[Retracted] Image Encryption Algorithm Based on Artificial Bee Colony Algorithm and Chaotic System

Zhou, Yanqi; Wang, Erfu; Song, Xiaomeng; Shi, Mengna

doi:https://doi.org/10.1155/2022/1444676

Security and Communication Networks

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Research Article | Open Access

Volume 2022 | Article ID 1444676 | https://doi.org/10.1155/2022/1444676

[Retracted] Image Encryption Algorithm Based on Artificial Bee Colony Algorithm and Chaotic System

Yanqi Zhou,¹Erfu Wang,¹Xiaomeng Song,¹and Mengna Shi¹

Academic Editor: Xingsi Xue

Received23 Jan 2022

Revised09 Mar 2022

Accepted01 Apr 2022

Published18 May 2022

Abstract

This article proposes an image encryption algorithm based on a chaotic bit-plane decomposition and optimization algorithm of a crossover operator artificial bee colony algorithm. Firstly, use the SHA-256 hash algorithm to calculate the plaintext image’s hash value as the starting value of the fractional Lorenz hyperchaotic system after operation. Utilize the chaotic sequence to permutate plaintext image in a bit plane to obtain the scrambled image. Secondly, block the scrambled image into four subimages of equal size, and count the hash value of each row of each block by the SHA-256 hash algorithm as the starting value of the Sine-Tent-Logistic chaotic system. Use the obtained chaotic sequence to substitute the images. Then, stitch the four sub-block images to get the final encrypted image, and the population is obtained. Finally, use the information entropy of ciphertext image as the fitness function of the artificial bee colony algorithm based on a crossover operator. Select the ciphertext image with the best information entropy from the population as the optimal encrypted image, and then, return the position value of the best honey source meanwhile. The experimental simulation and security analysis indicate that the scheme has an excellent encryption effect and ability to oppose various general attacks.

1. Introduction

With the advancement of network technology, the era of message integration is approaching, and information and data security is becoming increasingly important [1]. Because of its intuitive, vivid, and realistic qualities, image information has become a crucial carrier in human social interactions and information transfer. The safety of photographs is really important. Image encryption is a specific technology used to address image security concerns. Traditional encryption methods, such as DES [2] and AES [3], cannot meet the present needs of picture encryption due to the characteristics of a large number of data, strong correlation, and high redundancy. Therefore, lots of encryption algorithms have been proposed, such as compressed sensing theory [4, 5], chaos theory [6–10], and DNA coding theory [11–13].

For the past few years, many scholars used chaotic systems widely in image encryption systems because of their extreme sensitivity to starting conditions and unpredictability. They outperform other encryption algorithms in terms of encryption effectiveness. Generally, this algorithm contains two main steps: permutation and substitution. The permutation operation changes the location of the pixel to reduce the correlation between neighboring pixels. The purpose of the substitution operation is to alter the size of the pixel value, and the plaintext image becomes another different image. Pan et al. [14] came up with an encryption operation using a double logistic chaotic system to achieve image substitution prior to scrambling and finally obtain the encrypted image. However, the histogram distribution and information entropy of the encrypted image by this method are poor, making it vulnerable to attackers. Liu et al. [15] put forward a new hyperbolic sine chaotic system to figure out the problem of low security of the traditional chaotic system. The generated chaotic sequence has good pseudorandom characteristics and employs it for image substitution operation and row-column operation. The algorithm has a good encryption effect. However, compared with other encryption algorithms, the information entropy of encrypted images has some shortcomings.

However, the problems caused by chaotic systems still need to be solved. The loss of chaotic dynamic properties generated by the digital chaotic system’s finite precision impact is particularly damaging, as it directly undermines the chaotic system’s security in cryptographic systems. Peng et al. [16] put forward a high-dimensional chaotic map based on discrete memristor, and experimental simulation shows that the discrete memristor model can not only expand the hyperchaotic region, but also further improve the complexity of the system. Li et al. [17] discovered that the phase space of the Cat map executed on a digital computer has a severe regular pattern that differs significantly from the phase space in the infinite precision torus; Ma et al. [18] investigated the security of an image block encryption algorithm based on multiple chaotic maps and discovered it to be insecure.

