a sequence of length by using the subsequences , , , and .(x)Now obtained two sequences and of length using two different arrangements of , , , and . arrangement of sequences arrangement of sequences (xi)Compute the key streams and using the mathematical formula described as follows: where to . Step 3. Decimal Diffusion Level 1 The following analytical equation is used to diffuse the pixels of using and previously diffused pixels; where is a seed value used to diffuse the first pixel of . Step 4. Block Scrambling Process(i)First, divide into blocks, and the size of each block is ; then, the number of blocks is , where .(ii)Now, obtain a subsequence of length from the sequence, which is selected from the sequences and generate the sequence of random integers using the same procedure on , described in Step 2 (iv, v).(iii) is used to scramble the blocks of for times where indicates the number of scrambling rounds; after block scrambling of transform the into a 1-dimensional array . Step 5. Decimal Diffusion Level 2 The pixels of is diffused using the sum of previously diffused pixels and , with mathematical expression described as follows: where is the seed value used to diffuse the first pixel of . Step 6. Pixels Scrambling Process(i)Let be a random integers sequence of length , described in Step 2. (v).(ii) is used to scramble the pixels of for times where indicates the number of pixels scrambling rounds. Step 7. Cipher Image Transform the into matrix of size . Figure 2 illustrates the encryption process. Decryption is performed in the opposite direction as encryption.
4. Performance Evaluation
The performance of a presented cryptosystem is tested utilizing statistical and security analysis on distinct images such as Lena, Baboon, Pepper, Cameraman, and House. The parameters of the secret key are , , , , , , , , , , , , , , , and , and all experimental results are computed in MATLAB 2018b on a compatible computer with Windows 10, 8.00 GB RAM, and an Intel(R) Core(TM) i5-6300U CPU @ 2.5 GHz. Figure 3 demonstrates the encryption effect of our presented cryptosystem, and all experimental data are documented in tables, demonstrating that our cryptosystem’s performance is outstanding against any security and statistical threats.
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4.1. Key Space Analysis
The key space analysis helps us to decide whether or not our cryptosystem can withstand brute force attacks. The length of a key space is crucial to withstand a brute force assault because a prospective intruder would try every possible combination of key space to crack the cryptosystem. It is believed that if a cryptosystem’s key space is larger than , it is robust. Let , , , , , , , , , and be the secret key parameters, and each has a computational accuracy of , and , , and are the positive integers used at diffusion level, and is the number of discarding iterations where . Consequently, the length of the key size which is higher than , demonstrating that our presented encryption technique is resistant to brute force attacks. Table 1 shows the key space comparison to some existing encryption cryptosystems.
4.2. Histogram Analysis
Histogram analysis is the simplest and pictorial way to see the frequency distribution in plain and encrypted images. If the frequency distribution is uniform, then it is considered that the proposed encryption scheme is suitable for the secure transmission of data through the internet in daily use applications. In bit grayscale image, the range of pixels value is to 255. Histogram of plain and encrypted images (Lena, Baboon, Pepper, Cameraman, and House) can be seen in column (b, d) of Figure 3, and column (b) shows that the frequency distribution of pixels is not uniform. Still, on the other side, column (d) shows that the frequency distribution of pixels is uniform. The uniform distribution of pixels demonstrates that our proposed image encryption scheme can resist any statistical attack. A potential invader does not get any reliable information from the histogram of encrypted images.
4.3. Entropy
It is deemed that entropy is used to compute the uncertainty present in the source of information (digital image); if m is a source of information, the entropy of m can be computed described as follows:where p() is a frequency of the symbol , the entropy of a source of information is 8 according to entropy formula, if the frequency of symbol is uniformly distributed. The entropy of different encrypted images (Lena, Baboon, Pepper, Cameraman, and House) is nearly equal to 8, as shown in Table 2.
Sometimes global Shannon entropy does not measure the true randomness in the encrypted image; we estimate the local Shannon entropy [30] to overcome this drawback. In local Shannon entropy, the encrypted image divided into randomly selected K nonoverlapping blocks with pixels in each block. Compute the entropy of each block using (11). Finally, compute the mean of all the block’s entropy. The mathematical formula to calculate the local Shannon entropy is described as follows:
The benchmark of local Shannon entropy is 7.90. It is evident from the Table 2 that our proposed encryption scheme can produce adequate randomness in an encrypted image that a potential invader cannot get any reliable information about the plain image.
