# A Novel Image Encryption Algorithm Based on Improved Arnold Transform and Chaotic Pulse-Coupled Neural Network

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Analysis of Chaotic PCNN Model

#### 2.1. Uncoupled Linking PCNN Model

#### 2.1.1. Firing Period Analysis of Uncoupled Linking PCNN

- (1)
- When n = 0, which is the initial stage, ${U}_{ij}(0)={F}_{ij}(0)={S}_{ij}$, ${E}_{ij}(0)=0$, ${Y}_{ij}(0)=1$, neurons fire for the first time;
- (2)
- When n = 1, ${U}_{ij}(1)={F}_{ij}(1)={S}_{ij}+{e}^{-{a}_{F}}{S}_{ij}$, ${E}_{ij}(1)={V}_{E}$, ${Y}_{ij}(1)=0$, neurons extinguish;
- (3)
- When n = 2, ${U}_{ij}(2)={F}_{ij}(2)={S}_{ij}+{e}^{-{a}_{F}}({S}_{ij}+{e}^{-{a}_{F}}{S}_{ij})$, ${E}_{ij}(2)={e}^{-{a}_{E}}{V}_{E}$, ${Y}_{ij}(2)=\epsilon [{S}_{ij}+{e}^{-{a}_{F}}({S}_{ij}+{e}^{-{a}_{F}}{S}_{ij})-{e}^{-{a}_{E}}{V}_{E}]$.

_{1,}then:

_{2}, then:

_{3}, then:

_{4}…, ${n}_{m}=1+{n}_{m-1}+\frac{1}{{a}_{E}}\mathrm{ln}(\frac{{e}^{-{a}_{E}}c{S}_{ij}+{V}_{E}}{c\text{'}{S}_{ij}}),\begin{array}{cc}& m=4,5,\dots ,N\end{array}$. where $c=\frac{1-{e}^{-({n}_{m-1}+1){a}_{F}}}{1-{e}^{-{a}_{F}}}$, $c\text{'}=\frac{1-{e}^{-({n}_{m}+1){a}_{F}}}{1-{e}^{-{a}_{F}}}$.

#### 2.2. Uncoupled Linking Chaotic PCNN

#### 2.2.1. Basic Bifurcation Behavior Analysis of Chaotic PCNN

#### 2.2.2. Non-Periodicity Analysis of Sequence

#### 2.3. Improved Arnold Transform Image Scrambling Method

_{1}, y

_{1}) and traverses all the pixels of the image through Arnold transformation so that the pixels of the entire image are scrambled to achieve the effect of image encryption [17].

## 3. Scheme and Process of Image Encryption and Decryption

_{1}; set the initial parameter value of the chaotic system. See Section 4.2 for the initial parameter value;

_{2}is obtained by converting the sequence L3 into an integer in the range of [0, 255] using the modulo operation of Equation (11), where “sum” in Equation (11) refers to the sum of all pixel values of the input grayscale image;

_{1}with the key sequence I

_{2}to generate a new random sequence, as shown in Equation (12), and then convert the new random sequence into a pre-encrypted image Q1 of size W × H;

_{3}, and convert the key sequence I

_{3}into an image matrix M2 with a size of W × H;

Algorithm 1: Proposed decryption algorithm. |

Input: Image M1 of size W × H.Output: Encryption result Q3.1: Set the initial parameter value of the chaotic system, set fixed parameters $n=500$, $a=0.8$, $w=7.7$; controlling parameters ${a}_{F}=0.12$, ${V}_{Em}=17.2$, ${S}_{ij}=0.16$, ${a}_{E}=1.5$; initial value ${x}_{{}_{0}}=3$; |

2: Generate chaotic sequence L_{1} and L_{2} using proposed CPCNN map; 3: sum = sum(M1); 4: ${I}_{1}$ = reshape (M1, 1, W × H); |

