A New Hyperchaotic Image Encryption Scheme Based on DNA Computing and SHA-512

1. Introduction

With the rapid growth of smartphones and digital cameras, massive images are generated every day in the world. They are easily transmitted on the Internet, which is an insecure channel. It is extremely important to protect sensitive images from hacker attacks during communication. Steganographic algorithms [1-2], watermarking techniques [3-5], and encryption methods [6-10] are three important techniques in terms of protecting private images. Images are famous for special characteristics: data redundancy, high correlation, large storage space, and uneven energy distributions.Traditional text-based encryption algorithms could not be used to encrypt the image, such as AES [11], DES [12], and RSA [13].

Chaos has some special merits: ergodicity, pseudo randomness, and sensibility to initial states. It is widely applied for image encryption [14-20]. Many encryption schemes have been proven to be insufficiently secure and have been cracked [21-28]. The hyperchaotic system has some special characteristics. It has more system variables and parameters, more complicated structures, and better dynamic behaviors than the low-dimensional chaotic system. [29, 30]. Huang et al. [9] proposed a chaotic image encryption method that was based on the game of life and plaintext. Two scrambling matrices were constructed to scramble the plain image. A 4D hyperchaotic system and the game of life were employed to diffuse the image. Musanna and Kuma [14] proposed a novel image encryption method based on a fractional order chaotic system. The chaotic sequences were produced with the 3D chaotic system and the Fisher-Yates method. The plain image was segmented into different blocks of the same size. A 3D hyperchaotic system was employed to scramble the image. The generated chaotic sequences were adopted to diffuse the shuffling image. Fisher-Yates was applied to generate a chaotic matrix. Sun and Chen [18] designed a new cryptosystem based on SHA-256 and chaotic theory. Noise-like pixels were randomly inserted into the plaintext image and generated a hash value with SHA-256.The authors proposed a segmented coordinate descent scheme.

DNA molecules have some special characteristics, such as huge storage space, high parallelism, strong adaptability, and low power consumption. The DNA technique is very suitable for encrypting the image [31-36]. Mansoor et al. [31] presented an image encryption scheme based on chaos theory and DNA computing. The plain image was scrambled with the logistic map and the tent map. DNA computing was performed to encrypt the plain image. Chai et al. [35] introduced an image encryption scheme that was based on the DNA technique and the chaotic technique. The hash value was calculated on the basis of the plain image and applied to compute the initial states and system parameters. The DNA matrix was produced with chaotic sequences. A color image encryption method was designed in Ref. [36]. It adopted a hyperchaotic system and DNA computing. Firstly, four chaotic sequences were produced with a 2D hyperchaotic system. Then an image cube was constructed using image blocks. Finally, DNA operations were applied to the plain image.

Section 2 introduces related techniques. Chaotic sequences are generated in Section 3. Section 4 displays the proposed scheme. The simulation results are shown in Section 5. The security test is discussed in Section 6. Section 7 concludes the manuscript.

2. Related Techniques

2.1. Six-Dimensional Hyperchaotic System

In the manuscript the chaotic sequences are produced with hyperchaotic system. The dimension of the system is six. It is defined as [37]:

$$\left\{\begin{array}{l}\stackrel{.}{x_1}={\delta }_1(x_2-x_1)+x_4\\ {\stackrel{.}{x}}_2={\delta }_2{x}_1-x_1{x}_3+x_4\\ {\stackrel{.}{x}}_3=x_1{x}_2-x_3-x_4\\ {\stackrel{.}{x}}_4=-{\delta }_3(x_1+x_2)+x_5\\ {\stackrel{.}{x}}_5=-x_2-{\delta }_4{x}_5+x_6\\ {\stackrel{.}{x}}_6=-{\delta }_5(x_1+x_5)\end{array}\right.$$

(1)

where${\delta }_1$,${\delta }_2$,${\delta }_3$,${\delta }_4$, and${\delta }_5$are the system parameters.

Its Lyapunov exponents (LE) are 1.4620, 0.1433, 0.0725, 0.0449, 0 and -12.0700 if (${\delta }_1$,${\delta }_2$,${\delta }_3$,${\delta }_4$,${\delta }_5$) = (10,76,3,0.2,0.1) . There are four positive LEs in system (1), and it is a hyperchaotic system. The bifurcation diagram of system (1) is shown as Figure 1.

