# A 3D Cuboid Image Encryption Algorithm Based on Controlled Alternate Quantum Walk of Message Coding

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## Abstract

**:**

## 1. Introduction

## 2. Preliminary Works

#### 2.1. One-Dimensional Discrete Quantum Walks (1DQWs) on Circles

#### 2.2. Controlled Alternate Quantum Walks (CAQWs) on Circles

## 3. Proposed Method

#### 3.1. Generate Initial Key

#### 3.1.1. Initial Parameters ${d}_{1}~{d}_{8}$

#### 3.1.2. Sum of Plaintext Image Pixel Values $sumP$

#### 3.1.3. Image block size $Bsize$

#### 3.1.4. Supplementary Parameters $M1$, $N1$ and Number of Blocks $Bnum$

#### 3.2. Generating 3D Probability Magnitude Matrix, 3D Probability Distribution Matrix, and 3D Quantum Hash Sequence by Using Quantum Walks

#### 3.2.1. Generation of Initial Parameters for 1DQWs

- (1)
- Number of nodes on the circle

- (2)
- Coin parameter $\theta $

- (3)
- The initial state of the coin

#### 3.2.2. Generation of Initial Parameters of CAQWs

- (1)
- The number of nodes on the circle of CAQWs is equal to the block size $Bsize$.
- (2)
- Coin parameters ${\theta}_{0},{\theta}_{1}$

- (3)
- Initial state of the coin

#### 3.2.3. Generating Probability Distribution Sequences Using 1DQWs

#### 3.2.4. Message Encoding

#### 3.2.5. Controlled Alternate Quantum Walks for Message Encoding

#### 3.2.6. Three-Dimensional Probability Amplitude Matrix and Three-Dimensional Probability Distribution Matrix

#### 3.2.7. Three-Dimensional Quantum Hash Sequence

#### 3.3. Image Encryption Process

Algorithm 1. Sequence generation. |

Input: Original image(PlainImg)Output$:{d}_{1}~{d}_{8},Qum1D,message,fre,Qum2D,Btest$1 ${k}_{1}\sim {k}_{64}\leftarrow $SHA-512(PlainImg) 2 ${d}_{1}~{d}_{8}\leftarrow $Generate initial key parameters based on ${k}_{1}\sim {k}_{64}$ 3 Calculation of 1DQWs and 2DCAQWs 4 $Qum1D\leftarrow 1DQWs$($TC,t,\theta ,\alpha ,\beta $) // 1DQWs on a circle with node number $TC$ 5 $message\leftarrow encoding(Qum1D)$ // Quadratic encoding 6 $fre(2,Bsize\ast Bsize,i)\leftarrow \phi (F(x,y,t),message(i))i=1,\dots ,Bnum;$ 7 $Qum2D(Bsize,Bsize,i)\leftarrow \omega (P(x,y,t),message(i))i=1,\dots ,Bnum$ // Alternating quantum walks with $Bnum$ sub-message encoding control on a circle 8 $Btest\leftarrow dec2bin(Qum2D)$ // Complete diffusion operation using 3D quantum hash sequence Btest |

Algorithm 2. Encryption algorithms. |

Input: Original image(PlainImg), $Qum1D$, $message$, $fre$, $Qum2D$, $Btest$Output: Encrypted images($En$)1 Image block ($Bsize\ast Bsize\ast Bnum)\leftarrow $chunking the image 2 $Plain3D\leftarrow $Integrating image blocks as rectangles 3 $prm1Dt\leftarrow sum(reshape(Qum1D,lmess,Bnum))$; //Group summation 4 $\left[~,prm1Dindex\right]\leftarrow sort(prm1Dt)$ // Sort 5 $PrmP3D\leftarrow $ Displacement between layers according to its index sequence $prm1Dindex$ for the image rectangle $Plain3D$ 6 $Prm2\leftarrow $ Sorting and dislocation inside each depth layer using $Qum2D$ 7 for j $\leftarrow $ 1 to B_num 8 $\mathrm{BinP}3=\mathrm{dec}2\mathrm{bin}(Prm2$); 9 for i$\leftarrow $1 to x ∗ y do 10 $\mathrm{BinP}3\_\mathrm{t}\text{}\leftarrow \text{}$reshape$(\mathrm{BinP}3)$ 11 If A > B 12 Bin_P3$\leftarrow $$\mathrm{rot}90(\mathrm{BinP}3\_\mathrm{t}$ Upper)// Upper layer rotated 90 degrees counterclockwise 13 else 14 Bin_P3$\leftarrow $$\mathrm{rot}90(\mathrm{BinP}3\_\mathrm{t}$ Lower) //Lower layer rotated clockwise by 90 degrees 15 end 16 P3bin(i,:) = reshape(Bin_P3); 17 end 18 P3bin1 = bin2dec(P3bin); 19 P3bin2 = reshape(P3bin1, x, y); 20 end 21 $Ctest=bitxor(Btest,P3bin2)$ // Global xor 22 $En\leftarrow $Tiling the image block $Ctest$ |