5: for i = 1 to L1 do6: $L3\left(i\right)=(L1\left(i\right)-\mathrm{max}(L1))/(\mathrm{max}(L1)-\mathrm{min}(L1))$; 7: if sum $\ne $ 0 then8: ${I}_{2}=floor((L3\cdot sum\cdot {10}^{8})\mathrm{mod}256)$; 9: else10: ${I}_{2}=floor((L3\cdot {10}^{8})\mathrm{mod}256)$; 11: end if12: end for13: Q2 = reshape (${I}_{1}\oplus {I}_{2}$, W, H); 14: Using a 100 × 100 square sliding window and setting the step size to 30, the pre-encrypted image is traversed, and the entire pre-encrypted image is Arnold transformed to obtain a new scrambled image Q2; 15: M2 = reshape (L2, W, H); 16: index = zeros (W, H); 17: for i = 1 to W do18: [~, index (i, :)] =sort (M2 (i, :)); 19: end for |

20: count = W; 21: for i = 1 to W do22: for j = 1 to H do23: Q3 (i, j) = Q2 (index (i, j), count); 24: end for 25: count = count − 1; |

26: end for |

## 4. Experiment Environment and Results

#### 4.1. Experiment Environment

#### 4.2. Experiment Parameter Setting and Result

## 5. Security Analysis

#### 5.1. Statistical Characteristic Analysis of Ciphertext

#### 5.1.1. Histogram Statistical Analysis

#### 5.1.2. Correlation Analysis of Adjacent Pixels

#### 5.2. Information Entropy Analysis

#### 5.3. Tests for Randomness

^{−4}, 1−10

^{−4}], the test was successful [23]. Any p-value outside this range is considered a failed test. Table 5 shows the final test results of TestU01; it can be seen that the sequences produced by the chaotic system proposed in this paper can pass the TestU01 test. The experimental results show that the chaotic sequence generated by chaotic pulse-coupled neural network has good randomness and is safe and reliable.

#### 5.4. Key Space Analysis

^{−16}, then ${a}_{F}$, ${V}_{Em}$, ${S}_{ij}$, ${a}_{E}$, and ${x}_{{}_{0}}$ have available key spaces of 10

^{12}, 10

^{12}, 10

^{13}, 10

^{14}, and 10

^{14}, respectively. From cryptographic theories, only a key space larger than 2

^{128}can resist illegal brute force attacks. The key space of the encryption algorithm in this paper can reach above 10

^{130}, which is larger than 2

^{430}. Therefore, the encryption algorithm in this paper has a large enough key space and can effectively resist brute force attacks. The key space comparison results are shown in Table 7.

#### 5.5. Key Sensitivity Analysis

^{−10}. It can be seen from Figure 17 that the correct plaintext image cannot be restored as long as the key is slightly changed in the decryption process, and the image obtained by making a slight change to a single key value is very different from the plaintext image. Therefore, it can be shown that the algorithm in this paper has a very good key sensitivity in the decryption process.

^{−10}, the NPCR and UACI values are between the corresponding encrypted image and the original encrypted image, and the proportions of different pixels of the two encrypted images are shown in Table 8.

