# Hyper-Chaotic Color Image Encryption Based on Transformed Zigzag Diffusion and RNA Operation

^{*}

## Abstract

**:**

## 1. Introduction

- A novel 6D hyper-chaotic system is employed in this paper to produce chaotic matrix for permutation, diffusion, and RNA operation.
- A new 3D transformed Zigzag diffusion scheme is proposed to encrypt color images.
- RNA operation is modified specifically for color images.
- Extensive experiments and analyses demonstrate that the proposed HCZRNA could resist various types of attacks.

## 2. Preliminaries

#### 2.1. The 6D Hyper-Chaotic System

#### 2.2. 3D Transformed Zigzag Diffusion

#### 2.3. RNA Operation

## 3. Encryption and Decryption

#### 3.1. Encryption Scheme

#### 3.1.1. Initial Values Generation

- Step 1: divide the secret key K into 32 blocks, which could be expressed as $K=\{{k}_{1},{k}_{2},\cdots ,{k}_{32}\}$, each k is a 8-bits number.
- Step 2: K array that is generated in step 1 is calculated into four intermediate parameters ${d}_{1}$, ${d}_{2}$, ${d}_{3}$, ${d}_{4}$ by Equation (2) with four user-defined constants ${c}_{1}$, ${c}_{2}$, ${c}_{3}$ and ${c}_{4}$.$$\left\{\begin{array}{cc}\hfill {d}_{1}& ={c}_{1}+\frac{{k}_{1}\oplus {k}_{2}\oplus \cdots \oplus {k}_{8}}{256}\hfill \\ \hfill {d}_{2}& ={c}_{2}+\frac{{k}_{9}\oplus {k}_{10}\oplus \cdots \oplus {k}_{16}}{256}\hfill \\ \hfill {d}_{3}& ={c}_{3}+\frac{{k}_{17}\oplus {k}_{18}\oplus \cdots \oplus {k}_{24}}{256}\hfill \\ \hfill {d}_{4}& ={c}_{4}+\frac{{k}_{25}\oplus {k}_{26}\oplus \cdots \oplus {k}_{32}}{256}\hfill \end{array}\right.$$
- Step 3: The initial values ${x}_{1}$ to ${x}_{6}$ of 6D hyper-chaotic system could be obtained from the 4 intermediate parameters by Equation (3).$$\left\{\begin{array}{cc}\hfill {x}_{1}& =\frac{(({d}_{1}+{d}_{2})\times {10}^{8})\phantom{\rule{3.33333pt}{0ex}}mod\phantom{\rule{3.33333pt}{0ex}}256}{255}\hfill \\ \hfill {x}_{2}& =\frac{(({d}_{2}+{d}_{3})\times {10}^{8})\phantom{\rule{3.33333pt}{0ex}}mod\phantom{\rule{3.33333pt}{0ex}}256}{255}\hfill \\ \hfill {x}_{3}& =\frac{(({d}_{3}+{d}_{4})\times {10}^{8})\phantom{\rule{3.33333pt}{0ex}}mod\phantom{\rule{3.33333pt}{0ex}}256}{255}\hfill \\ \hfill {x}_{4}& =\frac{(({d}_{1}+{d}_{3})\times {10}^{8})\phantom{\rule{3.33333pt}{0ex}}mod\phantom{\rule{3.33333pt}{0ex}}256}{255}\hfill \\ \hfill {x}_{5}& =\frac{(({d}_{1}+{d}_{4})\times {10}^{8})\phantom{\rule{3.33333pt}{0ex}}mod\phantom{\rule{3.33333pt}{0ex}}256}{255}\hfill \\ \hfill {x}_{6}& =\frac{(({d}_{2}+{d}_{4})\times {10}^{8})\phantom{\rule{3.33333pt}{0ex}}mod\phantom{\rule{3.33333pt}{0ex}}256}{255}\hfill \end{array}\right.$$