#### 3.3.1. Generation of Image Cuboid

#### 3.3.2. Two Rounds of Scrambling

#### 3.3.3. Two Rounds of Diffusion

#### 3.4. Image Decryption Process

## 4. Simulation Results and Security Analysis

#### 4.1. Encryption and Decryption Results

#### 4.2. Histogram Analysis

#### 4.3. ${\chi}^{2}$ Test

#### 4.4. Correlation Coefficient Analysis

#### 4.5. Information Entropy Analysis

#### 4.6. Antidifferential Attack (NPCR and UACI Standard Evaluation)

#### 4.7. Anticlipping and Noise Attacks

#### 4.8. Key Space Analysis

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#### 4.9. Key Sensitivity Analysis

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**0.578947368421053**,

**0.290448343079922**);

**7**);

**0.426900584795322**,

**0.391812865497076**);

#### 4.10. Computational Complexity

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**A 3D Cuboid Image Encryption Framework Based on Message-Encoded Controlled Alternate Quantum Walks.

**Figure 9.**Simulation results (256 ∗ 256): (

**a**–

**e**) original images; (

**f**–

**j**) encrypted images; (

**k**–

**o**) decrypted images.

**Figure 10.**Simulation results (512 ∗ 512 and irregular): (

**a**–

**d**) original images; (

**e**–

**h**) encrypted images; (

**i**–

**l**) decrypted images.

**Figure 12.**Histogram Analysis (512 ∗ 512 and irregular): (

**a**–

**d**) original images; (

**e**–

**h**) encrypted image.

**Figure 17.**Decrypted images for varying degrees of cropping attacks: (

**a**) 1/16 crop; (

**b**) 1/8 crop; (

**c**) 1/4 crop; (

**d**) 1/2 crop; (

**e**–

**h**) decrypted images of (

**a**–

**d**).

**Figure 18.**Noise attack test results: (

**a**–

**c**) noisy cipher image of different noise level; (

**d**–

**f**) decrypted noisy image.

Image | Lena | Baboon | Peppers | White | Black | Avg. | Ref. [9] | Ref. [33] |
---|---|---|---|---|---|---|---|---|

Plaintext | 39,868.727 | 44,739.305 | 26,311.578 | 16,711,680 | 16,711,680 | - | - | - |

Ciphertext | 219.269 | 216.589 | 225.765 | 259.881 | 241.390 | 232.579 | 244.159 | 254.078 |

Image | Barbara | Cameraman | Livingroom | Irregular Lena | Avg. | Ref. [18] |
---|---|---|---|---|---|---|

Plaintext | 14,4101.119 | 418,530.147 | 276,815.883 | 74,075.357 | - | - |

Ciphertext | 238.454 | 250.778 | 245.376 | 250.828 | 246.359 | 260.400 |

Image | Plain Image | Cipher Image | Ref. [11] | ||||||
---|---|---|---|---|---|---|---|---|---|

H | V | D | H | V | D | H | V | D | |

5.1.09 | 0.9026 | 0.9406 | 0.9098 | 0.0005 | 0.0027 | −0.0023 | 0.0031 | −0.0016 | 0.0014 |

5.1.10 | 0.9016 | 0.8688 | 0.8311 | −0.0001 | 1.0819 × 10^{−6} | 0.0007 | 0.0044 | −0.0006 | −0.0052 |

5.1.11 | 0.9574 | 0.9480 | 0.8966 | −0.0046 | −0.0038 | −0.0003 | 0.0005 | −0.0019 | −0.0049 |

5.1.12 | 0.9576 | 0.9741 | 0.9408 | 0.0069 | 0.0039 | 0.0048 | 0.0068 | −0.0002 | −0.0055 |

5.1.13 | 0.8804 | 0.8497 | 0.7419 | −0.0024 | −0.0027 | −0.0019 | 5.76 × 10^{−6} | 0.0017 | −0.0072 |

5.1.14 | 0.9522 | 0.9018 | 0.8589 | 0.0015 | −0.0057 | 0.0038 | −0.0056 | 0.0001 | 0.0044 |