The chaotic system proposed in Ref [19] is formed by fusing and cascading three one-dimensional chaotic maps. Firstly, Tent map and Logistic map are fused to generate a new chaotic system; then, the Sine map and the map generated in the first step are employed for cascade operation. The chaotic system enhances the Lyapunov exponent of the chaotic system, effectively strengthens the chaotic state, prolongs the period efficiently, and delays the degradation as much as possible. The fractional hyper-Lorenz chaotic system, as presented in Ref [20], is also used in this article. The fractional hyperchaotic dynamic system has more complex and rich dynamic characteristics than the integer-order system, with the added benefit of increasing randomness and unpredictability. Moreover, the fractional hyperchaotic system can provide more key parameters for the encryption system, expanding the key space and improving the system’s security properties. Meanwhile, in view of the unique memory characteristics of the hyperchaotic system, it can effectively increase the complexity of chaotic sequence and make encryption more secure. Therefore, the chaotic systems used in this article can effectively avoid the degradation of chaotic systems under finite accuracy.

The permutation and substitution processes are two independent processes that are rarely linked to one another. Therefore, some researchers have presented a bit-plane decomposition image encryption algorithm. Bit-plane decomposition encryption is able to implement scrambling operation and change the value of pixels at the same time, allowing the scrambling and substitution processes to run concurrently. Zhou et al. [21] presented an encryption algorithm using Fibonacci p-code to decompose the plaintext image into a certain number of bit-plane images, shuffled all the bit-plane images using the key, adjusted the size of the bit plane using 2D p-Fibonacci transform, and encrypted all the bit planes, and finally, all bit planes are merged to get the last encrypted image. The experimental simulation suggests the arithmetic has an encryption effect, but as is shown from the histogram of the encrypted image that the histogram of the encryption algorithm is not flat, contains some spikes, and is likely to be subject to statistical attacks by the attacker. Rathore and Pal [22] put forward an image encryption arithmetic based on the combination of bit plane decomposition and chaotic system. The proposed method used the sequence generated by two-dimensional Henon chaos to substitute the plaintext image, decomposed the substituted image into eight-bit-plane images, and reconstructed the sequence generated by two-dimensional Henon chaos into a binary key matrix. Eight different binary edge planes are obtained by edge detection and then XOR with the corresponding bit planes after substitution. Finally, the eventual encrypted image is gained by merging the eight-bit planes. But the simulation results indicate that the encryption effect cannot achieve the optimal consequence, and the correlation between the encryption algorithm and plaintext is not good. When it is subjected to a selective plaintext assault or a differential attack, it is vulnerable to information leaking.

In recent years, many researchers have applied the heuristic evolutionary algorithm to the field of image encryption. Choosing the appropriate fitness function to optimize the generated key or ciphertext image is optimal. Abdullah et al. [23] put forward an encryption scheme based on a hybrid genetic algorithm and chaotic system. The logistic mapping function is used to encrypt the image, and multiple encrypted images create the starting population of the genetic algorithm. The genetic algorithm optimizes the encrypted image and chooses the optimal encryption image based on its low correlation coefficient and high information entropy. Ghazvini et al. [24] came up with a hybrid image encryption method based on GA and chaos. The encryption scheme contains three major steps: scrambling stage, substitution stage, and optimization stage employing GA. Mozaffari put forward a parallel image encryption algorithm based on bit-plane decomposition [25]. The original gray image is transformed into a binary image by local binary mode and bit-plane decomposition method. Use the multipopulation GA to complete the scrambling and replacement operation through crossover and mutation operation. Finally, gain the final encrypted image by merging. It is hard to select parameters of a genetic algorithm, which requires a sequence of processes such as selection, crossover, and mutation. If the parameters are not carefully chosen, they’ll slip into local convergence, causing premature convergence and so on. Dua et al. [26] generate a mask sequence with the help of differential evolution technique, which is further converted to DNA and utilized in the DNA diffusion process. Saravanan and Sivabalakrishnan adopted an improved meta-heuristic algorithm termed as CI-WOA for optimizing the parameters [27]. Liu et al. proposed to use a hybrid chaos system and an artificial fish swarm neural network to encrypt images, which is used to train the hybrid random array and remove its chaotic periodicity, allowing the neural network sequence to be obtained [28].