2.2. DNA Coding Rules

There are four nucleic acid bases in a DNA sequence. They are C (cytosine), A (adenine), T (thymine), and G (guanine). Watson and Crick proposed the DNA pairing rules [32]. A and C are complementary to T and G. Two binaries11, 10, 01 and 00 can be adopted to encode four bases. There are eight DNA coding rules, which are presented in Table 1. These coding rules could be used to encode and decode DNA sequences.

Suppose a pixel value is 156 and its binary is 10011100. If the encoding and decoding rules are 3 and 7 respectively, then the encoding and decoding results are shown in Figure 2.

2.3. Modern Dna Complementary Rules

The modern DNA complementary rules are satisfied as [33]:

where W(y) represents a base pair of y and is different from y.

Six modern DNA complementary rules are displayed [35]:

  Rule 1. T→A , A→G , G→C , C→T

  Rule 2. T→A , A→C , C→G , G→T

  Rule 3. T→C , C→A , A→G , G→T

  Rule 4. T→C , C→G , G→A , A→T

  Rule 5. T→G , G→A , A→C , C→T

  Rule 6. T→G , G→C , C→A , A→T

2.4. DNA Computing

Some algebraic and biology operations are proposed for DNA computing. There are DNA addition operation, DNA subtraction operation, and DNA XOR operation. Due to the existence of eight DNA coding rules, there are eight DNA addition, DNA subtraction, and DNA XOR rules. They are shown respectively in Table 2, Table 3 and Table 4.

3. Generation of Chaotic Sequences

The plain image with size M × N is computed with SHA-512, and the hash value W with 512-bit is generated. If an image is changed one bit and a new image is generated, then they will produce totally different hash values [38]. The initial states of system (1) are calculated using secret keys and the hash value W. W is split into many 8-bit data blocks. Each one is changed into a decimal number and is represented as w₁, w₂, …, w₆₄ .

The initial values will be achieved as follows:

$$w{v}_i=\mathrm{mod}(w_{9i-8}+w_{9i-7}+w_{9i-6}+w_{9i-5}+w_{9i-4}+w_{9i-3}+w_{9i-2}+w_{9i-1}+w_{9i},256)$$

(3)

$$x_1(0)={\displaystyle \frac{w{v}_1}{255}}+{\displaystyle \frac{{\displaystyle {\sum }_{l=55}^{64}{w}_l}}{10\times 255}}+s{k}_1$$

(4)

$$x_j(0)={\displaystyle \frac{w{v}_j}{255}}+x_{j-1}(0)+s{k}_j$$

(5)

where sk_i is the secret key, i$\in${1, 2, 3, 4, 5, 6}; j$\in${2, 3, 4, 5, 6}.

2. First iterate system (1) 600 times.

3. Continue to perform 4MN iterations.

4. New sequences z₁, z₂, z₃, z₄, z₅, z₆ and z₇ are achieved.

$$z_1(j)=\mathrm{mod}((abs(100x_1(j))-floor(abs(100x_1(j))))\times {10}^{12},4MN)$$

(6)

$$z_2(j)=\mathrm{mod}((abs(100x_2(j))-floor(abs(100x_2(j))))\times {10}^{12},8)+1$$

(7)

$$z_3(j)=\mathrm{mod}(floor(abs(x_3(j))\times {10}^{15}),4)$$

(8)

$$z_4(j)=\mathrm{mod}(floor(abs(x_4(j))\times {10}^{15}),6)+1$$

(9)

$$z_5(j)=\mathrm{mod}(floor(abs(x_5(j))\times {10}^{15}),3)+1$$

(10)

$$z_6(j)=\mathrm{mod}((abs(x_5(j))-floor(abs(x_5(j))))\times {10}^{15},8)+1$$

(11)

$$z_7(j)=\mathrm{mod}(floor(abs(x_6(j))\times {10}^{15}),256)$$

(12)

where j=1, 2, …, 4MN; abs means an absolute function; mod symbols a modulus function; floor(z) receives an integer equal to or less than z.

4. Generation of Chaotic Sequences

1. Divide the plain image with an 8-bit pixel into four 2-bit parts. It becomes a new image P₁ with size M × 4N. P₁(i, j)$\in${0, 1, 2, 3}, 1≤ i ≤M and 1≤ j ≤4N.

2. Exchange pixel P₁(i, j) with pixel P₁(i′, j′).

$$i^{\text{'}}=floor({\mathrm{z}}_1(i,j)/4N)+1$$

(13)

$$j^{\text{'}}=z_1(i,j)\mathrm{mod}4N+1$$

(14)

where 1≤ i ≤M and 1≤ j ≤4N.