5.2.08 | 0.9289 | 0.8784 | 0.8472 | −0.0014 | −0.0057 | 0.0011 | 0.0028 | 0.0006 | −0.0027 |

5.2.09 | 0.9059 | 0.8701 | 0.8122 | −0.0013 | −0.0035 | −0.0017 | 0.0006 | −0.0123 | 0.0001 |

5.2.10 | 0.9409 | 0.9274 | 0.8948 | 0.0019 | 0.0002 | −0.0026 | −0.0024 | −0.0005 | 0.0031 |

5.3.01 | 0.9756 | 0.9811 | 0.9669 | −0.0004 | −0.0006 | 0.0017 | 0.0015 | 4.43 × 10^{−5} | −0.0017 |

5.3.02 | 0.9154 | 0.9053 | 0.8608 | 0.0004 | −0.0020 | 0.0015 | −0.0024 | −0.0011 | 0.0008 |

7.1.01 | 0.9635 | 0.9194 | 0.9080 | −0.0015 | −0.0033 | 0.0021 | −0.0021 | 0.0011 | 0.0067 |

7.1.02 | 0.9468 | 0.9478 | 0.9032 | −0.0008 | 0.0009 | −0.0007 | 0.0084 | 0.0017 | 0.0021 |

7.1.03 | 0.9410 | 0.9306 | 0.8998 | −0.0016 | 0.0005 | 0.0033 | 0.0045 | 8.67 × 10^{−5} | −0.0056 |

7.1.04 | 0.9762 | 0.9646 | 0.9546 | 0.0007 | −0.0006 | 0.0003 | −0.0031 | 0.0073 | 0.0002 |

7.1.05 | 0.9449 | 0.9193 | 0.8983 | 0.0005 | 0.0024 | −0.0008 | −0.0040 | −0.0004 | −0.0002 |

7.1.06 | 0.9378 | 0.9004 | 0.8791 | 0.0025 | 0.0023 | −0.0001 | 0.0002 | −0.0014 | −0.0057 |

7.1.07 | 0.8872 | 0.8735 | 0.8296 | −0.0003 | 0.0004 | 7.5782 × 10^{−5} | 0.0020 | −0.0009 | −0.0160 |

7.1.08 | 0.9563 | 0.9335 | 0.9245 | −0.0007 | 0.0010 | 0.0014 | 0.0083 | −0.0008 | 0.0057 |

7.1.09 | 0.9667 | 0.9277 | 0.9160 | −0.0010 | −0.0008 | 0.0004 | 0.0007 | −0.0005 | 0.0063 |

7.1.10 | 0.9659 | 0.9497 | 0.9341 | 0.0001 | −0.0008 | 0.0007 | 0.0058 | −0.0001 | 0.0050 |

7.2.01 | 0.9670 | 0.9468 | 0.9463 | −0.0011 | 0.0002 | −0.0018 | −0.0008 | −0.0031 | −0.0044 |

boat.512 | 0.9371 | 0.9722 | 0.9220 | −0.0006 | −0.0013 | −0.0005 | 0.0024 | −6.53 × 10^{−7} | −0.0036 |

gray21.512 | 0.9965 | 0.9998 | 0.9963 | −0.0010 | 0.0008 | 0.0004 | 0.0011 | 0.0007 | −0.0056 |

ruler.512 | 0.4332 | 0.5068 | −0.0207 | −0.0015 | 0.0022 | −0.0002 | 0.0007 | 0.0120 | 3.59 × 10^{−5} |

Mean | 0.9215 | 0.9095 | 0.8581 | −0.0014 | 0.0019 | 0.0014 | 0.0013 | 0.0020 | 0.0042 |

