A Novel Image Encryption Algorithm Based on a Fractional-Order Hyperchaotic System and DNA Computing

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Introduction
Images, as one of the most popular media types, are widespread over various networks.How to prevent images from illegal copying and distribution in the era of the Internet is a critical issue.Therefore, image encryption has become one of the hottest research topics of information security in recent years.Although there exist some classical schemes such as Data Encryption Standard (DES), Advanced Encryption Standard (AES), and International Data Encryption Algorithm (IDEA) for information security [1], they usually cannot be directly applied to image encryption to yield satisfactory results due to some intrinsic properties of images such as bulky data capacity, strong correlation, and high redundancy [2][3][4].In contrast, the chaos-based image encryption has attracted much attention for research purposes and has been demonstrated to be effective and secure in recent years [5][6][7][8][9].
Chaotic systems have the following properties: pseudorandomness, extreme sensitivity to the initial values and system parameters, ergodicity, and unpredictability, which make it very suitable for image encryption [10].Typically, chaos-based image encryption framework includes chaotic sequence generation, pixel position permutation, and pixel value diffusion.One-dimensional (1D) chaotic systems have simple forms and are easy to implement, and thus some researchers used them to encrypt images.For example, the authors used two 1D chaotic Logistic maps to generate the pseudorandom sequence for image encryption in [11].Boriga et al. presented a new 1D chaotic map for real-time 2 Complexity image encryption [12].However, since the 1D chaotic systems usually have only one variable and a few parameters, along with relatively simple structures and chaotic orbits, it is easy to estimate the orbits and to predict the initial values and/or parameters by little information extracted from them [13].Therefore, in order to improve the security of image encryption, chaotic systems with two or more dimensions have been applied to image encryption.Fridrich put forward symmetric ciphers with two-dimensional (2D) chaotic maps and the experimental results demonstrated good diffusion properties with respect to the key and the plain image [14].Hua et al. proposed an image encryption algorithm using 2D Sine Logistic modulation map that has better properties of chaos when compared with some existing chaotic systems [15].Using the chaotic three-dimensional (3D) cat map extended from 2D Arnold's cat map [16] and 3D Chen's chaotic system [17], Chen et al. proposed a symmetric image encryption scheme for alternative permutation and diffusion [5].The Lyapunov exponent (LE) is a type of measurement methodology for chaotic level, and a chaotic system is said to be hyperchaotic if it has two or more positive LEs [18].Since hyperchaotic systems have more advantages such as richer dynamic phenomena and higher randomness than common chaotic systems, lots of hyperchaotic systems have been employed to encrypt images [19][20][21][22][23].For example, Norouzi et al. used the key stream generated by a hyperchaotic system to perform one round diffusion on the image to attain good results [20].A novel image encryption algorithm based on genetic recombination and hyperchaotic Lorenz system was put forward by Wang and Zhang [21].Yuan et al. proposed a parallel image cryptosystem by combining the Logistic map and a five-dimensional (5D) hyperchaotic system [23].
Most of the above-mentioned literature uses integralorder chaotic systems for image encryption.It has been reported that fractional-order hyperchaotic systems, as a counterpart of integral-order chaos, show higher nonlinearity and degrees owing to the complex geometrical interpretation of fractional derivatives for the nonlocal effects either in time or in space [24,25].Therefore, the fractionalorder hyperchaotic systems have great potential in information security.Wang et al. applied the fractional-order hyperchaotic Lorenz system to color image encryption.To enhance the security of images, both system parameters and derivative order were embedded in the scheme [25].The 3D fractional-order Lorenz system and Chen chaotic systems were employed to encrypt images by Wu et al. and Zhao et al., respectively [3,26].Huang et al. used a four-dimensional (4D) fractional-order hyperchaotic neural network system to cipher color images, and the experiments demonstrated the effectiveness of the system [27].