#### 5.6. Anti-Attack Ability Analysis

#### 5.6.1. Anti-Differential Attack Analysis

#### 5.6.2. Anti-Salt and Pepper, Gaussian Noise Attack Analysis

#### 5.6.3. Analysis of Anti-Shearing Attack Ability

#### 5.6.4. Noise Processing of Decrypted Images

#### 5.7. Analysis of Speed

#### 5.8. Algorithm Comparative Analysis

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 8.**The (

**a**) structure diagram and (

**b**) flow chart of improved Arnold transform image scrambling method.

**Figure 11.**The encryption and decryption effect of the algorithm in this paper. (

**a**) Plain-text image of Lena image (original image), (

**b**) encrypted image, (

**c**) decrypted image, (

**d**) plain-text image of Cameraman image, (

**e**) encrypted image, (

**f**) decrypted image, (

**g**) plain-text image of White image, (

**h**) encrypted image, (

**i**) decrypted image, (

**j**) plain-text image of Black image, (

**k**) encrypted image, (

**l**) decrypted image.

**Figure 13.**Lena scatter distribution map of adjacent pixels before and after image encryption. (

**a**) Horizontal, (

**b**) vertical, and (

**c**) diagonal directions of the plain Lena image and (

**d**) horizontal, (

**e**) vertical, and (

**f**) diagonal directions of the encrypted Lena image.

**Figure 14.**Cameraman scatter distribution map of adjacent pixels before and after image encryption. (

**a**) Horizontal, (

**b**) vertical, and (

**c**) diagonal directions of the plain Cameraman image and (

**d**) horizontal, (

**e**) vertical, and (

**f**) diagonal directions of the encrypted Cameraman image.

**Figure 15.**White scatter distribution map of adjacent pixels before and after image encryption. (

**a**) Horizontal, (

**b**) vertical, and (

**c**) diagonal directions of the plain White image and (

**d**) horizontal, (

**e**) vertical, and (

**f**) diagonal directions of the encrypted White image.

**Figure 16.**Black scatter distribution map of adjacent pixels before and after image encryption. (

**a**) Horizontal, (

**b**) vertical, and (

**c**) diagonal directions of the plain Black image and (

**d**) horizontal, (

**e**) vertical, and (

**f**) diagonal directions of the encrypted Black image.

**Figure 17.**Decryption result after changing key. (

**a**,

**e**) Decryption image with correct key; (

**b**,

**f**) decrypted image with ${V}_{E\mathrm{m}}{+10}^{-10}$; (

**c**,

**g**) decrypted image with ${a}_{E}{+10}^{-10}$; (

**d**,

**h**) decrypted image with ${a}_{F}{+10}^{-10}$.

**Figure 18.**Comparison diagram before and after differential processing. (

**a**,

**e**) Plain-text image, (

**b**,

**f**) encrypted image, (

**c**,

**g**) plain-text image after differential processing, and (

**d**,

**h**) encrypted image after differential processing.

**Figure 19.**Ciphertext image with salt and pepper noise and its decryption graph. (

**a**,

**d**) Encrypted image with 0.1 salt and pepper noise; (

**b**,

**e**) plain-text image; (

**c**,

**f**) decrypted image.

**Figure 20.**The ciphertext image with Gaussian noise and its decryption graph. (

**a**,

**d**) Encrypted image with Gaussian noise; (

**b**,

**e**) plain-text image; (

**c**,

**f**) decrypted image.

**Figure 21.**The cropped ciphertext image and its decryption renderings. Encrypted image with (

**a**,

**i**) 1/16 data loss, (

**b**,

**j**) 1/8 data loss, (

**c**,

**k**) 1/4 data loss, and (

**d**,

**l**) 1/2 data loss; decrypted image with (

**e**,

**m**) 1/16 data loss, (

**f**,

**n**) 1/8 data loss, (

**g**,

**o**) 1/4 data loss, and (

**h**,

**p**) 1/2 data loss.

**Figure 22.**De-noising result. (

**a**) Original image; (

**b**) encrypted image with 0.7 salt and pepper noise; (

**c**) decrypted image of (

**b**); (

**d**) denoising for decrypted image (

**c**). (

**e**) Original image; (

**f**) encrypted image with 0.8 salt and pepper noise; (

**g**) decrypted image of (

**f**); (

**h**) denoising for decrypted image (

**g**).

Image Size N | Scrambling Period T | Image Size N | Scrambling Period T |
---|---|---|---|

10 | 30 | 32 | 24 |

14 | 24 | 64 | 48 |

16 | 12 | 128 | 96 |

18 | 12 | 256 | 192 |

25 | 50 | 512 | 384 |

Image (256 × 256) | Original Original ${\mathit{\chi}}^{2}$ | Encrypted Image ${\mathit{\chi}}^{2}$ | Result |
---|---|---|---|