Because of the simplicity of the chaotic system, the key space is too minor to stand up to exhaustive attack; the correlation between encryption scheme and plaintext is low, resulting in a weak ability to resist selected plaintext attack, known-plaintext attack, and differential attack; the artificial bee colony algorithm is a simple and efficient optimization algorithm to simulate bee behavior. Role switching is a unique mechanism of the artificial bee colony algorithm. The mutual conversion and perfect cooperation between different bee species enable bees to find a better location of honey source in any environment. The artificial bee colony algorithm has a positive and negative feedback mechanism. When the hired bees find a high-quality honey source, they can recruit more observation bees to follow. If there are fewer honey sources, the number of observation bees recruited will also be reduced. In this way, the positive and negative feedback cooperate with each other to find the optimal honey source more efficiently. Particle swarm optimization has only positive feedback mechanism, and the effect and efficiency of optimization will be reduced. Compared with other optimization algorithms, the artificial bee colony algorithm has a small number of parameters, which can reduce the impact of artificial parameter setting as much as possible, and this article presents an image encryption method based on chaotic bit-plane decomposition and optimization algorithm of the crossover operator artificial bee colony algorithm. The main contributions of this article are as follows:(1)A ciphertext image optimization algorithm based on artificial bee colony is proposed(2)Introducing crossover operator into the artificial bee colony algorithm can generate new individuals and enrich population diversity(3)Calculating the hash value for pixel values of each block and each line of the scrambled image strengthens the connection between bit-plane scrambling and diffusion(4)The hash value of the plaintext image is calculated as key, which enhances the relationship between encryption and plaintext, and enhances the ability to resist selective plaintext attack

The rest of the article arranges as follows: Section 2 introduces the basic principles of the algorithm, Section 3 concretely presents the encryption method proposed in this article, the experimental results and security analysis are given in Section 4, and the conclusion is presented in Section 5.

2. Preliminary Works

Because of the randomness, ergodicity, uncertainty, and sensitivity to initial conditions and parameters, chaotic systems are extensively employed in privacy communication systems. This article uses the fractional-order super Lorenz chaotic system and the Sine-Tent-Logistic chaotic system. The artificial bee colony algorithm is a heuristic optimization algorithm, which can be used in this article to optimize the encrypted image to obtain an encrypted image with a better encryption effect.

2.1. The Fractional Hyper-Lorenz Chaotic System

The fractional-order chaotic dynamic system has more complicated and rich dynamic features than the integer-order system, with the added benefit of enhancing randomness and unpredictability. In addition, a fractional-order chaotic system can offer more significant parameters for the encryption system, thus increasing the key space and further enhancing the security characteristics of the system. The complexity of the chaotic sequence may be effectively increased, and the encryption is more safe, thanks to the fractional chaotic system’s unique memory features. Many academics have employed fractional-order chaotic systems to encrypt photos in recent years. Kaur et al. [29] presented a new opto-digital color picture encryption scheme based on compound chaotic mappings, the reality-preserving fractional Hartley transformation, and the piecewise linear chaotic map for image pixel replacement, optical processing, and permutation. The presented picture encryption technique has a greater level of protection and heightened sensitivity to keys. Kaur et al. [30] proposed a multiple order optical transformation encryption scheme for two-dimensional image encryption based on chaos. The transform coefficients are calculated in the collective time-frequency domain using the multiparameter fractional Fourier transform of chaotic ordering. Two piecewise linear chaotic maps (PWLCMs) are used to generate multiple transformation orders along two dimensions. The two chaotic sequences generated by PWLCM are substituted in the proposed transform (C-MOFRFT) and then permutated in the C-MOFRFT domain based on an integrated chaotic mapping. The mathematical model of the fractional hyper-Lorenz chaotic system [20] used in this article is as follows:where represent the state variables of the system, when , and the system is in a chaotic state.

2.2. The Sine-Tent-Logistic Chaotic System

Integer-order chaotic systems include one-dimensional chaotic systems and hyperchaotic systems. One-dimensional chaotic systems generally contain the Logistic chaotic map, Sine chaotic map, and Tent chaotic map. Most of these one-dimensional chaotic systems only comprise one variable and some parameters, making them vulnerable to attackers and resulting in information leakage. In this article, we use the chaotic system proposed in Ref [19]. The mathematical model of the Sine-Tent-Logistic(STL) chaotic system is

The Lyapunov index of the STL chaotic system is greater than 0 at any value in the parameter . The chaotic sequence generated has good pseudorandom characteristics.