Most image encryption algorithms are performed on pixel-level or bit-level data.With the development of bioinformatics, some image encryption algorithms based on deoxyribonucleic acid (DNA) have emerged since Adleman completed the first experiment on DNA computing [28], due to the properties of DNA: massive parallelism, huge storage, and ultralow power consumption [29][30][31][32][33][34].Typically, DNA-based image encryption consists of three steps: DNA encoding, DNA operations, and DNA decoding.The bit stream of images is encoded as DNA sequences with some encoding rules in the step of DNA encoding.Then, different DNA operations such as addition, subtraction, and exclusive OR (XOR) are performed on DNA.The types of both encoding and operations are usually determined by chaotic sequences.Finally, the results of DNA operations are decoded to bits with the counterpart of corresponding encoding rules.Zhang et al. used Logistic maps and two DNA operations (addition and complement) to encrypt image blocks, but the blocks led to low robustness against noise [33].In the RGB image encryption scheme by Liu et al., DNA addition and complement operations were carried out on each channel of RGB image with the DNA sequence matrix generated from Logistic map [34].Zhan et al. jointly used a hyperchaotic system and DNA computing to encrypt images, where the hyperchaotic sequence was applied to all steps.However, two important evaluation standards, that is, the number of pixels change rate (NPCR) and the unified average changing intensity (UACI), still need to be improved [29].
Motivated by the above analysis, this paper aims at proposing a novel image encryption algorithm that incorporates the fractional-order hyperchaotic Lorenz (FOHCL) system and fractional-order hyperchaotic Lorenz DNA (FOHCLDNA) computing in order to improve the security of image encryption.Specifically, the proposed FOHCLDNA is mainly composed of six stages: (1) the FOHCL is firstly applied to generating the pseudorandom sequence for encryption; (2) global pixel diffusion, global pixel permutation, and 2D permutation are carried out on pixels; (3) global bit permutation and 3D permutation are conducted on bits; (4) the bit stream of image is encoded as DNA sequence according to the encoding rules decided by the hyperchaotic sequence; (5) one of the DNA operations (addition, subtraction, or XOR) is carried out on each acid base, and, at the same time, global DNA permutation and 3D permutation further improve the security.Both the types of DNA operations and the orders of DNA permutation are determined by the hyperchaotic sequence; (6) the encrypted DNA sequence is decoded to bit stream, followed by bit-topixel decoding.Finally, the encrypted image is obtained.The main contributions of this paper are four aspects: (1) different from most existing literature that uses integral-order chaotic or hyperchaotic systems, the proposed FOHCLDNA uses a fractional-order hyperchaotic system for image encryption; (2) a simple but effective pixel diffusion is proposed; (3) permutation is carried out at different levels, that is, pixels, bits, and acid bases, while both the DNA encoding rule and DNA operation for each acid base are determined by corresponding hyperchaotic sequence; (4) extensive experiments demonstrate that the FOHCLDNA is promising for image encryption.The novelty of this paper is threefold: (1) it is a good attempt to integrate fractional-order hyperchaotic system and DNA computing to enhance the security of image encryption; (2) the simple pixel diffusion can spread the little change in one pixel to all other pixels; (3) several permutation operations performed at different levels can further improve the security.
The remainder of this paper is organized as follows.A brief description of the fractional-order hyperchaotic