#### 3.1.2. Hyper-Chaotic Matrices Generation

#### 3.1.3. Permutation

#### 3.1.4. Diffusion

#### 3.1.5. RNA Operation

- Step 1: RNA operation is initiated from creating two encrypted codons tables, called ${T}_{00}$ and ${T}_{01}$. In which, ${T}_{00}$ and ${T}_{01}$ are shuffled tables from codons truth, as in Table 2. The shuffle orders are generated according to indexes sequences calculated from Equation (7). After sorting with these two indexes sequences, the original codons truth table could be shuffled to two different encrypted codons tables ${T}_{00}$ and ${T}_{01}$.Subsequently, by the complementary rules of RNA, additional tables ${T}_{10}$ and ${T}_{11}$ could be generated from ${T}_{00}$ and ${T}_{01}$. Hence, four encrypted codons tables are generated.
- Step 2: for each element in ${D}_{mat}$, binary number conversion is processed, which is recorded as B.$$B=\left\{{b}_{i,j,m}\right\}.\phantom{\rule{1.em}{0ex}}i,j=1,2,\cdots ,N;m=1,2,3$$Each ${b}_{i,j,m}$ could be expressed as eight binary numbers, which could be depicted as ${b}_{0}^{i,j,m}{b}_{1}^{i,j,m}{b}_{2}^{i,j,m}{b}_{3}^{i,j,m}{b}_{4}^{i,j,m}{b}_{5}^{i,j,m}{b}_{6}^{i,j,m}{b}_{7}^{i,j,m}$.
- Step 3: divide ${b}_{i,j,m}$ into four pieces, each two bits are one piece, which are recorded as:$$\begin{array}{c}\hfill b{t}_{1}^{i,j,m}={b}_{0}^{i,j,m}{b}_{1}^{i,j,m}\\ \hfill b{t}_{2}^{i,j,m}={b}_{2}^{i,j,m}{b}_{3}^{i,j,m}\\ \hfill b{t}_{3}^{i,j,m}={b}_{4}^{i,j,m}{b}_{5}^{i,j,m}\\ \hfill b{t}_{4}^{i,j,m}={b}_{6}^{i,j,m}{b}_{7}^{i,j,m}\end{array}$$Additionally, combine three channels’ $bt$s at the same coordinate together:$$\begin{array}{c}\hfill b{t}_{1}^{i,j}=b{t}_{1}^{i,j,1}b{t}_{1}^{i,j,2}b{t}_{1}^{i,j,3}\\ \hfill b{t}_{2}^{i,j}=b{t}_{2}^{i,j,1}b{t}_{2}^{i,j,2}b{t}_{2}^{i,j,3}\\ \hfill b{t}_{3}^{i,j}=b{t}_{3}^{i,j,1}b{t}_{3}^{i,j,2}b{t}_{3}^{i,j,3}\\ \hfill b{t}_{4}^{i,j}=b{t}_{4}^{i,j,1}b{t}_{4}^{i,j,2}b{t}_{4}^{i,j,3}\end{array}$$Therefore, each $b{t}^{i,j}$ has six bits that could transfer to RNA codons according to Table 1. Exchange each two bits in $bt$s to RNA base one-by-one according to the principle of row priority, $bt$s could be coded to codons. And put them into a one-dimension sequence $BS$ as Equation (14).$$BS=\{b{t}_{1}^{1,1},b{t}_{2}^{1,1},b{t}_{3}^{1,1},b{t}_{4}^{1,1},b{t}_{1}^{1,2},b{t}_{2}^{1,2},\cdots ,b{t}_{4}^{2,1},b{t}_{1}^{2,2},\cdots ,b{t}_{3}^{N,N},b{t}_{4}^{N,N}\}.$$
- Step 4: convert key to binary format. 256-bit key could be changed into a binary sequence $BK$.$$\begin{array}{cc}\hfill key& =[ke{y}_{0},ke{y}_{1},\cdots ,ke{y}_{31}]\hfill \\ \hfill ke{y}_{i}& =ke{y}_{i,0},ke{y}_{i,1},\cdots ,ke{y}_{i,7}\hfill \\ \hfill BK& =[ke{y}_{1,0},ke{y}_{1,1},\cdots ,ke{y}_{1,7},ke{y}_{2,0},ke{y}_{2,1},\cdots ,ke{y}_{31,7}]\hfill \end{array}$$Walk through sequence $BS$, and find corresponding index $id$ of each codon in $BS$ from Table 2. For each codon in $BS$, check 2-bits table number z in sequence $BK$.$$z=B{K}_{n\phantom{\rule{3.33333pt}{0ex}}mod\phantom{\rule{3.33333pt}{0ex}}2048}B{K}_{(n+1)\phantom{\rule{3.33333pt}{0ex}}mod\phantom{\rule{3.33333pt}{0ex}}2048}$$Take the codon ${T}_{z}\left(id\right)$ to replace the origin codon $BS\left(n\right)$.When iterations termination, an encrypted sequence is generated.
- Step 5: decode each base in encrypted sequence $BS$ to binary format by Table 1, put all of the binary digits back to original coordinates by reversing operations in Step 3. Additionally, change binary matrix into 2-bit matrix. The cipher image is generated.