2.3. Artificial Bee Colony Algorithm

Artificial bee colony algorithm is a bionic swarm intelligence optimization algorithm based on a bee honey collection mechanism. In 2005, Professor Karaboga of Erciyes University in Turkey first proposed the artificial bee colony algorithm model [31]. It has been one of the hotspots of bionic intelligent algorithm research in recent years. Each nectar source symbolizes a feasible solution to the optimization problem in the implementation phase of the algorithm, and the pollen number of nectar sources is the fitness function value in the optimization problem. Let the feasible solution of the optimization problem be a -dimensional vector , and the population with feasible solutions is expressed as ; specific processes of the artificial bee colony algorithm is described as follows:(1)Colony initialization: The algorithm generates a certain number of food sources randomly at the initial stage in the feasible range. The following formula determines the initial location of the nectar source: where represents the dimension of the , is a random digit between 0 and 1, and denote the maximum and minimum of the dimension of .(2)Hiring bee stage: Use equation (4) to seek the location of the nectar source near the initial food source, start a local search, judge the fitness of each nectar source, and judge the quality of the nectar source by the fitness value. Equation (5) employs a greedy approach to save the better solution: where represents the adjacent food source, represents the current food source, indicates the rate of change of food sources, and represents the fitness function.(3)Observation bee stage: The observation bees select the excellent nectar sources searched by the employed bees according to the probability and then further search for the better food sources in the neighborhood. Observing bees conduct a local search near the food source according to equation (4) to generate new individuals with higher quality and use the greedy mechanism to save better solutions. Among them, the selection probability is(4)Investigation bee stage: Abandon a food source if it does not renew after several collection. The corresponding hired bees or observation bees will transform into observation bees. The Scout bees construct a new food supply at random, move the nectar source around, and continue to search for the best answer in the global range.

3. Proposed Encryption and Optimization Algorithm

This article first uses the fractional Lorenz hyperchaotic system to scramble each bit plane of the plaintext image and then recombines the 8-bit planes into a scrambled image after scrambling. Split the scrambled image into blocks, and then, each piece is substituted by the Sine-Tent-Logistic chaotic system to generate an initial population containing multiple encrypted images. Apply the artificial bee colony algorithm based on a crossover operator to optimize the initial population and the entropy of the encrypted image used as the fitness function of the artificial bee colony algorithm. Several repetitions are used to get the best ciphertext image with the most information entropy; the position value of the best honey source returns to facilitate subsequent decryption meanwhile. The specific encryption process is described below.

3.1. Key Generation

The key of this article consists of two parts. The first part is the key needed in bit-plane scrambling. Calculate the hash value of the plaintext image according to the SHA-256 hash algorithm. The result of the SHA-256 hash algorithm is a 256 bit hash value. Every 4 bit binary number is transformed into a hexadecimal number and finally obtained a string of 64 hexadecimal numbers: . After an initial value substitution iteration, the fractional hyper-Lorenz chaotic system will generate four groups of different chaotic sequences , and . Because this article needs to scramble 8-bit planes, eight groups of chaotic sequences are required. It is necessary to generate 8 initial values, which are divided into two groups and substituted into the fractional hyperchaotic system to generate 8 chaotic sequences. So, it is necessary to divide the key₁ into eight blocks via the following equation:

Because the is an 8-bit hexadecimal number that must be quantized to create a decimal number between 0 and 1, use it as the initial value of the fractional hyper-Lorenz chaotic system and substitute it into the chaotic system to iteratively obtain the chaotic sequence by the following equation:where is a function in MATLAB that can convert hexadecimal numbers to decimal numbers, and the is the maximum value in eight-bit hexadecimal number. Assign to . By substituting them into the chaotic system as the initial values of two groups of chaotic systems for iterative operation, eight groups of chaotic sequences to scramble the bit plane of the original plaintext image are obtained.

The second part is the key needed in the substitution process after bit-plane scrambling. Get a scrambled image after the bit plane permutation operation finishes. The scrambled image divides into four sub-block images of equal size; adopt the SHA-256 algorithm to calculate the hash value of all pixel values of each block and each line to calculate the via the following equation:where represents all pixel values of the row of the block image, and hash calls the SHA-256 hash algorithm adopted in this article to get a 64 bit hexadecimal number string. Use the function to convert the hexadecimal number into a decimal number, and quantize it to 0‐1 as the initial value of the Sine-Tent-Logistic chaotic system.

3.2. Bit-Plane Scrambling Algorithm Based on Fractional Lorenz Hyperchaotic System

Any non-negative integer can be described by a string of n-bit binary sequences. In gray images, the range of pixel value is between [0, 255], so each pixel value can be described by a series of 8-bit binary sequences. A gray image can be decomposed into 8-bit-plane images [32]. The bit plane is composed of the bit of the binary of each pixel value. The bit-plane decomposition of the original gray image yielded eight-bit-plane images in Figure 1.

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Bit-plane scrambling not only realizes the global scrambling of image pixel position but also changes the pixel value. It can smooth out the histogram of scrambled photos, reducing the risk of information leaking.

The pseudocode of bit-plane scrambling algorithm based on the fractional Lorenz hyperchaotic system is shown in Algorithm 1.