Images | Size | Plain Image | Cipher Image | ||
---|---|---|---|---|---|

Ref. [6] | Ref. [12] | Ours | |||

5.1.09 | 256 ∗ 256 | 6.7093 | 7.9973 | 7.9971 | 7.9970 |

5.1.10 | 256 ∗ 256 | 7.3118 | 7.9973 | 7.9974 | 7.9977 |

5.1.11 | 256 ∗ 256 | 6.4523 | 7.9970 | 7.9969 | 7.9974 |

5.1.12 | 256 ∗ 256 | 6.7057 | 7.9975 | 7.9972 | 7.9974 |

5.1.13 | 256 ∗ 256 | 1.5483 | 7.9972 | 7.9969 | 7.9974 |

5.1.14 | 256 ∗ 256 | 7.3424 | 7.9970 | 7.9974 | 7.9977 |

5.2.08 | 512 ∗ 512 | 7.5237 | 7.9992 | 7.9993 | 7.9994 |

5.2.09 | 512 ∗ 512 | 6.9940 | 7.9994 | 7.9993 | 7.9993 |

5.2.10 | 512 ∗ 512 | 5.7056 | 7.9993 | 7.9993 | 7.9993 |

5.3.01 | 1024 ∗ 1024 | 7.5237 | 7.9998 | 7.9998 | 7.9998 |

5.3.02 | 1024 ∗ 1024 | 6.8303 | 7.9998 | 7.9998 | 7.9997 |

7.1.01 | 512 ∗ 512 | 6.0274 | 7.9993 | 7.9991 | 7.9993 |

7.1.02 | 512 ∗ 512 | 4.0045 | 7.9993 | 7.9992 | 7.9994 |

7.1.03 | 512 ∗ 512 | 5.4957 | 7.9993 | 7.9993 | 7.9994 |

7.1.04 | 512 ∗ 512 | 6.1074 | 7.9992 | 7.9993 | 7.9993 |

7.1.05 | 512 ∗ 512 | 6.5632 | 7.9993 | 7.9992 | 7.9993 |

7.1.06 | 512 ∗ 512 | 6.6953 | 7.9992 | 7.9993 | 7.9993 |

7.1.07 | 512 ∗ 512 | 5.9916 | 7.9993 | 7.9993 | 7.9992 |

7.1.08 | 512 ∗ 512 | 5.0534 | 7.9994 | 7.9993 | 7.9992 |

7.1.09 | 512 ∗ 512 | 6.1898 | 7.9993 | 7.9992 | 7.9994 |

7.1.10 | 512 ∗ 512 | 5.9088 | 7.9993 | 7.9993 | 7.9993 |

7.2.01 | 1024 ∗ 1024 | 5.6415 | 7.9998 | 7.9998 | 7.9998 |

boat.512 | 512 ∗ 512 | 7.1914 | 7.9993 | 7.9994 | 7.9994 |

gray21.512 | 512 ∗ 512 | 4.3923 | 7.9992 | 7.9994 | 7.9993 |

ruler.512 | 512 ∗ 512 | 0.5000 | 7.9993 | 7.9992 | 7.9993 |

Mean of 256 ∗ 256 | 256 ∗ 256 | - | 7.9972 | 7.9972 | 7.9974 |

Mean of 512 ∗ 512 | 512 ∗ 512 | - | 7.9993 | 7.9993 | 7.9993 |

Mean of 1024 ∗ 1024 | 1024 ∗ 1024 | - | 7.9998 | 7.9998 | 7.9998 |

Image | NPCR | UACI | ||||||
---|---|---|---|---|---|---|---|---|

$\mathit{\alpha}=0.05$ | Ref. [6] | Ref. [7] | Ref. [8] | Proposed | Ref. [6] | Ref. [7] | Ref. [8] | Proposed |

256 ∗ 256 | Theoretical NPCR | 99.5693 | Theoretical UACI | 33.2824~33.6447 | ||||

5.1.09 | 99.6109 | 99.603 | 99.5124 | 99.5787 | 33.4475 | 33.552 | 33.5214 | 33.3580 |

5.1.10 | 99.5972 | 99.636 | 99.6121 | 99.6565 | 33.4846 | 33.453 | 33.4215 | 33.5306 |

5.1.11 | 99.5865 | 99.942 | 99.5943 | 99.5860 | 33.4482 | 33.586 | 33.4014 | 33.5725 |

5.1.12 | 99.6323 | 99.792 | 99.5811 | 99.5992 | 33.4453 | 33.453 | 33.4158 | 33.5240 |

5.1.13 | 99.6216 | 99.792 | 99.5963 | 99.5802 | 33.4531 | 33.520 | 33.4236 | 33.4265 |

5.1.14 | 99.6078 | 99.621 | 99.5945 | 99.5831 | 33.4293 | 33.440 | 33.3951 | 33.5393 |

512 ∗ 512 | Theoretical NPCR | 99.5893 | Theoretical UACI | 33.3730~33.5541 | ||||

5.2.08 | 99.6105 | 99.960 | 99.5878 | 99.6136 | 33.5035 | 33.692 | 33.3978 | 33.5362 |

5.2.09 | 99.6033 | 99.876 | 99.5812 | 99.6022 | 33.4674 | 33.548 | 33.4182 | 33.3835 |