#### 3.2. Decryption

- Step 1: redo the processes that are listed in Section 3.1.1 and Section 3.1.2 to generate hyper-chaotic matrices ${S}_{1}$, ${S}_{2}$, and ${S}_{3}$.
- Step 2: convert the cipher image to a binary format, and reconstruct three channels’ pixels at each coordinate into four 6-bit binary arrays by using Euqation (12) and (13). Change 6-bit arrays into codons from codons truth Table 2, and put them in a one-dimension sequence $B{S}^{\prime}$ as Equation (14).
- Step 3: generate key binary sequence $BK$ through Equation (15) and encrypted codons tables $\{{T}_{00},{T}_{01},{T}_{10},{T}_{11}\}$ by redoing Step 1 in Section 3.1.5.
- Step 4: Check each 2-bits z in $BK$ and find corresponding table ${T}_{z}$ from $\{{T}_{00},{T}_{01},{T}_{10},{T}_{11}\}$. Walk through $B{S}^{\prime}$ and find each codon’s corresponding index $i{d}^{\prime}$ in Table ${T}_{z}$. Replace codon in $B{S}^{\prime}$ to codon $i{d}^{\prime}$ in codons truth Table 2. After all codons are replaced, convert them into binary formats and 8-bit numbers, matrix ${D}_{mat}^{\prime}$ is obtained.
- Step 5: split matrix ${D}_{mat}^{\prime}$ and place triangles on cube surfaces as the process shown in Section 2.2. Redo Section 3.1.4 with modified Equation (17) two rounds, and then walk through pixels with reversed Zigzag path. Take Figure 4 in Section 2.2 as an example, the traversal road of decryption is shown in Figure 7. If we put all the pixels together, the order of traversal is depicted in Figure 8.$$\begin{array}{cc}\hfill {C}_{i,j,m}^{\prime}& =({D}_{i,j,m}^{\prime}\oplus ({T}^{\prime}+{x}_{i,j,m;2}))mod256\hfill \\ \hfill {P}_{i,j,m}^{\prime}& =({C}_{i,j,m}^{\prime}\oplus (T+{x}_{i,j,m;1}))mod256\hfill \end{array}$$
- Step 6: after the process in Step 5, return the triangles in the cube to theirs original coordinates on a image. Additionally, the reverse processes in Section 3.1.3, reshape ${S}_{1}$ to construct sorted sequence. Find image pixels’ corresponding coordinates through sorted sequence and recover. The decrypted image is generated.

## 4. Experimental Results

#### 4.1. Key Space

#### 4.2. Sensitivity of Keys

#### 4.3. Histogram

#### 4.4. Correlation

#### 4.5. Information Entropy

#### 4.6. Differential Attack

#### 4.7. Robustness

#### 4.8. Running Time

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

6D | 6 Dimensional |

RNA | Ribonucleic acid |

HCZRNA | Hyper-chaotic color image encryption mechanism based |

on transformed Zigzag diffusion and RNA operations | |

RGB | Red Greeen Blue |

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**Figure 3.**Color image to cube.The first row is three channels of a color image. The second row is the triangles generated from image. Additionally, the third row is the placement of triangles on a cube.