	Require: Plain image ,
	Ensure: Scrambled image
(1)	/Calculate the hash value of the plaintext image/
(2)
(3)	/Calculate the initial value of the fractional chaotic system/
(4)	Calculate the initial value of the chaotic sequence according to equation (7) to equation (9)
(5)	/Calculate fractional chaotic sequence/
(6)	Calculate chaotic sequence by using , and the given chaotic parameters into equation (1) to calculate, Pre-iterative chaotic mapping 1000 times to eliminate the adverse consequences caused by transient reactions, continue to iterate times
(7)
(8)	Decompose into bit planes to get 8 bit planes
(9)	fordo
(10)	Arrange the bit plane into a one-dimensional vector in column first
(11)	Arrange the chaotic sequence in in ascending order, and generate a sequence used to record the position of each element in the sorted sequence in the original sequence
(12)	Re-arrange the vector in the order of the sequence to obtain a new one-dimensional vector after transformation
(13)	Adjust the size of the newly generated to make it consistent with the size of the plaintext image
(14)	end for
(15)	Combine the obtained 8 scrambled bit-plane images to gain the final scrambled image

3.3. Substitution Algorithm Based on Sine-Tent-Logistic Chaotic System

The chaotic substitution process changes the pixel value of the original image information to generate another different image. This section uses the Sine-Tent- Logistic system to perform substitution processing on the scrambled image and repeats the operation times to generate the initial population required for the subsequent artificial bee colony optimization operation. The specific substitution steps are as follows:(1)The scrambled image received in the preceding step is divided into four blocks, and the size of each block is .(2)According to Section 3.1, calculate the hash value of each block and each row after block and employ it as the chaotic initial value of Sine-Tent-Logistic after quantization.(3)In order to get three different chaotic sequences, set three different parameters of the chaotic system—, substituting parameters and into the Sine-Tent-Logistic chaotic system. Pre-iterate the chaotic map 1000 times to eliminate the adverse consequences caused by transient reactions and continue to iterate times. Get the chaotic sequence according to the following equation:

In the light of the following equation, the pixel values of each block are substituted by the obtained three chaotic sequences, and the substituted image of each block is obtained:where is the scrambled image of each block. Finally, the four substituted images are recombined according to the following equation, and the final encrypted image is obtained:

To generate the initial population for artificial bee colony optimization, go back to Step 2 and repeat for times. It ensures that the hash value of each element in each block and line can be used as the initial value of chaos, and the initial population containing different encrypted images is obtained.

3.4. Image Optimization Algorithm of Artificial Bee Colony Algorithm Based on Crossover Operator

After the permutation and substitution process is completed, individuals of encrypted images will be generated to form the initial population optimized by the artificial bee colony algorithm. Each encrypted image in the population does not have visual visibility, so it is necessary to select specific indicators in the ciphertext image that can judge the quality of its encryption. In this article, the information entropy of the ciphertext image is selected and calculated as the fitness function to judge the quality of the ciphertext. This article presents a crossover operator shown in Figure 2, which can continuously generate new encrypted images by crossing two existing encrypted images, enhance the diversity of the population. After continuous optimization operations, the encrypted image with the largest information entropy is finally obtained. The specific optimization steps are as follows:(1)Initialization stage: Section 3.3 generates an initial population with different encrypted images. Calculate the information entropy of each encrypted image, and take it as the fitness function. The formula is as follows: According to the order of fitness function from small to large, to sort each encrypted images in the to obtain , each individual is regarded as a honey source, and search optimization is performed by changing the position of the honey source.(2)Hiring bee stage: Search for the location of the food source near the current nectar source , represents the location of other nectar sources. Update the position of the hired bee according to equation (4), and then, determine whether to choose the updated hired bee or the initially hired bee according to the greedy algorithm of equation (5). This step is equivalent to the selection stage, which can save more of the better-encrypted images in the population. Generate a new population .(3)Introducing crossover operator: Save images with the highest information entropy in . Then according to the crossover operator shown in Figure 2, individuals in are crossed to generate the remaining encrypted images. Calculate the fitness value of the newly generated individual according equation (13). According to the greedy calculation method of equation (5), the better individuals and their positions are selected for preservation and generate a new population .(4)Observation bee stage: Calculate the probability according to equation (6). The follower bees choose the honey source to follow according to the follower bees. According to equation (4), the local search is carried out, and the better individuals are selected. Calculate the fitness value of the newly generated individual. According to the greedy calculation method of equation (5), the better individuals and their positions are selected for preservation and generate a new population .(5)Investigation bee stage: Abandon a food source if it does not renew after several collection. The corresponding hired bees or observation bees will transform into observation bees. The Scout bees randomly create a new food source, change the location of the nectar source, and continue to seek out the optimal solution in the global range.(6)Selection stage: Calculate the information entropy of each encrypted image in , where the encrypted image with the best fitness function is the optimal encrypted image obtained in this article and output its corresponding position information to facilitate subsequent decryption.

