# Generation of Boxes and Permutations Using a Bijective Function and the Lorenz Equations: An Application to Color Image Encryption

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Elliptic Curve

#### 2.2. Lorenz Equations

#### 2.3. Correlation Coefficient

#### 2.4. Entropy

#### 2.5. Discrete Fourier Transform

#### 2.6. Goodness-of-Fit Test

#### 2.7. Parameters NPCR, UACI and AC

#### 2.8. Parameters of Homogeneity, Contrast, Energy and Median Filter

#### 2.9. Median Filter

## 3. Model Development

#### 3.1. Calculation of the Multiplicative Inverse

**Theorem**

**1.**

**Proof of Theorem**

**1.**

#### 3.2. SP Parameter

#### 3.3. Algorithm for the Generation of Permutations

#### 3.4. Cipher Procedure

#### 3.5. Generation of the S-Boxes, Permutation, and Schedule Keys

- p = 988464ba59685284506433ccd3f83450166fda2d2ec7109a5c0679434e9dfb46b3a447043b406c4115af9a2c7fdc17bc9b6668f07d80d7142f534a1dc64ef400b9b2100acb691
- q = 2621192e965a14a114190cf334fe0d14059bf68b4bb1c42697019e50d3a77ed1ace911d9c1e6fa9ecb772f44670d29e05a69fc249b108ed06114d9e01b7662721ecf151ce2329
- a = 31662dbf1d0c1691728e2c47e26720c3d0f760b216aa800eb153c54ae3e0c522345eb09
- b = 308
- k = 870eebe8cf19ece84593dd9deaec2ebab1380c94c240fe8fc1d45836fff18114c42308e5aafef0ee4d1a643b179415eb34d8b2118e51ad727b63efc5dba104179bcece5a0d7cb

- ${x}_{1}$ = 28b1f61561824dac022aa29d37df70295a2d7f34f6965e032d85b35b6e4c8403a47922b96753ba338061a05eee530f5759043d58aa09d69ae8b2377b640c01e484ac14d27d693
- ${x}_{2}$ = 288189d9988f9d839ae797195f3a4b512b36773156affd0b64a5ed740c9ea059233eab4765397a0a5de87ea46a20d208cf8988d433e4d703792e2f950ad6a0a631f0d424e6951

#### 3.6. Images Used to Evaluate ICLEBF

## 4. Damage in Encrypted Images

#### 4.1. Noise Generated by the Variable ${\chi}^{2}$

#### 4.2. Additive and Multiplicative Noises

#### 4.3. Gaussian Noise

#### 4.4. Occlusion Noise

## 5. Results

#### 5.1. Correlation and Entropy

#### 5.2. Results of NPCR, UACI, AC, Energy, Contrast and Homogeneity

#### 5.3. Discrete Fourier Transform and Goodness-of-Fit Test

#### 5.4. Tests on Black or White Images

#### 5.5. Result of Encrypted Images with Noise

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ICLEBF | Image Cipher utilizing Lorenz Equation and a Bijective Function |