Deoxyribonucleic Acid (DNA)
Computing.Deoxyribonucleic acid (DNA) is a kind of biological macromolecule, and the knowledge of DNA sequence is widely used in genetic engineering, biotechnology, and identification.An individual DNA sequence is mainly composed of carbohydrate and four different nucleic acid bases: A (Adenine), G (Guanine), C (Cytosine), and T (Thymine), where T and A; C and G are complementary pairs.The number of DNA coding combinations is 4! = 24 in total, which only have eight kinds of DNA bases legally to meet the DNA complementary rules, as shown in Table 1 [29,39].In the theory of binary system, 0 and 1 are complementary pairs such as 00 (0) and 11 (3), 01 (1) and 10 (2).Each pixel value of grayscale image in binary sequence is 8 bits, and if every 2 bits is represented by a letter, the representation of a pixel would be a 4-length nucleotide string.For example, for a pixel value 161 in decimal, its binary combination is "10010011," and the corresponding DNA sequence is "GCAT" by adopting the first encoding rule.
If any other DNA coding rules are used to code the same binary sequence, the result will definitely be different.

Image Encryption Scheme
3.1.Hyperchaotic Sequence Generation.Since fractionalorder hyperchaotic systems have good properties for image encryption, we use the FOHCL system described in Section 2.1 for generating the hyperchaotic sequence.The generating process is comprised of three steps.
Step 1.To eliminate the adverse effects, the FOHCL system is firstly iterated  0 times and then the generated sequence is removed.Step 2. The FOHCL system continues to iterate  = ⌈(30ℎ+ 3(ℎ+)+13)/4⌉ times, where ⌈⋅⌉ denotes the ceiling operation and ℎ and  denote the width (column) and the height (row) of the image to encrypt, respectively.For the th iteration, four state values denoted by   = { Step 3.After the whole iteration, the fractional-order hyperchaotic sequences  can be obtained by concatenating all the   ( = 1, 2, . . ., ) as ( The purposes of the generated sequence  for encryption are two aspects: (1) sorting subsequence of  to get the index of original data for permutation; (2) using subsequence of  to change the values of images for diffusion.In our scheme, for the first purpose, we directly use the original values of  for sorting while, for the second purpose, we map the hyperchaotic subsequence of  with  values to the integer range of [0, 255] by where   is the th integer in the generated integer sequence, mod is the modulo operation, | ⋅ | is the absolute value operation, and ⌊⋅⌋ denotes flooring operation [29].

Global Pixel Diffusion.
In our scheme, we carry out a simple two-step diffusion for image on pixels.Specifically, for a given image  of size ℎ × , we can transform the image into a 1D pixel sequence  = {  },  = 1, 2, . . ., , where  = ℎ × .Suppose we have an initial key  0 and a key sequence  = {  ∈ [0, 255]},  = 1, 2, . . ., ; the first-step diffusion can be described as follows: and the second-step diffusion can be formulated as follows: and in both ( 4) and ( 5), ⊗ is XOR operation and  is the result of pixel diffusion.

Global Permutation and 2D and 3D
Permutation.In this approach, several permutations are carried out at different levels, that is, pixel level, bit level, and DNA level.For an image of size ℎ × , global pixel/bit/DNA permutation means permuting all pixels/bits/DNA with corresponding hyperchaotic subsequences.Since pixel-level data is a 2D plane of size ℎ × , we can permute the image firstly by row and then by column, which is called 2D permutation.Bit-level and DNA-level data are a 3D cube of size ℎ ×  × 8 and ℎ ×  × 4, respectively, and we can permute the image by row, column, and depth, respectively, which is called 3D permutation in this paper.Specifically, the global permutation can be summarized as follows.
Step 2. Extract a subsequence with the length of  from the hyperchaotic sequence .Sort the subsequence in ascending order to get the index sequence   ,  = 1, 2, . . ., .
Step 3.According to   , rearrange the vector V to get the new vector V  by The 3D permutation is to permute planes at different directions in 3D spaces.The operations in each direction are very similar.For simplicity, here we only give the operation in the direction of width as follows.
Step 1. Extract a subsequence of length  = ℎ from the hyperchaotic sequence .Sort the subsequence in ascending order to get the index sequence   ,  = 1, 2, . . ., .
Step 2. According to   , rearrange the plane  to get the new vector   by It is clear that the 2D permutation is a special case of the 3D permutation.The proposed scheme extracts subsequence with  = ℎ + , ℎ +  + 8, and ℎ +  + 4 from  for 2D pixel permutation, 3D bit permutation, and 3D DNA permutation, respectively.Note that the proposed global permutation and 3D permutation at bit-level data or DNA-level data can change the positions of bits or DNAs to lead the values of pixels changed, and thus they can simultaneously permute and diffuse the images at bit-level or DNA-level data.