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The pseudocode for image optimization based on the artificial bee colony algorithm of the crossover operator is shown in Algorithm 2.

	Require: Population: , Maximum number of iterations: , investigation_size, observation_size
	Ensure: Optimal encrypted image and its position
(1)	According to equation (13)
(2)
(3)	Rearrange according
(4)	fordo
(5)	/Hiring bee stage/
(6)	fordo
(7)
(8)
(9)
(10)
(11)
(12)	Substitute into equation (4) to update its position
(13)	Boundary value processing
(14)	Search for the corresponding encrypted image according to
(15)	equation (13) which in
(16)	According to equation (5) for select the better individuals
(17)	end for
(18)	/Introducing crossover operator/
(19)	Sort and save the top in
(20)	The remaining according to Figure 2 to cross other images
(21)	Calculate the fitness According to equation (13), select the better individuals according to equation (5)
(22)	/Observation bee stage/
(23)	fordo
(24)	according to equation (6)
(25)
(26)
(27)
(28)
(29)
(30)
(31)	repeat 11–16
(32)	end for
(33)	/Investigation bee stage/
(34)	fordo
(35)	Judge whether to give up the honey source
(36)	end for
(37)	end for
(38)	Calculate the fitness of each encrypted image in , select the encrypted image with the maximum fitness, and return its corresponding position

The encryption flow diagram is shown in Figure 3.

4. Simulation Results and Analysis

An excellent encryption system may withstand various assaults, such as statistical, differential, and selective plaintext attacks, among others. To verify the efficacy and reliability of the presented encryption algorithm, we use MATLAB software for simulation. Input images are grayscale with dimensions . The plain images are tested as shown in Figure 4. Analyze the encrypted images through a series of security analysis methods, for instance, key space analysis, statistical analysis, information entropy analysis, difference analysis, and so on. The experiment consequence shows that the proposed encryption scheme has outstanding encryption characteristics.

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4.1. Statistical Analysis

4.1.1. Key Space Analysis

The high-security encryption system in which key space should be large sufficient to resist all kinds of brute force attacks. The key space of the image encryption algorithm composes the key space of scrambling and substitution. The literature suggests that the encryption system can stand up to all sorts of attacks by exhaustive search only when the key space of the encryption system is not less than [33], so that it can reach the level of sufficient security. The accuracy of this article is double precision . The parameters needed for the chaotic system are as follows: , and the key space is . The initial value of the chaotic system is hashed by SHA-256 algorithm. The size of key space provided by the SHA-256 algorithm is . Therefore, the key space of this article is , which is much larger than , so the encryption algorithm presented in this article has sufficient key space to stand up to any violent attack.

4.1.2. Histogram Analysis

The histogram of the digital images can directly indicate the allocation of pixel values in the image, the original plaintext information contains rich image information, so the distribution diagram of plaintext images is uneven. The ciphertext information encrypted by encryption technology is similar to noise information, and the image information can not reflect the content of the image. As a result, the ciphertext’s histogram should be flat, the fluctuation amplitude should be modest, and the distribution should be close to uniform, allowing it to withstand statistical attacks. Figure 5 shows the plaintext image, plaintext image histogram, ciphertext image, and ciphertext image histogram information of Cameraman and House.

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As can be seen from the figures, the histogram of the original plaintext image fluctuates greatly, and the encrypted image’s histogram is very flat and approximately evenly distributed, which does not provide any favorable information. Therefore, the assailant will not be able to deduce any plaintext messages from the encrypted image and will be able to defend against the statistical analysis attack.

4.1.3. Correlation Analysis

The correlation between two neighboring pixels is called the correlation coefficient. It is one of the significant indexes in image encryption analysis. It can be used to show the amount to which image pixels have been substituted. The correlation in the original plaintext image is usually near to one, and it should be close to zero after encryption, suggesting that the encryption influence is significant. The calculation formula of the correlation coefficient is [34]where represent two neighbor pixels, and is the whole amount of pixels chosen in the plaintext image. Compute the correlation coefficients of all adjacent elements in the image, and average them to get the correlation coefficients in the horizontal direction, vertical direction, and diagonal direction of the image. Figures 6 and 7 are the distribution maps of neighbor pixels in horizontal, vertical, and diagonal directions of Cameraman plaintext image and encrypted Cameraman image, respectively. Table 1 is the correlation coefficient values of adjacent pixels of four plaintext images and encrypted images in all directions. As indicated in Table 2, the correlation coefficient obtained by the provided encryption algorithm is compared to other literature. As a result, the suggested encryption algorithm disrupts encrypted image correlation while providing strong uniform distribution performance.