5.2.10 | 99.6101 | 99.654 | 99.6100 | 99.6147 | 33.4253 | 33.454 | 33.4263 | 33.4199 |

7.1.01 | 99.6136 | 99.957 | 99.6028 | 99.6310 | 33.4885 | 33.648 | 33.4474 | 33.5258 |

7.1.02 | 99.6040 | 99.918 | 99.6078 | 99.6151 | 33.4508 | 33.465 | 33.4326 | 33.4589 |

7.1.03 | 99.6101 | 99.849 | 99.5811 | 99.6124 | 33.4352 | 33.273 | 33.4836 | 33.4771 |

7.1.04 | 99.6178 | 99.991 | 99.5946 | 99.5930 | 33.5024 | 33.202 | 33.4782 | 33.4340 |

7.1.05 | 99.5979 | 99.942 | 99.5937 | 99.6136 | 33.4739 | 33.830 | 33.4716 | 33.4699 |

7.1.06 | 99.6269 | 99.670 | 99.5912 | 99.6041 | 33.4764 | 33.627 | 33.4365 | 33.4511 |

7.1.07 | 99.6193 | 99.983 | 99.6014 | 99.5995 | 33.4310 | 33.609 | 33.4313 | 33.4637 |

7.1.08 | 99.5979 | 99.818 | 99.6013 | 99.6170 | 33.4997 | 33.375 | 33.4460 | 33.4841 |

7.1.09 | 99.6113 | 99.874 | 99.6148 | 99.6109 | 33.4630 | 33.530 | 33.3856 | 33.4925 |

7.1.10 | 99.6166 | 99.697 | 99.6097 | 99.6079 | 33.4701 | 33.438 | 33.3941 | 33.4423 |

boat.512 | 99.6227 | 99.715 | 99.6101 | 99.6212 | 33.4448 | 33.374 | 33.3973 | 33.4156 |

gray21.512 | 99.6067 | 99.643 | 99.6034 | 99.5946 | 33.5113 | 33.507 | 33.4089 | 33.4367 |

ruler.512 | 99.6124 | 99.637 | 99.5945 | 99.6056 | 33.4620 | 33.415 | 33.4635 | 33.4649 |

1024 ∗ 1024 | Theoretical NPCR | 99.5994 | Theoretical UACI | 33.4183~33.5088 | ||||

5.3.01 | 99.6067 | 99.950 | 99.6032 | 99.6294 | 33.5013 | 33.508 | 33.4392 | 33.5005 |

5.3.02 | 99.6015 | 99.982 | 99.6108 | 99.6128 | 33.4255 | 33.514 | 33.4547 | 33.4263 |

7.2.01 | 99.6005 | 99.980 | 99.6036 | 99.6088 | 33.4438 | 33.487 | 33.4301 | 33.4511 |

Mean | 99.6098 | 99.8192 | 99.5957 | 99.6076 | 33.4634 | 33.5 | 33.4329 | 33.4674 |

Std | 0.0103 | 0.2665 | 0.0200 | 0.0173 | 0.0268 | 0.0459 | 0.0328 | 0.0518 |

Pass/All | 25/25 | 25/25 | 22/25 | 25/25 | 25/25 | 17/25 | 25/25 | 25/25 |

Area | 1/16 | 1/8 | 1/4 | 1/2 |
---|---|---|---|---|

PSNR | 21.314192 | 18.380856 | 15.342343 | 12.250188 |

Noise Level | 0.01 Level | 0.05 Level | 0.1 Level |

PSNR | 28.971428 | 22.408105 | 19.258843 |

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## Share and Cite

**MDPI and ACS Style**

Liu, P.; Zhou, S.; Yan, W.Q. A 3D Cuboid Image Encryption Algorithm Based on Controlled Alternate Quantum Walk of Message Coding. *Mathematics* **2022**, *10*, 4441.
https://doi.org/10.3390/math10234441

**AMA Style**

Liu P, Zhou S, Yan WQ. A 3D Cuboid Image Encryption Algorithm Based on Controlled Alternate Quantum Walk of Message Coding. *Mathematics*. 2022; 10(23):4441.
https://doi.org/10.3390/math10234441

**Chicago/Turabian Style**

Liu, Pai, Shihua Zhou, and Wei Qi Yan. 2022. "A 3D Cuboid Image Encryption Algorithm Based on Controlled Alternate Quantum Walk of Message Coding" *Mathematics* 10, no. 23: 4441.
https://doi.org/10.3390/math10234441