**Figure 4.**Three-dimensional (3D) transformed Zigzag diffusion. (

**a**) is the Zigzag diffusion process on the front side of cube. (

**b**) is the Zigzag diffusion process on the back side of cube.

**Figure 5.**3D transformed Zigzag path. For all triangles on the cube, 3D transformed Zigzag diffusion is implemented through this order.

**Figure 9.**Histogram. image (

**a**,

**c**,

**e**) are the histograms of three channels of Baboon, and image (

**b**,

**d**,

**f**) are the histograms of corresponding channels of encrypted Baboon.

**Figure 10.**Correlations.The first row is correlations of plaintext images, and the second row is correlations of cipher images.

**Figure 11.**Cropping attack tests. The first row is cipher images with $12.5\%$, $25\%$ and $50\%$ data loss, and the second row is decrypted images from the first row.

**Figure 12.**Noise attack tests. The first row is cipher images with $1\%$, $5\%$, $10\%$ salt and pepper noise, and the second row is decrypted images from the first row.

RNA Bases | A | C | G | U |
---|---|---|---|---|

Binary | 00 | 01 | 10 | 11 |

# | Bin. | Codon | # | Bin. | Codon | # | Bin. | Codon | # | Bin. | Codon |
---|---|---|---|---|---|---|---|---|---|---|---|

0 | 000000 | AAA | 16 | 010000 | CAA | 32 | 100000 | GAA | 48 | 110000 | UAA |

1 | 000001 | AAC | 17 | 010001 | CAC | 33 | 100001 | GAC | 49 | 110001 | UAC |

2 | 000010 | AAG | 18 | 010010 | CAG | 34 | 100010 | GAG | 50 | 110010 | UAG |

3 | 000011 | AAU | 19 | 010011 | CAU | 35 | 100011 | GAU | 51 | 110011 | UAU |

4 | 000100 | ACA | 20 | 010100 | CCA | 36 | 100100 | GCA | 52 | 110100 | UCA |

5 | 000101 | ACC | 21 | 010101 | CCC | 37 | 100101 | GCC | 53 | 110101 | UCC |

6 | 000110 | ACG | 22 | 010110 | CCG | 38 | 100110 | GCG | 54 | 110110 | UCG |

7 | 000111 | ACU | 23 | 010111 | CCU | 39 | 100111 | GCU | 55 | 110111 | UCU |

8 | 001000 | AGA | 24 | 011000 | CGA | 40 | 101000 | GGA | 56 | 111000 | UGA |

9 | 001001 | AGC | 25 | 011001 | CGC | 41 | 101001 | GGC | 57 | 111001 | UGC |

10 | 001010 | AGG | 26 | 011010 | CGG | 42 | 101010 | GGG | 58 | 111010 | UGG |

11 | 001011 | AGU | 27 | 011011 | CGU | 43 | 101011 | GGU | 59 | 111011 | UGU |

12 | 001100 | AUA | 28 | 011100 | CUA | 44 | 101100 | GUA | 60 | 111100 | UUA |

13 | 001101 | AUC | 29 | 011101 | CUC | 45 | 101101 | GUC | 61 | 111101 | UUC |

14 | 001110 | AUG | 30 | 011110 | CUG | 46 | 101110 | GUG | 62 | 111110 | UUG |

15 | 001111 | AUU | 31 | 011111 | CUU | 47 | 101111 | GUU | 63 | 111111 | UUU |

Image | Size ($\mathit{h}\times \mathit{w}\times \mathit{c}$) | Image | Size ($\mathit{h}\times \mathit{w}\times \mathit{c}$) |
---|---|---|---|