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4.1.4. Information Entropy Analysis

Information entropy can estimate the allocation of gray values in an image. The more even the gray allocation is, the greater the information entropy of the image is. Plaintext image information distribution is uneven, making plaintext image data information easy to access by attackers. However, the distribution of encrypted images is uniform, and the unpredictability of images is strong. For completely ideal random images, their information entropy is 8. Assuming that represents an information source, the information entropy can be calculated by the following formula [38]:where indicates the total number of states of the information source and indicates the probability of a symbol appearing. Test several images to get the information entropy analysis shown in Table 3. The method put forward in this article is compared with other algorithms, and the comparison consequence is described in Table 4.

Tables 3 and 4 suggest that the information entropy of encrypted ciphertext information is closer to the ideal value of 8, and the pixels in the ciphertext are uniformly allocated. For the Cameraman image, the encryption scheme proposed in this article has greater information entropy, which proves that the attacker can obtain less useful information from the gray distribution, is more secure, and is not easy to disclose information, and has better performance than other encryption schemes.

4.2. Sensitivity Analysis

4.2.1. Key Sensitivity Analysis

Key sensitivity plays a crucial role in resisting violent attacks. On the one hand, a seemingly insignificant modification in the security key will result in completely different encrypted images. On the other hand, decrypting the encrypted image with a slightly altered key will not yield the correct decoded image.

To verify the sensitivity of the encryption key, use the set chaotic parameters to encrypt the plaintext image to obtain Figure 8(a). Then, a parameter in the Sine-Tent-Logistic chaotic system is changed and increased by ; ensure that other key parameters in the encryption algorithm remain unchanged, and then, encrypt the plaintext image again to get Figure 8(b). Make a difference between the two encrypted images to obtain Figure 8(c). The image illustrates that, while the encryption key changes slightly, the encrypted images generated by different encryption keys differ dramatically.

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In order to verify the decryption key sensitivity, parameter in the Sine-Tent-Logistic chaotic system is changed and increased by ; ensure that the decryption algorithm’s other key parameters remain unaltered. Figure 9 illustrates the decryption effect, which indicates that although the decryption key has a tiny modification, the ciphertext image cannot successfully decrypt and restore to the original image.

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4.2.2. Differential Attack

The sensitivity of the encryption scheme to plaintext determines its power to oppose the differential attack. The sensitivity of the encryption scheme to the original image can be evaluated by two indexes: pixel change rate and normalized average change intensity of pixel value . and separately indicate the proportion and degree of change in the pixel value of the encrypted image after altering a certain pixel value of the plaintext image at random. If the pixel values of the plaintext in ciphertext changes lead to a complete different image, which indicates that the method has a powerful ability to oppose the differential attack. For two plaintext images with only one pixel changed, let the pixels of in their ciphertext image be and separately; if , define ; or else, . The calculation formulae of and are [41]

The formulae for calculating the ideal expected value of and are [42]where and are the numbers of rows and columns of image pixels, respectively, and is the bit depth of image color. For an 8-bit grayscale image, is 8. The theoretical values of and are 99.6094 and 33.4635 , respectively. We simulate the Cameraman image by randomly selecting a pixel value at a fixed location in the image and assigning it a value between 0 and 255, and then calculate the and values using the formula mentioned above. The analysis of NPCR and UACI of different encrypted images and the comparison of differential attack of Cameraman are shown in Tables 5 and 6. As shown, the experimental results in this article are closer to the theoretical values, and the invention has strong sensitivity and strong resistance to difference.

4.3. Noise Attack

In the actual process of information transmission, since the transmission channel is insecurity, the data are susceptible to attack by noise, resulting in leakage of the information data and so on. Excellent encryption algorithm has better noise immunity attack performance and robustness. Use peak signal-to-noise ratio as an index to estimate the robustness of the algorithm. The mathematical formula is [45]where is the size of the image, is the plaintext, is the encrypted image, and represents the maximum pixel value in the image. Substituting the original plaintext and decrypted image into the above formula to calculate the value, the larger the , the better the decryption effect.