FOHCLDNA: The Proposed Image Encryption Scheme.
With the above-mentioned description, the flowchart of the proposed FOHCLDNA is shown in Figure 2, and the details are as follows.
Step 1.Let ℎ ×  denote the size of the input image .Concatenate the four sequences generated by (1) to compose one hyperchaotic sequence  using (2).
Step 2 (conduct pixel-level encryption).Extract the first ℎ+ items from  to carry out row and column permutation (2D permutation) on  to obtain  0 .Extract the next ℎ ×  items from  to carry out global pixel permutation on  0 to obtain  1 .Extract the next ℎ ×  + 1 items from  to compose a new sequence  0 and then map  0 to the integer range of [0, 255] by (3) to obtain sequence  1 .Use the first item in  1 as the initial value and the rest as the key to carry out global pixel diffusion on  1 to obtain  2 .
Step 3.  2 is encoded to a bit sequence  0 .
Step 4 (conduct bit-level encryption).Use the next ℎ ×  × 8 items from  to perform global bit permutation on  0 to obtain  1 .Use the next ℎ +  + 8 items from  to perform 3D permutation on  1 to obtain  2 .
Step 5 (DNA encoding).Map the next ℎ ×  × 4 items in  to the integer range of [0, 255] by (3) to obtain sequence  2 .Encode the th pair bits in  2 with the DNA rule decided by (8) to obtain  0 : where  2  denotes the th item in the sequence  2 . Step where op denotes the type of DNA operation.When op equals 1, 2, and 3, the corresponding DNA operation is ++, −−, and ⊗⊗, respectively.Extract the next ℎ ×  × 4 items from  to carry out global DNA permutation on  2 to obtain  3 .
Decode the th DNA in  2 with the DNA rule decided by (10) to obtain a binary sequence  3 : Note that for a specified acid base, the encoding rule to generate it and the decoding rule to decode it may be different because of the difference between  2 and  4 .
Step 8.The binary sequence  3 is converted to the cipher image .
The decryption process is the reverse version of the encryption process.
The proposed FOHCLDNA enhances the security of images in several aspects.Firstly, the hyperchaotic sequence with high nonlinearity and complex dynamics generated by the FOHCL is used throughout the process of image encryption.Secondly, the global pixel diffusion can spread little change in one pixel to all the other pixels, leading to a good result of diffusion.Thirdly, permutations are performed on different levels, that is, pixels, bits, and acid bases.Specially, for each two adjacent bits in a pixel and each acid base, a unique DNA rule determined by the hyperchaotic sequence is used to perform DNA encoding and DNA decoding, respectively.The operation type of each acid base is also decided by the sequence when the image is operated with the DNA mask image.All these properties enhance the security of images.

Experimental Settings.
In order to evaluate the performance of the proposed FOHCLDNA, we compare it with some state-of-the-art schemes, such as the hyperchaotic and DNA sequence-based method (HC-DNA) [29], the image encryption using cipher diffusion in crisscross pattern (CDCP) [40], and a class hyperchaos-based scheme (CHC) [41].The parameters for the compared schemes are set as given by the authors.We set the parameters of the FOHCLDNA as follows.The initial values for the FOHCL system are  0 1 = 12,  0 2 = 22,  0 3 = 31, and  0 4 = 4.And

Security Key Analysis
. Key space and sensitivity to secret key are two essential points in encryption.A good encryption scheme should have an enough large key space and be extremely sensitive to any small changes in its security key.Both a large key space and extreme sensitivity can resist brute-force attacks.

Key Space.
Basically, the security keys of the proposed FOHCLDNA are composed of 4 initial values, that is, ( 0 1 ,  0 2 ,  0 3 ,  0 4 ).If the precision of each initial value is 10 −16 , the key space size is 10 16×4 = 10 64 ≈ 2 212 .From the view of cryptology, the size of the key space larger than 2 100 is capable of providing a high-level security [1,42].Therefore, the key space of the FOHCLDNA is large enough to resist brute-force attacks from current computers.In addition, the fractional orders of the FOHCL can also be used as keys to further enhance the key space.