Lena | 4.2981 × 10^{5} | 232.6328 | pass |

Cameraman | 1.5196 × 10^{6} | 211.2109 | pass |

White | 1.6712 × 10^{7} | 209.8203 | pass |

Black | 1.6711 × 10^{7} | 259.1016 | pass |

Peppers | 3.1639 × 10^{4} | 246.3125 | pass |

Plane | 1.7322 × 10^{5} | 219.4922 | pass |

Image (256 × 256) | Correlation Coefficient | |||||
---|---|---|---|---|---|---|

Unencrypted | Encrypted | |||||

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

Lena | 0.9204 | 0.9546 | 0.8944 | 0.0035 | 5.3876 × 10^{−4} | 3.2753 × 10^{−4} |

Cameraman | 0.9756 | 0.9851 | 0.9601 | −0.0020 | 0.0016 | 0.0013 |

White | -- | -- | -- | 0.0037 | −7.5285 × 10^{−4} | −7.6476 × 10^{−4} |

Black | -- | -- | -- | −1.4710 × 10^{−5} | 0.0065 | −0.0042 |

Peppers | 0.9648 | 0.9697 | 0.9388 | 0.0015 | 0.0029 | 0.0019 |

Plane | 0.9387 | 0.9320 | 0.8832 | 0.0018 | −0.0057 | −1.1486 × 10^{−4} |

Image (256 × 256) | Information Entropy | |
---|---|---|

Original | Encrypted | |

Lena | 7.7758 | 7.9974 |

Cameraman | 6.9749 | 7.9977 |

White | 0 | 7.9977 |

Black | 0 | 7.9972 |

Peppers | 7.5798 | 7.9973 |

Plane | 6.7334 | 7.9976 |

Battery | Parameters | Number of Statistics | Result |
---|---|---|---|

SmallCrush | Standard | 15 | Pass |

Alphabit | Standard | 17 | Pass |

Rabbit | Standard | 40 | Pass |

FIPS_140_2 | Standard | 16 | Pass |

BlockAlphabit | Standard | 17 | Pass |

Image | GVD Score | Image | GVD Score |
---|---|---|---|

Lena | 0.9415 | Peppers | 0.9674 |

Cameraman | 0.9700 | Plane | 0.9512 |

White | 1.000 | Black | 1.000 |

Method | Year | Key Space |
---|---|---|

C.H. et al. [26] | 2018 | 2^{106} |

G.D. et al. [27] | 2018 | 2^{186} |

R.Z. et al. [28] | 2019 | 2^{199} |

Xiaohong et al. [7] | 2021 | 2^{212} |

Xw et al. [29] | 2021 | 2^{100} |

Xiang H et al. [30] | 2021 | 2^{128} |

Khalil Noura et al. [6] | 2021 | 2^{262} |

Wang X et al. [4] | 2021 | 2^{420} |

Our algorithm | 2022 | 2^{430} |

Image (256 × 256) | Initial Value | NPCR | UACI | Different Pixel Proportions |
---|---|---|---|---|

Lena | ${a}_{F}{+10}^{-10}$ | 99.6368 | 33.3494 | 99.64 |

${V}_{E\mathrm{m}}{+10}^{-10}$ | 99.6170 | 33.4318 | 99.62 | |

${S}_{ij}{+10}^{-10}$ | 99.6002 | 33.3646 | 99.60 | |

${a}_{E}{+10}^{-10}$ | 99.6063 | 33.5540 | 99.61 | |

${x}_{0}{+10}^{-10}$ | 99.6231 | 33.3861 | 99.62 | |

Cameraman | ${a}_{F}{+10}^{-10}$ | 99.6445 | 33.3839 | 99.64 |

${V}_{E\mathrm{m}}{+10}^{-10}$ | 99.5956 | 33.5159 | 99.60 | |

${S}_{ij}{+10}^{-10}$ | 99.6078 | 33.4547 | 99.61 | |

${a}_{E}{+10}^{-10}$ | 99.6109 | 33.4457 | 99.61 | |

${x}_{0}{+10}^{-10}$ | 99.6445 | 33.4119 | 99.64 | |

White | ${a}_{F}{+10}^{-10}$ | 99.6429 | 33.6027 | 99.64 |

${V}_{E\mathrm{m}}{+10}^{-10}$ | 99.6689 | 33.5871 | 99.67 | |

${S}_{ij}{+10}^{-10}$ | 99.5865 | 33.5613 | 99.59 | |

${a}_{E}{+10}^{-10}$ | 99.6262 | 33.5986 | 99.63 | |

${x}_{0}{+10}^{-10}$ | 99.5895 | 33.2934 | 99.62 | |

Black | ${a}_{F}{+10}^{-10}$ | 99.6078 | 33.6249 | 99.61 |

${V}_{E\mathrm{m}}{+10}^{-10}$ | 99.6063 | 33.5503 | 99.61 | |

${S}_{ij}{+10}^{-10}$ | 99.5712 | 33.4333 | 99.57 | |

${a}_{E}{+10}^{-10}$ | 99.6048 | 33.4588 | 99.60 | |

${x}_{0}{+10}^{-10}$ | 99.5895 | 33.4558 | 99.59 | |

Average | -- | 99.614015 | 33.4841 | 99.6145 |

**Table 9.**Comparison of different pixels of encrypted images before and after differential processing (%).

Image | Different Pixel Proportions | Image | Different Pixel Proportions |
---|---|---|---|