To verify the antinoise ability of the encryption algorithm presented in this article, add salt-peppers noise or Gaussian noise with different intensities to the encrypted image, and its value is calculated after decryption to judge whether it is robust or not.

4.3.1. Salt-Peppers Noise Attack

Add salt-peppers noise with the densities of 0.001, 0.005, 0.01, and 0.1 to the Barbara encrypted image; their scores are calculated as 37.9783, 31.0693, 27.6775, and 17.8212, respectively. The decrypted images are illustrated in Figure 10. It can be seen from Figure 10 that the decrypted image after adding noise can still clearly understand the information in the original plaintext image. The PSNR results after adding salt-peppers noise are shown in Table 7.

(a)

(b)

(c)

(d)

4.3.2. Gaussian Noise Attack

Add Gaussian noise with the densities of 0.005, 0.01, 0.05, and 0.1 to the Barbara encrypted image, and their scores are calculated as 15.0117, 13.7699, 11.2407, and 10.4507 respectively. Figure 11 shows the decrypted images. The results demonstrate that decrypted images can still be recognized despite a degree of noise attack.

(a)

(b)

(c)

(d)

4.4. Data Loss Attack

Encrypted images may lose information due to clipping attacks or transmission through network and storage, which causes difficulties in the receipt of ordinary images. For purpose of verifying the anticutting performance, we replaced different sizes and positions of the encrypted image with zero value. After the cropped encrypted image is decrypted, compare the decrypted image with the plaintext information. If the similarity between the two images is high, it proves that the scheme has a strong anticutting attack performance.

Cut out 1/16, 1/8, 1/4, and 1/2 of the Barbara encrypted image at different positions, and their scores are calculated as 19.7757, 16.8110, 13.8381, and 10.9201 respectively. The decrypted images are illustrated in Figure 12. The PSNR results after data loss are shown in Table 8.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

5. Conclusion

This article presented an image encryption scheme based on the artificial bee colony optimization algorithm. The significance of the encryption algorithm was to introduce the artificial bee colony optimization algorithm(ABC) after the bit-plane scrambling and block substitution. The ABC algorithm discovers the ciphertext image with the best encryption effect from initial population. A crossover operator is introduced in the ABC algorithm. This method considerably increases the population’s diversity throughout the optimization phase. Compared with other optimization algorithms, the ABC algorithm has its own special mechanisms—role switching, and positive and negative feedback mechanism and has a minimal number of parameters; it limits the influence of artificial parameter setting as much as possible and has significant robustness, which increases optimization efficiency and the final result. In the encryption phase, compared with other research studies, the algorithm improves the algorithm’s link with plaintext and its capacity to resist plaintext attack. The scrambling and diffusion encryption operations are intertwined, making the technique more safe. The experimental simulation results indicated that the encryption algorithm proposed in this article could efficiently cut down the correlation between adjacent pixels and improve the ciphertext information entropy, and the ciphertext image has high randomness and dispersion. Simultaneously, the encryption algorithm had eminent security and could oppose general representative attacks.

In future work, when moving algorithms to hardware systems, we have to deal with time consumption and storage issues, and there are three aspects that can be further improved. Firstly, different heuristic optimization algorithms can be combined to form a more efficient optimization algorithm applied in the field of image encryption; secondly, in the selection of fitness function, multiple evaluation indexes can be used for mixed calculation, which can make the optimization more perfect; finally, the optimization algorithm can be used to optimize the parameters or initial values of the chaotic system, which can make the obtained chaotic sequence more random, and the obtained chaotic sequence can be used to encrypt the image more secure.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

Erfu Wang conceived the study; Yanqi Zhou and Miaomeng Song helped with the methodology; Yanqi Zhou helped with software; Yanqi Zhou, Miaomeng Song, and Mengna Shi validated the study; Miaomeng Song and Mengna Shi curated the data and reviewed the literature; Yanqi Zhou prepared the original draft; Yanqi Zhou and Erfu Wang reviewed and edited the manuscript; Erfu Wang helped with the project administration; and Erfu Wang and Yanqi Zhou carried out the funding acquisition. All authors have read and agreed to the published version of the manuscript.

Acknowledgments

This work was supported by the Natural Science Foundation of Heilongjiang Province, China (No. LH2019F048), the Outstanding Youth Fund Project of Heilongjiang University: The Research on parallel compressed sensing encryption Algorithm based on sequence Signal Generator, Heilongjiang University (No. YJSCX2020-062HLJU), and the Natural Science Foundation of China (No. 61801173).

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Copyright © 2022 Yanqi Zhou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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