NPCR | Number of Pixels Change Rate |

UACI | Unified Average Changing Intensity |

AC | Avalanche Criteria |

SP | Similarity Parameter |

SPN | Substitution Permutation Network |

RSA | Rivest–Shamir–Adleman |

## References

- Alexan, W.; ElBeltagy, M.; Aboshousha, A. RGB Image Encryption through Cellular Automata, S-Box and the Lorenz System. Symmetry
**2022**, 14, 443. [Google Scholar] [CrossRef] - Sani, R.H.; Behnia, S.; Akhshani, A. Creation of S-box based on a hierarchy of Julia sets: Image encryption approach. Multidimens. Syst. Signal. Process.
**2022**, 33, 39–62. [Google Scholar] [CrossRef] - Hayat, U.; Ullah, I.; Azam, N.A.; Azhar, S. A Novel Image Encryption Scheme Based on Elliptic Curves over Finite Rings. Entropy
**2022**, 24, 571. [Google Scholar] [CrossRef] - Murtaza, G.; Azam, N.A.; Hayat, U. Designing an Efficient and Highly Dynamic Substitution-Box Generator for Block Ciphers Based on Finite Elliptic Curves. Secur. Commun. Netw.
**2021**, 2021, 3367521. [Google Scholar] [CrossRef] - Zhou, S.; Zhao, Z.; Wang, X. Novel chaotic colour image cryptosystem with deep learning. Chaos Solitons Fractals
**2022**, 161, 112380. [Google Scholar] [CrossRef] - Li, T.; Yan, W.; Chi, Z. A new image encryption algorithm based on optimized Lorenz chaotic system. Concurr. Comput.
**2022**, 34, e5902. [Google Scholar] [CrossRef] - Ren, H.; Niu, S.; Chen, J.; Li, M.; Yue, Z. A Visually Secure Image Encryption Based on the Fractional Lorenz System and Compressive Sensing. Fractal Fract.
**2022**, 6, 302. [Google Scholar] [CrossRef] - Rashmi, P.; Supriya, M.C.; Hua, Q. Enhanced Lorenz-Chaotic Encryption Method for Partial Medical Image Encryption and Data Hiding in Big Data Healthcare. Secur. Commun. Netw.
**2022**, 2022, 9363377. [Google Scholar] [CrossRef] - Tang, M.; Zeng, G.; Yang, Y.; Chen, J. A hyperchaotic image encryption scheme based on the triple dislocation of the Liu and Lorenz system. Optik
**2022**, 261, 169133. [Google Scholar] [CrossRef] - Bhat, J.; Saqib, M.; Moon, A.H. Fuzzy extractor and chaos enhanced elliptic curve cryptography for image encryption and authentication. Int. J. Syst. Assur. Eng. Manag.
**2022**, 13, 697–712. [Google Scholar] [CrossRef] - Elsaid, S.A.; Alotaibi, E.R.; Alsaleh, S. A robust hybrid cryptosystem based on DNA and Hyperchaotic for images encryption. Multimed. Tools Appl.
**2022**, 82, 1995–2019. [Google Scholar] [CrossRef] - Li, M.; Wang, M.; Fan, H.; An, K.; Liu, G. A novel plaintext-related chaotic image encryption scheme with no additional plaintext information. Chaos Solitons Fractals
**2022**, 158, 111989. [Google Scholar] [CrossRef] - Ametepe, A.F.X.; Ahouandjinou, A.S.; Ezin, E.C. Robust encryption method based on AES-CBC using elliptic curves Diffie–Hellman to secure data in wireless sensor networks. Wirel. Netw.
**2022**, 28, 991–1001. [Google Scholar] [CrossRef] - Banik, A.; Laiphrakpam, D.S.; Agrawal, A.; Patgiri, R. Secret image encryption based on chaotic system and elliptic curve cryptography. Digit. Signal Process
**2022**, 129, 103639. [Google Scholar] [CrossRef] - Silva-García, V.M.; Flores-Carapia, R.; Rentería-Márquez, C.; Luna-Benoso, B.; Chimal-Eguía, J.C. Image cipher applications using the elliptical curve and chaos. Int. J. Appl. Math. Comput. Sci.
**2020**, 30, 377–391. [Google Scholar] [CrossRef] - Shen, C.; Panda, S.; Vogelstein, J.T. The Chi-Square Test of Distance Correlation. J. Comput. Graph. Stat.
**2022**, 31, 254–262. [Google Scholar] [CrossRef] - General de la Nación, A. Manual de digitalización de documentos. Bol. Arch. General Nación
**2022**, 9, 41–117. [Google Scholar] - Stinson, D.R.; Patterson, M. Cryptography: Theory and Practice, 4th ed.; CRC Press: Boca Raton, FL, USA, 2018; pp. 278–295. [Google Scholar]
- Underwood, R.G. Cryptography for Secure Encryption, 1st ed.; Springer: Cham, Switzerland, 2022; pp. 271–296. [Google Scholar]
- Zheng, Z. Modern Cryptography, 1st ed.; Springer: Singapore, 2022; Volume 1, pp. 229–251. [Google Scholar]
- Silva-García, V.M.; Flores-Carapia, R.; González-Ramírez, M.D.; Vega-Alvarado, E.; Villarreal-Cervantes, M.G. Cryptosystem Based on the Elliptic Curve With a High Degree of Resistance to Damage on the Encrypted Images. IEEE Access