Lena | 99.65 | Cameraman | 99.59 |

White | 99.62 | Black | 99.58 |

Peppers | 99.65 | Plane | 99.61 |

Original Image (256 × 256) | NPCR | UACI |
---|---|---|

Lena | 99.6460 | 33.4397 |

Cameraman | 99.5880 | 33.5050 |

White | 99.6170 | 33.4276 |

Black | 99.5758 | 33.5291 |

Peppers | 99.6506 | 33.4559 |

Plane | 99.6063 | 33.4554 |

Average | 99.61395 | 33.468783 |

Ideal value | 99.609375 | 33.463542 |

Image | Year | Image Size | Time (s) |
---|---|---|---|

Lena (Our method) | 2022 | 256 × 256 | 0.1690 |

Cameraman (Our method) | 2022 | 256 × 256 | 0.1740 |

JinLong et al. [36] | 2021 | 256 × 256 | 0.6563 |

Wang et al. [37] | 2021 | 256 × 256 | 0.2523 |

Wenying Wen et al. [38] | 2020 | 256 × 256 | 2.1328 |

Farah M et al. [39] | 2020 | 256 × 256 | 1.1202 |

Lena (Our method) | 2022 | 512 × 512 | 0.7080 |

Cameraman (Our method) | 2022 | 512 × 512 | 0.6640 |

Wenying Wen et al. [38] | 2020 | 512 × 512 | 18.1354 |

José, A. et al. [40] | 2019 | 512 × 512 | 10.4200 |

Lena (Our method) | 2022 | 1024 × 1024 | 2.2990 |

Cameraman (Our method) | 2022 | 1024 × 1024 | 2.1700 |

Image (256 × 256) | Method | Year | Info Entropy | Correlation Coefficient | ||
---|---|---|---|---|---|---|