3. Convert the matrix P₁ into a binary sequence Q= [Q(1), Q(2), …,Q(4MN)].

4. Encode binary sequences Q and z₃ into DNA sequences U and z₃ according to the chaotic sequence z₂.

5. Modern DNA complementary rules are performed on the DNA sequence U.

$$G(i)=\left\{\begin{array}{l}Rule1(U(i))\hspace{1em}if{z}_4(i)=1\\ Rule2(U(i))\hspace{1em}if{z}_4(i)=2\\ Rule3(U(i))\hspace{1em}if{z}_4(i)=3\\ Rule4(U(i))\hspace{1em}if{z}_4(i)=4\\ Rule5(U(i))\hspace{1em}if{z}_4(i)=5\\ Rule6(U(i))\hspace{1em}if{z}_4(i)=6\end{array}\right.$$

(15)

where i = 1, 2, …, 4MN.

6. DNA computation is executed on the DNA sequences G and z₃. The sequence H is decided by the chaotic sequence z₅.

$$H(j)=\left\{\begin{array}{l}G(j)+z_3(j)\hspace{1em}if{z}_5(j)=1\\ G(j)-z_3(j)\hspace{1em}if{z}_5(j)=2\\ G(j)\oplus z_3(j)\hspace{1em}if{z}_5(j)=3\end{array}\right.$$

(16)

where j = 1, 2, …, 4MN.

Decode the DNA sequence H into binary sequences J and convert J to the decimal sequence K.

Diffuse the sequence K and obtain the sequence C.

$$C(l)=\left\{\begin{array}{l}K(l)\oplus \mathrm{mod}({\displaystyle \sum _{i=1}^{MN}P(i),}256)\oplus \mathrm{mod}({\displaystyle \sum _{j=1}^{64}w(j),}256)\oplus z_7(l)\hspace{1em}ifl=1\\ K(l)\oplus \mathrm{mod}({\displaystyle \sum _{t=1}^{6}s{k}_t\times {10}^{15}},256)\oplus C(l-1)\oplus z_7(l)\hspace{1em}ifl=2\\ K(l)\oplus C(l-1)\oplus C(l-2)\oplus z_7(l)\hspace{1em}ifl\in [3,MN]\end{array}\right.$$

(17)

Sequence C is converted into matrix D and finally the encrypted image is obtained.

The procedure of the presented scheme is briefly described in Figure 3.

The decryption method is the opposite of the encryption procedure. It is not discussed in detail.

5. Simulation Results

The proposed algorithm runs on MATLAB, 16G of RAM, 3.2 GHz CPU. The experimental images are ‘Lena’, ‘Baboon’, ‘Peppers’ and ‘Testpat’ with size 256×256. The simulation results are displayed in Figure 4. Their histograms are revealed in Figure 5.

6. Security Test

6.1. Key Space Test

If the key space of an encryption scheme is more than 2¹⁰⁰, then it will be able to withstand exhaustive attacks [38]. In this manuscript, the secret key is composed of the hash value W and the given values sk_i, i=1, 2, …, 6. Supposing the computation accuracy of 10^-15 [39], the key space is almost 2²⁵⁶×(10¹⁵)⁶ ≈ 2²⁵⁶×2²⁹⁹ = 2⁵⁵⁵. It is far more than 2¹⁰⁰ and the designed algorithm could withstand exhaustive attack.

6.2. Histogram Test

The theoretical histogram of an encrypted image is flat and noise-like. Attackers could not find anything from the histogram. Figure 5 shows the uniform and flat pixel value distribution in encrypted image. It represents that the proposed algorithm could withstand count attack. The Chi square distribution [40] is usually used for histogram analysis. It is defined as:

$${\chi }^2={\displaystyle \sum _{l=0}^{255}\frac{{(F_l-A)}^2}{A}}$$

(18)

$$A={\displaystyle \frac{M\times N}{256}}$$

(19)

where F_l is the rate of the gray level l.

If the confidence level is set as 0.05 and the value of ${\chi }^2$is less than${\chi }_{0.05}^2$(255) =293.25 [41], then the distribution of the histogram will be very consistent. The result of the Chi square distribution is shown in Table 5.

From Table 5, it reveals that four images can pass the Chi-square examination. Therefore, the histogram distribution of the encrypted image is different from that of the plain image. The designed algorithm is capable of resisting the histogram analysis attack.