Lena | $256\times 256\times 3$ | Baboon | $512\times 512\times 3$ |

Peppers | $256\times 256\times 3$ | Splash | $512\times 512\times 3$ |

Image | Lena | Pepper | Baboon | Splash |
---|---|---|---|---|

Difference | $99.59\%$ | $99.62\%$ | $99.61\%$ | $99.60\%$ |

Image | Channels | Plaintext | Ciphertext | ||
---|---|---|---|---|---|

$\mathbf{\alpha}$ | $\mathbf{\beta}$ | $\mathbf{\alpha}$ | $\mathbf{\beta}$ | ||

Lena | R | 65,306 | 255 | 248 | 15 |

G | 30,665 | 175 | 258 | 16 | |

B | 91,939 | 303 | 232 | 15 | |

Pepper | R | 57,413 | 239 | 249 | 15 |

G | 119,411 | 345 | 238 | 15 | |

B | 151,644 | 389 | 237 | 15 | |

Baboon | R | 165,679 | 407 | 520 | 22 |

G | 285,616 | 534 | 532 | 23 | |

B | 159,885 | 399 | 541 | 23 | |

Splash | R | 1,211,325 | 1100 | 566 | 23 |

G | 1,541,948 | 1241 | 495 | 22 | |

B | 2,958,482 | 1720 | 504 | 22 |

Image | Channels | Plaintext | Ciphertext | ||||
---|---|---|---|---|---|---|---|

Horizontal | Vertical | Diagonal | Horizontal | Vertical | Diagonal | ||

Lena | R | 0.9512 | 0.9755 | 0.9444 | 0.0046 | 0.0024 | 0.0051 |

G | 0.9512 | 0.9679 | 0.9276 | −0.0027 | −0.0007 | 0.0002 | |

B | 0.9512 | 0.9479 | 0.9021 | −0.0023 | 0.0014 | 0.0004 | |

Baboon | R | 0.9218 | 0.8624 | 0.8531 | 0.0003 | 0.0001 | 0.0015 |

G | 0.9218 | 0.7591 | 0.7299 | −0.0010 | 0.0004 | 0.0020 | |

B | 0.9218 | 0.8782 | 0.8411 | 0.0005 | −0.0022 | 0.0012 |

Image | Channels | Plaintext | Ciphertext | |||
---|---|---|---|---|---|---|

HCZRNA | Ref. [44] | Ref. [45] | ||||

Baboon | R | Horizontal | 0.9218 | 0.0003 | 0.0054 | −0.0073 |

Vertical | 0.8624 | 0.0001 | −0.0042 | −0.0059 | ||

Diagonal | 0.8531 | 0.0015 | −0.0177 | −0.0136 | ||

G | Horizontal | 0.9218 | −0.0010 | −0.0055 | 0.0046 | |

Vertical | 0.7591 | 0.0004 | 0.0119 | −0.0077 | ||

Diagonal | 0.7299 | 0.0020 | 0.0046 | −0.0044 | ||

B | Horizontal | 0.9218 | 0.0005 | −0.0021 | −0.0067 | |

Vertical | 0.8782 | −0.0022 | 0.0104 | −0.0111 | ||

Diagonal | 0.8411 | 0.0012 | −0.0021 | 0.0122 |

Image | Channels | Lena | Peppers | Baboon | Splash |
---|---|---|---|---|---|

Plaintext | R | 7.2353 | 7.3369 | 7.7067 | 6.9481 |

G | 7.5683 | 7.4394 | 7.4744 | 6.8845 | |

B | 6.9176 | 7.0219 | 7.7522 | 6.1265 | |

Ciphertext | R | 7.9973 | 7.9972 | 7.9993 | 7.9993 |

G | 7.9970 | 7.9970 | 7.9993 | 7.9994 | |

B | 7.9972 | 7.9972 | 7.9993 | 7.9993 |

Image | Channel | Plaintext | HCZRNA | Ref. [29] | Ref. [23] | Ref. [44] | Ref. [45] | Ref. [46] |
---|---|---|---|---|---|---|---|---|