Sensitivity to Secret Key.
The extreme sensitivity of an image encryption algorithm implies that even one bit changed in the keys will lead to a completely different encrypted image.In other words, if the security key changes a little, the decrypted image will be completely different from the input image.
To demonstrate the sensitivity to secret key of the FOHCLDNA, we decrypt the cipher images twice.In the first run, we use the exact encryption keys ( 0 1 = 12,  0 2 = 22,  0 3 = 31,  0 4 = 4) to decrypt the cipher images, while, in the second run, we attempt to decrypt the cipher images with slightly different keys ( 0 1 = 12+10 −15 ,  0 2 = 22,  0 3 = 31, and  0 4 = 4).We conduct the experiments on the images of Lena, Circuit, Peppers, and Plane, and the results are shown in Figure 3.As we can see, the decrypted images with the slightly different keys are completely different from those decrypted with the correct keys, showing that the proposed FOHCLDNA has high sensitivity to secret key.

Statistical Analysis.
A good cryptosystem should have the ability to resist all kinds of statistical attacks.Hence statistical analysis is another widely used and effective way to analyze a cryptosystem.Typical statistical analysis includes histogram analysis, information entropy, and correlation analysis.

Histogram Analysis.
For image encryption, histogram is a popular tool to measure the distribution of pixel values in the plain image and the cipher image.The histogram of a plain image is usually unevenly distributed while that of cipher image by a good encryption scheme should be close to a uniform distribution.To put it another way, as far as the effectiveness of encryption schemes, the flatter the histogram of the cipher image is, the better the encryption scheme is.
The histograms of the plain images and their corresponding cipher images are shown in Figure 4.It can be seen from Figure 4 that the histograms of the plain images are irregularly distributed while all those of cipher images are very close to a uniform distribution.The results demonstrate that the proposed FOHCLDNA can resist histogram attacks.

Information Entropy. Information entropy (IE) is used
to reflect the complexity of a system.For the 8-bit grayscale images used in the experiments, their intensity has 2 8 kinds of possible values ([0, 255]).The IE can be defined as where () is the probability that the pixel value  appears [29].
When each pixel of cipher image has the same probability, that is, 1/256, IE reaches the ideal value 8.The IEs of input images and cipher images are shown in Table 6.It can be seen that the IEs of input images are far below 8, while those of cipher images are very close to the ideal value.Among the encryption schemes, the FOHCLDNA achieves 4 out of 8 optimal values while all the IEs by HC-DNA are less than those by any other schemes.It is demonstrated that the FOHCLDNA is secure enough to resist entropy attacks.

Correlation Analysis.
Two adjacent pixels in a natural image usually have high correlation.A good image encryption algorithm should be capable of reducing such correlation dramatically.Correlation coefficient  is a popular metric to measure the correlation that can be formulated as follows [25]:  where  and  are the gray levels of two neighbouring pixels in an image and  denotes the total number of pixels involved in the image.
To analyze the correlation of the image encryption schemes, we firstly calculate the correlation coefficients for all input images and cipher images in different directions, that is, horizontal  ℎ , vertical  V , and diagonal   , respectively [29], as listed in Table 7. From this table, it can be found that the correlation coefficients of all the input images are close to 1 in all directions, while those of the cipher images are round 0, showing that the encryption schemes can dramatically reduce the correlation of the adjacent pixels of the images.Specifically, the FOHCLDNA outperforms the rest schemes on 7 out of 24 correlation coefficients, whereas the HC-DNA achieves the optimal value only three times.
To have a further correlation analysis, we randomly select 4000 pairs of adjacent pixels in horizontal direction from each input image and corresponding cipher image by the FOHCLDNA, respectively, to show their adjacent-pixel distribution maps in Figure 5.It can be seen that the values of input images are distributed near the diagonal of coordinate plane, indicating strong correlation of input images.However, the correlation is completely destroyed by the FOHCLDNA so that the values of cipher images are distributed over almost the whole plane, showing very weak correlation in cipher images.