6.3. Key Sensitive Test

Key sensitive test plays an important role in security analysis. The hash value W is produced with a plain image. Since the initial conditions of the cryptosystem are generated with hash value and secret keys, the encrypted result is tightly connected with the original image. The Baboon image is used to display the simulation performances. A bit is changed in W and forms a new W₁.

One of the keys is marginally revised (γ=10^-15) and the other keys remain the same. The new key is applied to encrypt and decrypt the image

From Figs. 6 and 7, it shows that even if there is a tiny discrepancy between two images or secret keys, the obtained results will be diverse. The discrepancy of the encrypted images C1 and C2 is about 99.6%. Figure 8 reveals that the incorrect key could not correctly decrypt the image. It proves that the proposed scheme could withstand sensitive analysis.

Figure 6. Sensitive analysis to the hash value in the encryption process: (a) plain image, (b) encrypted image C1, (c) encrypted image C2, (d) |C1-C2|, (e) - (h) are the corresponding histograms of (a) - (d).

Figure 6. Sensitive analysis to the hash value in the encryption process: (a) plain image, (b) encrypted image C1, (c) encrypted image C2, (d) |C1-C2|, (e) - (h) are the corresponding histograms of (a) - (d).

Figure 7. Sensitive analysis of the keys (γ=10^-15) in the encryption procedure: (a) |Enc(sk₁+γ)-C1|, (b) |Enc(sk₃-γ)-C1|; (c) |Enc(sk₅+γ)-C1|, (d) |Enc(sk₆-γ)-C1|.

Figure 7. Sensitive analysis of the keys (γ=10^-15) in the encryption procedure: (a) |Enc(sk₁+γ)-C1|, (b) |Enc(sk₃-γ)-C1|; (c) |Enc(sk₅+γ)-C1|, (d) |Enc(sk₆-γ)-C1|.

Figure 8. Sensitive analysis of secret keys (γ=10^-15) in the decryption procedure: (a) encrypted image C1, (b) the right decrypted result, (c) |Dec(C1, sk₁-γ)|, (d) |Dec(C1, sk₃+γ)|.

Figure 8. Sensitive analysis of secret keys (γ=10^-15) in the decryption procedure: (a) encrypted image C1, (b) the right decrypted result, (c) |Dec(C1, sk₁-γ)|, (d) |Dec(C1, sk₃+γ)|.

6.4. Correlation Test

The correlation coefficient (CR) can be defined as [35]:

$$C{R}_{AB}={\displaystyle \frac{{\displaystyle {\sum }_{i=1}^L(A_i-E(A))(B_i-E(B))}}{\sqrt{({\displaystyle {\sum }_{i=1}^L{(A_i-E(A))}^2})({\displaystyle {\sum }_{i=1}^L{(B_i-E(B))}^2})}}}$$

(20)

where A_i and B_i represents the adjacent pixel; L is the total number of pixels.

Some neighboring pixel pairs are randomly chosen to compute the CR values in vertical (V), horizontal (H), and diagonal (D) directions. Figure 9 illustrates the correlation of adjacent pixels in the Baboon image and the corresponding encrypted image. Table 6 shows the CR values in three directions, and Table 7 presents comparisons of the CR values using different algorithms.

Table 6 reveals that the correlation of neighboring pixels in three directions is approaching 0. It symbols that the proposed scheme could effectively defend on the correlation analysis attack. Table 7 indicates that the suggested scheme is more effective than some existing methods.

6.5. Shannon Entropy Test

The global Shannon entropy (GSE) is often applied to test randomness. It can be computed as follows:

$$\mathrm{GS}E(n)=-{\displaystyle \sum _{j=0}^{255}p(n_j){\log }_{2}p(n_j)}$$

(21)

where n_j means the j-th information source; p(n_j) symbolises the probability of n_j. For an 8-bit grayscale image, the theoretical value of GSE is 8.

The local Shannon entropy (LSE) is proposed by Wu et al. [46] and adopted to measure randomness from a local perspective. It is calculated as

$$L\mathrm{S}{E}_{r,L}(B)={\displaystyle \sum _{i=1}^r\frac{GIE(B_i)}{r}}$$

(22)

where B_i is a randomly chosen and non-overlapped data block. There are L pixels in each block. In this paper, (r, L) is appointed to (30,1936). If the LSE value lies within the interval [$v_{left}^{\alpha }$,$v_{right}^{\alpha }$], then the proposed algorithm will pass the Shannon entropy test [15]. The GSE and LSE values are organized in Table 8.

Table 8 indicates that the average value of GSE is about 7.9971. Most LSE values could pass three kinds of tests. It reveals that the designed scheme effectively resists Shannon entropy analysis.