4.4. Chi-Square Analysis
Chi-square is a quantitative method of assessing pixel uniformity. Chi-square is a statistical technique that is described mathematically as follows:where indicates the actual frequency of pixel, the range of pixels value is 0 to 255 because we are working on the 8-bit grayscale image. The numerical value for the level of significance 0.05. The distribution of pixels in encrypted images is highly uniform if the Chi-square score of the encrypted image is lower than as much as possible. The Chi-square score of distinct encrypted images (Lena, Baboon, Pepper, Cameraman, and House) is listed in Table 2, which proved that our proposed encryption scheme uniformly distribute the pixels value 0 to 255 in different rounds of encryption.
4.5. Majority Logic Criteria (Texture Analysis of the Image)
To evaluate the presented encryption scheme’s efficiency on digital images, the MLC [31, 32] tool is utilized, which is a collection of five statistical tests such as correlation, contrast, entropy, energy, and homogeneity.
4.5.1. Contrast
The brightness deterioration of plain images throughout the encryption process is calculated using contrast analysis. The better the encryption technology, the higher the contrast value. The numerical equation is as follows:where indicates the grayscale co-occurrences matrix.
4.5.2. Correlation
The range of correlation values is . In plain images, the correlation value of a pixel to its neighboring pixels is one or nearly equal to 1. The correlation value one or almost equal to 1 shows that the correlation of a pixel to its neighboring pixels is very strong. On the other hand, the correlation value 0 or negative shows that the correlation of a pixel to its neighboring pixels is weak. The purpose of any encryption scheme is to break the correlation of a pixel to its adjacent pixels. Mathematical equation which is used to measure correlation is described as follows:where indicates the grayscale co-occurrences matrix, we also find the correlation of a pixel to its adjacent pixels in the horizontal, vertical, and diagonal direction. The correlation of a pixel to its adjacent pixels is listed in Table 3 and the correlation of a pixel to its neighboring pixel in a horizontal, vertical, and diagonal direction of plain and encrypted images (Lena, Baboon, Pepper, Cameraman, and House) listed in Table 4. Figure 4 shows the correlation of plain and encrypted images of Lena in a horizontal, vertical, and diagonal direction. The experimented results of correlation demonstrate that the performance of the proposed encryption scheme is excellent and able to resist any attack.
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4.5.3. Energy
To execute the energy analysis, we use a gray-level co-occurrence matrix. The numerical expression of energy is given as follows:where indicates the grayscale co-occurrences matrices.
4.5.4. Homogeneity
Homogeneity analysis is used to compute the closeness of the grayscale level of co-occurrence matrices in (GLCM). The encryption method is deemed to be excellent if the homogeneity value is lower as much as possible. The mathematical expression of homogeneity is described as follows:where indicates the grayscale co-occurrences matrices.
The experimental MLC score of plain and encrypted images is listed in Table 3. MLC scores indicate that the encryption technique is excellent for digital images for secure transmission through the internet.
4.6. MSE/MAD/PSNR/SSIM Analysis
A cryptosystem’s encryption quality may be assessed using mean square error (MSE) and mean absolute difference (MAD) tests. MSE and MAD are mathematically expressed as follows:where and indicate the dimensions of plain image and encrypted image , the cryptosystem is deemed efficient and secure if MSE and MAD values are higher.
The peak signal to noise ratio (PSNR) is inversely proportional to the square root of MSE, as seen in equation (19). The PSNR value should be as low as feasible for the best encryption quality. The PSNR is calculated using the following mathematical formula:
The structural similarity index measurement (SSIM) evaluates how much an encrypted image deteriorates throughout the encryption process. The SSIM mathematical equation is as follows:where and are the average pixel values and , and are the variance of corresponding images and . Also, represents the covariance between and , and and are two predetermined constants used to ensure stability. The MSE, MAD, PSNR, and SSIM experimental findings are presented in Table 5, and the experimental results show that the proposed cryptosystem can withstand any statistical attack.
4.7. Data Loss Attack Analysis
A data loss attack means a potential invader artificially dethrones the pixels of a particular area of an encrypted image or loses some portion of the image during transmission through the internet. A cryptosystem is immaculate if it convalesces the critical information from the encrypted image, which loses the data. It is clear from Figure 5 that our proposed cryptosystem can retrieve helpful information from the encrypted images which lose data during communication. In Figure 5, to the figures which lose a block of data size to demonstrate the data loss attack and lose data from all sides, and to represent the figures after performing the decryption process on the figures which lose data respectively.