Horizontal | Vertical | Diagonal | ||||

Lena | Li et al. [41] | 2020 | 7.9894 | 0.0044 | 0.0015 | 0.0019 |

Wang et al. [42] | 2020 | 7.9969 | 0.0006 | 0.0082 | 0.0032 | |

Kamrani et al. [43] | 2020 | 7.9945 | -- | -- | -- | |

Hosny et al. [22] | 2021 | 7.9972 | 0.0069 | 0.0479 | 0.0075 | |

Xw et al. [29] | 2021 | 7.9971 | −0.0017 | −0.0132 | 0.0084 | |

Zhang et al. [44] | 2021 | 7.9969 | 0.0040 | −0.0012 | −0.0021 | |

Farhan et al. [45] | 2021 | 7.9971 | −0.0004 | −0.0028 | 0.0040 | |

Wang et al. [37] | 2021 | 7.9960 | 0.0023 | 0.0020 | 0.0073 | |

Xiang et al. [30] | 2021 | 7.9972 | 0.0013 | -0.0041 | −0.0044 | |

JinLong et al. [36] | 2021 | 7.9858 | 0.0031 | 0.0076 | −0.0026 | |

Proposed | 2022 | 7.9974 | −0.0035 | 5.3876 × 10^{−4} | 3.2753 × 10^{−4} | |

Cameraman | Niu et al. [46] | 2020 | 7.9971 | −0.0070 | 0.0083 | 0.0013 |

Kamrani et al. [43] | 2020 | 7.9947 | -- | -- | -- | |

Wu et al. [47] | 2021 | 7.9935 | −0.0036 | 0.0048 | 0.0073 | |

JinLong et al. [36] | 2021 | 7.9868 | −0.0252 | −0.0060 | −0.0078 | |

Proposed | 2022 | 7.9977 | 0.0020 | 0.0016 | 0.0013 | |

Peppers | Hua, Z. et al. [48] | 2019 | 7.9971 | 0.0196 | 0.0165 | 0.0210 |

Minjun et al. [49] | 2020 | 7.9970 | 0.00476 | −0.009531 | 0.007338 | |

Wang et al. [37] | 2021 | 7.9964 | −0.0037 | 0.0035 | −0.0057 | |

Xw et al. [29] | 2021 | 7.9971 | −0.0062 | −0.0236 | −0.0047 | |

Wu et al. [47] | 2021 | 7.9941 | −0.0170 | −0.0334 | −0.0073 | |

Hosny et al. [22] | 2021 | 7.9970 | 0.0211 | 0.0129 | 0.0013 | |

Proposed | 2022 | 7.9973 | 0.0015 | 0.0029 | −0.0019 | |

Plane | Wu et al. [50] | 2018 | 7.9970 | 0.0028 | 0.0041 | 0.0010 |

Hua, Z. et al. [48] | 2019 | 7.9971 | 0.0055 | 0.0014 | 0.0083 | |

Xw et al. [29] | 2021 | 7.9972 | −0.0043 | −0.0236 | −0.0047 | |

Hosny et al. [22] | 2021 | 7.9972 | 0.0229 | 0.0103 | 0.0100 | |

Wang et al. [37] | 2021 | 7.9959 | 0.0054 | 0.0027 | 0.0028 | |

Proposed | 2022 | 7.9976 | −0.0018 | −0.0057 | −1.1486 × 10^{−4} |

Image (256 × 256) | Method | Year | NPCR (%) | UACI (%) | ${\mathit{\chi}}^{2}$ |
---|---|---|---|---|---|

Ideal value | 99.609375 | 33.463542 | Minimum | ||

Lena | Kamrani et al. [43] | 2020 | 99.7864 | 30.3256 | -- |

Li et al. [41] | 2020 | 99.66 | 33.42 | -- | |

Minjun et al. [49] | 2020 | 99.6114 | 33.4523 | -- | |

Hosny et al. [22] | 2021 | 99.6246 | 33.4226 | 264.8750 | |

Zhang et al. [44] | 2021 | 99.62 | 33.50 | -- | |

Wang et al. [37] | 2021 | 99.5894 | 33.4629 | -- | |

Xw et al. [29] | 2021 | -- | -- | 266.6797 | |

Proposed | 2022 | 99.6460 | 33.4397 | 232.6328 | |

Cameraman | Kamrani et al. [43] | 2020 | 99.791 | 27.6376 | -- |

Zhang et al. [44] | 2021 | 99.63 | 33.56 | -- | |

Wang et al. [37] | 2021 | 99.5879 | 33.4553 | -- | |

Proposed | 2022 | 99.5880 | 33.5050 | 211.2109 | |

Peppers | Minjun et al. [49] | 2020 | 99.6115 | 33.4245 | -- |

Hosny et al. [22] | 2021 | 99.6033 | 33.4274 | 268.4766 | |

Xw et al. [29] | 2021 | -- | -- | 260.3906 | |

Proposed | 2022 | 99.6506 | 33.4559 | 246.3125 | |

Plane | Minjun et al. [49] | 2020 | 99.6043 | 33.2875 | -- |

Xw et al. [29] | 2021 | -- | -- | 252.1172 | |

Proposed | 2022 | 99.6063 | 33.4554 | 219.4922 |

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## Share and Cite

**MDPI and ACS Style**

Ye, J.; Deng, X.; Zhang, A.; Yu, H.
A Novel Image Encryption Algorithm Based on Improved Arnold Transform and Chaotic Pulse-Coupled Neural Network. *Entropy* **2022**, *24*, 1103.
https://doi.org/10.3390/e24081103

**AMA Style**

Ye J, Deng X, Zhang A, Yu H.
A Novel Image Encryption Algorithm Based on Improved Arnold Transform and Chaotic Pulse-Coupled Neural Network. *Entropy*. 2022; 24(8):1103.
https://doi.org/10.3390/e24081103

**Chicago/Turabian Style**

Ye, Jinhong, Xiangyu Deng, Aijia Zhang, and Haiyue Yu.
2022. "A Novel Image Encryption Algorithm Based on Improved Arnold Transform and Chaotic Pulse-Coupled Neural Network" *Entropy* 24, no. 8: 1103.
https://doi.org/10.3390/e24081103