Analysis of Resisting Differential Attacks.
According to the theory of cryptography, a good image encryption algorithm should also be very sensitive to the plain images; that is, a little change (e.g., a bit change) in a plain image can lead to a completely different cipher image.An image encryption scheme that has such a property can effectively resist differential attacks.
The number of pixels change rate (NPCR) and the unified average changing intensity (UACI) are two important metrics for differential attack analysis.NPCR is defined as the variation ratio of two cipher images when the value of a pixel in the input image is slightly changed.UACI indicates the average intensity of the differences between the same cipher images.Mathematically, NPCR and UACI between  two cipher images  1 and  2 can be formulated as ( 13) and ( 14), respectively: where ℎ and  are the height and the width of the image, respectively, and   is defined as follows: Generally speaking, the more NPCR gets close to 100% and the bigger UACI is, the more encryption scheme becomes effective in resisting differential attacks.For a 256-level gray image, the maximum theoretical values for NPCR and UACI are 99.6094% and 33.4635%, respectively [29].We randomly change one bit in the plain images to compute one value of NPCR and UACI.We repeat the process 10 times and report the average values of NPCR and UACI in Tables 8 and 9, respectively.
It can be shown that although the values by the FOHCLDNA are not as good as those by CDCP and CHC, they are very close to the maximum theoretical values.The FOHCLDNA apparently outperforms HC-DNA in terms of NPCR and UACI.The results show that the global pixel diffusion is effective and the FOHCLDNA has the capability of resisting differential attacks.

Conclusions
In this paper, we propose a novel image encryption algorithm based on a fractional-order hyperchaotic Lorenz system and DNA computing (FOHCLDNA).The fractional-order hyperchaotic Lorenz system is adopted to generate the pseudorandom sequence that is utilized throughout the process of encryption.Besides pixel-level and bit-level operations, DNA operations such as DNA addition, DNA subtraction, and DNA XOR are also introduced to the algorithm.A simple pixel diffusion is used to spread the slight change in one pixel to all other pixels.Several types of permutation are carried out on different level data.Through the results of extensive experiments and corresponding security analysis,

Figure 2 :
Figure 2: The flowchart of the proposed FOHCLDNA algorithm.

Figure 4 :
Figure 4: Histograms of the plain images and their corresponding cipher images.The first column is plain images.The second column is the histograms of the plain images.The third column is cipher images.And the fourth column is the histograms of the cipher images.

Figure 5 :
Figure 5: The adjacent-pixel distribution maps of the input images and the corresponding cipher images in horizontal direction.

Table 1 :
Encoding and decoding rules of DNA.
6(conduct DNA-level encryption).Use the next ℎ++4 items from  to perform 3D permutation on  0 to obtain  1 .Map the next ℎ ×  × 4 items in  to the integer range of [0, 255] by (3) to obtain sequence  3 .Create a mask DNA matrix  of size ℎ ×  × 4 using the next ℎ ×  × 4 items in .For the th DNA in  1 , conduct DNA operation with th DNA in  to obtain  2 .The operation type is decided by op

Table 5 :
Testing images.0 are set to 10000.All the fractional orders   ,  = (1, 2, 3, 4) are set to a fixed value 0.98.Eight publicly accessed images with different sizes are used to test the proposed FOHCLDNA, as listed in Table5.All the experiments were conducted by Matlab 8.6 (Mathworks, Natick, MA, USA) on a 64-bit Windows 7 (Microsoft, Redmond, WA, USA) with 32 GB memory and 3.4 GHz I7 CPU.

Table 6 :
The IE of the testing images.

Table 7 :
The correlation coefficients  of the testing images.

Table 8 :
The average NPCR (%) of running the schemes 10 times.