4.8. Noise Attack Analysis
Normally, images possess information, but during the transmission, the image is polluted with some noise. The appearance of noise will annihilate the original information from the image. The pepper and salt noise effect on our encryption algorithm is examined in this paper on Lena’s image, and different salt and pepper noise ratios represent the attack intensity. In Figure 6, we use the pepper and salt noise ratios 0.01, 0.03, 0.05 on the figure , , and , respectively. The decryption process are employed on the , , and and as a result get figures , , and , respectively. Figure 6 demonstrates that our proposed system is flawless for dealing with polluted images and retrieving important information as much as possible.
4.9. Differential Attacks
The two main criteria for evaluating the robustness [33] of a cryptosystem against a slight change in any pixel or secret key parameter are (number of pixel changing rate) and (unified averaged changed intensity). The mathematical equations for NPCR and UACI are as follows:where and are the dimensions of an image, also and are two encrypted images corresponding to plain images differ by single pixel. Benchmark values of the NPCR and UACI for different sizes of images are listed in Table 6.
4.10. Plaintext Sensitivity Analysis
A cryptosystem is plaintext sensitive if a slight change in randomly selected pixel of plain image will produce a different encrypted image; that encrypted image does not provide any important clue about plain image. Method to test the plaintext sensitivity, let and be two plain images; all the pixels of and are the same except one randomly selected pixel. The original key is used to encrypt the plain images and , get two encrypted images and , respectively. NPCR and UACI between two encrypted images and , are computed using equations (21) and (22). A cryptosystem is sensitive to plaintext if the value of and is greater or closer to . The procedure, as mentioned earlier of plaintext sensitivity employed on the image (Lena, Baboon, Pepper, Cameraman, and House) and compute the and score, is listed in Table 7. The plaintext sensitivity score of different images demonstrates that our cryptosystem can resist any differential attack.
4.11. Key Sensitivity Analysis
The cryptosystem is regarded as extremely key sensitive; if the encrypted image acquired after a subtle change in any of the secret key parameters provided in Table 8 is distinctive from the encrypted image generated without any change in secret key parameters. To test the key sensitivity, first, encrypt plain image using the original secret key parameters listed in Table 8, then encrypt the same plain image using a minor modification of in any secret key parameter, yielding two encrypted images and . and are computed between to encrypted images and using the equations (21) and (22). We performed a key sensitivity analysis on the Lena image by making minor changes to each secret key parameter one at a time. The resulting and scores are shown in Table 8, indicating that our proposed cryptosystem is extremely key sensitive and resistant to differential attacks.
4.12. Randomness Test for Cipher
Our proposed cryptosystem creates enough randomness in the pixel of encrypted images; a potential invader can never acquire reliable information about the plain image. The Nist test in [34] is used to check the randomness present in the pixels of encrypted images. The Table 9 exhibits the two Nist test results, which confirm that the security of our proposed cryptosystem is excellent to resist any attack.
4.13. Speed Analysis
The speed and security of any encryption scheme are the key characteristics in the application of real life. The analysis, as mentioned earlier, exhibits that the security of our proposed encryption scheme is immaculate. We test the speed of our encryption scheme in MATLAB 2018b on a compatible computer with Windows 10, 8.00 GB RAM, and an Intel(R) Core(TM) i5-6300U CPU @ 2.5 GHz. Table 10 shows the speed analysis of our proposed scheme with other encryption schemes developed using multidimensional chaotic maps. The speed analysis confirmed that our encryption scheme is unassailable and efficient. Also, the statistical test results of the baboon image are compared in Table 11 with the existing encryption scheme.
5. Conclusion
The novelty of this paper is to generate a unique key stream and scrambling process (blocks of plain image and pixels of plain images) using different chaotic maps, which is further utilized to propose a cryptosystem for the security of digital images during transmission from a potential invader. The aforementioned cryptosystem is employed on the standard grayscale images to see its effectiveness. The simulations and results of the experimental analysis demonstrate that the presented cryptosystem is exceptional in preserving high security and privacy requirements. We can confidently assert that the presented scheme is ideally suited to multimedia communications and online systems.
Data Availability
The data used to support the study are included in the paper.
Conflicts of Interest
The authors declare that they have no conflicts of interest.