6.6. Differential Attack Test

UACI and NPCR are often used for the differential attack test [34].

$$NPCR={\displaystyle \frac{1}{M\times N}}{\displaystyle \sum _{x=1}^M{\displaystyle \sum _{y=1}^{N}D(x,y)\times 100\%}}$$

(23)

$$UACI={\displaystyle \frac{1}{M\times N}}{\displaystyle \sum _{x=1}^M{\displaystyle \sum _{y=1}^N\frac{|E_1(x,y)-E_2(x,y)|}{255}\times 100\%}}$$

(24)

$$D(x,y)=\left\{\begin{array}{l}0,if{E}_1(x,y)=E_2(x,y)\\ 1,else\end{array}\right.$$

(25)

where E₁ and E₂ are two encrypted images.

33.4635% and 99.6094% are the theoretical values of UACI and NPCR [7]. Table 9 reveals that the values of two indicators are close to the ideal values. This means that the designed algorithm is capable of resisting differential attacks.

6.7. Encryption Quality Test

ID and DUH can be used to measure encryption image quality [8]. Encryption quality will be much better if the values of ID and DUH are smaller. They can be obtained as [47]:

$$H_D=abs(H_P-H_C)$$

(26)

$$M_D={\displaystyle \frac{1}{256}}{\displaystyle \sum _{l=0}^{255}{H}_D(l)}$$

(27)

$$ID={\displaystyle \sum _{k=0}^{255}|H_D(k)-M_D|}$$

(28)

$$DUH={\displaystyle \frac{{\displaystyle {\sum }_{j=0}^{255}|H_C(j)-M\times N/256|}}{M\times N}}$$

(29)

where H_C and H_P represent the histograms of encrypted image C and plain image P.

The DUH and ID values are listed in Table 10. It shows that the designed method produces smaller values of two indicators than the existing schemes. It reveals that the proposed algorithm has high encryption quality.

6.8. Texture Test

Homogeneity, contrast, and energy are three indexes which are used for texture test [8]. They are defined as follows:

$$Homogeneity={\displaystyle \sum _{u,v}\frac{GLCM(u,v)}{1+|u-v|}}$$

(30)

$$Contrast={\displaystyle \sum _{u,v}|u-v{|}^{2}GLCM(u,v)}$$

(31)

$$Energy={\displaystyle \sum _{u,v}GLCM{(u,v)}^2}$$

(32)

where GLCM(u, v) means the gray-level co-occurrence matrix.

The lower values of homogeneity and energy denote a higher security, and the higher value of contrast will be much better.

Three values are listed in Table 11.

As shown in Table 11, the designed algorithm performs well. It proves that the proposed method will be able to withstand the texture test.

6.9. Robustness Test

6.9.1. Cropping attack test

Images may be subject to noise or data loss attacks during transmission. Robustness reflects the ability of a scheme to resist interference. The encrypted image of Baboon is cropped 3.125%, 6.25%, 12.5%, and 25% data. The decrypted results are shown in Figure 10.

Figure 10 shows that even if there is a lot of noise in the decrypted result, it is still identifiable. It reveals that the designed algorithm passes the cropping attack test.

6.9.2. Noise attack test

The encrypted image of Baboon is suffered from noise attack. The densities of salt & pepper noise are 0.5%, 2%, 10%, and 30%. Figure11 shows the decrypted images. It means that the recovered images are able to be recognized. It symbolises that the proposed scheme is robust enough to pass noise attack test.

Figure 11. Robustness against noise attack: (a)-(d) are the decrypted images with 0.5%, 2%, 10%, and 30% noise.

Figure 11. Robustness against noise attack: (a)-(d) are the decrypted images with 0.5%, 2%, 10%, and 30% noise.

7. Discussion

In the manuscript a novel hyperchaotic image encryption scheme is proposed. Chaotic sequences are generated with 6D chaotic systems. The encrypted result is closely related to the plain image with SHA-512. The initial values are produced with a hash value and a secret key. Each pixel is divided into four equal segments. Permutation is performed based on chaotic sequences. DNA coding, modern DNA complementary rules, DNA computing and DNA decoding are executed on the new image with size M × 4N. The image is performed on diffusion operation, and encrypted image is achieved finally. The experimental results display the effects of the proposed scheme. Secure test shows that the designed method possesses a huge key space, low pixel correlation, high encryption quality, and sensitivity. It also concludes that the designed algorithm could resist common attacks and is much better than some existing methods.