Lena | R | 7.2353 | 7.9973 | 7.9971 | 7.9973 | - | - | 7.9973 |

G | 7.5683 | 7.9970 | 7.9971 | 7.9972 | - | - | 7.9975 | |

B | 6.9176 | 7.9972 | 7.9971 | 7.9971 | - | - | 7.9975 | |

Baboon | R | 7.7067 | 7.9993 | 7.9926 | - | 7.9993 | 7.9993 | 7.9970 |

G | 7.4744 | 7.9993 | 7.9926 | - | 7.9993 | 7.9993 | 7.9978 | |

B | 7.7522 | 7.9993 | 7.9926 | - | 7.9993 | 7.9992 | 7.9987 |

**Table 10.**The mean number of pixel change rate (NPCR) and unified average changing intensity (UACI) of cipher images.

Image | NPCR(%) | UACI(%) | ||||
---|---|---|---|---|---|---|

R | G | B | R | G | B | |

Lena | 99.6619 | 99.6272 | 99.6460 | 33.6177 | 33.6048 | 33.6422 |

Peppers | 99.6481 | 99.6404 | 99.6239 | 33.7208 | 33.5701 | 33.6435 |

Baboon | 99.6159 | 99.6769 | 99.6115 | 33.5196 | 33.5203 | 33.5049 |

Splash | 99.6219 | 99.6934 | 99.6253 | 33.4983 | 33.5114 | 33.4816 |

Image | Channel | HCZRNA | Ref. [29] | Ref. [23] | Ref. [44] | Ref. [45] | Ref. [46] |
---|---|---|---|---|---|---|---|

Lena | R | 99.6619 | 99.60 | 99.6323 | - | - | 99.615 |

G | 99.6272 | 99.61 | 99.6109 | - | - | 99.62 | |

B | 99.6460 | 99.61 | 99.6338 | - | - | 99.617 | |

Baboon | R | 99.6159 | 99.6083 | - | 99.6037 | 99.6037 | 99.6140 |

G | 99.6769 | 99.6065 | - | 99.6048 | 99.6017 | 99.6073 | |

B | 99.6115 | 99.6094 | - | 99.6059 | 99.6043 | 99.6292 |

Image | Channel | HCZRNA | Ref. [29] | Ref. [23] | Ref. [44] | Ref. [45] | Ref. [46] |
---|---|---|---|---|---|---|---|

Lena | R | 33.6177 | 33.56 | 33.4683 | - | - | 33.4732 |

G | 33.6048 | 33.45 | 33.4341 | - | - | 33.3428 | |

B | 33.6422 | 33.49 | 33.4991 | - | - | 33.4647 | |

Baboon | R | 33.5196 | 33.4939 | - | 33.4427 | 29.9630 | 33.4843 |

G | 33.5203 | 33.4295 | - | 33.4605 | 28.5708 | 33.4690 | |

B | 33.5049 | 33.4856 | - | 31.9747 | 31.2574 | 33.4965 |

Data Loss | MSE | PSNR |
---|---|---|

$12.5\%$ | 1195 | 17.35 |

$25\%$ | 2315 | 14.48 |

$50\%$ | 4502 | 11.60 |

Salt and Pepper Noise | MSE | PSNR |
---|---|---|

$1\%$ | 610 | 20.28 |

$5\%$ | 2684 | 13.84 |

$10\%$ | 4490 | 11.61 |

Image Size | Encryption | Decryption |
---|---|---|

$64\times 64\times 3$ | 0.59 | 0.44 |

$128\times 128\times 3$ | 2.39 | 1.77 |

$256\times 256\times 3$ | 9.36 | 6.96 |

$512\times 512\times 3$ | 37.54 | 28.49 |

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## Share and Cite

**MDPI and ACS Style**

Zhang, D.; Chen, L.; Li, T.
Hyper-Chaotic Color Image Encryption Based on Transformed Zigzag Diffusion and RNA Operation. *Entropy* **2021**, *23*, 361.
https://doi.org/10.3390/e23030361

**AMA Style**

Zhang D, Chen L, Li T.
Hyper-Chaotic Color Image Encryption Based on Transformed Zigzag Diffusion and RNA Operation. *Entropy*. 2021; 23(3):361.
https://doi.org/10.3390/e23030361

**Chicago/Turabian Style**

Zhang, Duzhong, Lexing Chen, and Taiyong Li.
2021. "Hyper-Chaotic Color Image Encryption Based on Transformed Zigzag Diffusion and RNA Operation" *Entropy* 23, no. 3: 361.
https://doi.org/10.3390/e23030361