**2020**, 8, 218777–218792. [Google Scholar] [CrossRef] - Ali, F.; Rather, B.A.; Fatima, N.; Sarfraz, M.; Ullah, A.; Alharbi, K.A.M.; Dad, R. On the Topological Indices of Commuting Graphs for Finite Non-Abelian Groups. Symmetry
**2022**, 14, 1266. [Google Scholar] [CrossRef] - Aldaya, A.C.; Sarmiento, A.J.C.; Sánchez-Solano, S. SPA vulnerabilities of the binary extended Euclidean algorithm. J. Cryptogr. Eng.
**2017**, 7, 273–285. [Google Scholar] [CrossRef] - Cohen, S.D.; Kapetanakis, G.; Reis, L. The existence of F
_{q}-primitive points on curves using freeness. Comptes Rendus Math.**2022**, 360, 641–652. [Google Scholar] [CrossRef] - Yu, J.; Li, C.; Song, X.; Guo, S.; Wang, E. Parallel Mixed Image Encryption and Extraction Algorithm Based on Compressed Sensing. Entropy
**2021**, 23, 278. [Google Scholar] [CrossRef] [PubMed] - Zeng, J.; Wang, C.; Ye, G. A Novel Hyperchaotic Image Encryption System Based on Particle Swarm Optimization Algorithm and Cellular Automata. Secur. Commun. Netw.
**2021**, 2021, 6675565. [Google Scholar] [CrossRef] - Chai, X.; Fu, J.; Gan, Z.; Lu, Y.; Zhang, Y. An image encryption scheme based on multi-objective optimization and block compressed sensing. Nonlinear Dyn.
**2022**, 108, 2671–2704. [Google Scholar] [CrossRef] - Shannon, C.E. A mathematical theory of communication. Bell Syst. Tech. J.
**1948**, 27, 379–423. [Google Scholar] [CrossRef] [Green Version] - Panchikkil, S.; Manikandan, V.; Zhang, Y.D. An efficient spatial transformation-based entropy retained reversible data hiding scheme in encrypted images. Optik
**2022**, 261, 169211. [Google Scholar] [CrossRef] - Kowalska, K.A.; Fogliano, D.; Coello, J.G. On the Revision of NIST 800-22 Test Suites; Cryptology ePrint Archive, Paper 2022/540; Crypta Labs: London, UK, 2022; Available online: https://eprint.iacr.org/2022/540 (accessed on 11 December 2022).
- Liu, Z.; Shen, J.; Barfield, R.; Schwartz, J.; Baccarelli, A.A.; Lin, X. Large-Scale Hypothesis Testing for Causal Mediation Effects with Applications in Genome-wide Epigenetic Studies. J. Am. Stat. Assoc.
**2022**, 117, 67–81. [Google Scholar] [CrossRef] - Bourgade, P.; Mody, K.; Pain, M. Optimal Local Law and Central Limit Theorem for β-Ensembles. Commun. Math. Phys.
**2022**, 390, 1017–1079. [Google Scholar] [CrossRef] - Pandurangi Ramacharya, B.; Patil, M.R.; Keralkar, S. Fast partial image encryption with fuzzy logic and chaotic mapping. Evol. Intel.
**2022**, 1, 1–17. [Google Scholar] [CrossRef] - Arab, A.A.; Rostami, M.J.B.; Ghavami, B. An image encryption algorithm using the combination of chaotic maps. Optik
**2022**, 261, 169122. [Google Scholar] [CrossRef] - Poojary, A.; Kiran Kumar, V.; Nagesh, H. FPGA implementation novel lightweight MBRISI cipher. J. Ambient. Intell. Humaniz. Comput.
**2022**, 1, 1–13. [Google Scholar] [CrossRef] - Kiran, P.; Parameshachari, B. Resource Optimized Selective Image Encryption of Medical Images Using Multiple Chaotic Systems. Microprocess. Microsyst.
**2022**, 91, 104546. [Google Scholar] [CrossRef] - Iqbal, N.; Hanif, M.; Rehman, Z.U.; Zohaib, M. On the novel image encryption based on chaotic system and DNA computing. Multimed. Tools Appl.
**2022**, 81, 8107–8137. [Google Scholar] [CrossRef] - Asif, M.; Asamoah, J.K.K.; Hazzazi, M.M.; Alharbi, A.R.; Ashraf, M.U.; Alghamdi, A.M. A Novel Image Encryption Technique Based on Cyclic Codes over Galois Field. Comput. Intell. Neurosci.
**2022**, 2022, 1–9. [Google Scholar] [CrossRef] - Guo, S.; Wang, G.; Han, L.; Song, X.; Yang, W. COVID-19 CT image denoising algorithm based on adaptive threshold and optimized weighted median filter. Biomed. Signal Process. Control
**2022**, 75, 103552. [Google Scholar] [CrossRef] [PubMed] - Fink, E.; Clarke, P.; Spoerk, M.; Khinast, J. Unsupervised real-time evaluation of optical coherence tomography (OCT) images of solid oral dosage forms. J. Real-Time Image Process.
**2022**, 19, 881–892. [Google Scholar] [CrossRef] - Gallian, J.A. Contemporary Abstract Algebra, 10th ed.; CRC Press: Boca Raton, FL, USA, 2021; pp. 19–21. [Google Scholar]
- Lone, M.A.; Qureshi, S. RGB image encryption based on symmetric keys using Arnold transform, 3D chaotic map and affine hill cipher. Optik
**2022**, 260, 168880. [Google Scholar] [CrossRef] - Shetty, N.P.; Muniyal, B.; Kaithi, R.R.; Yemma, S.C.R. Comparison of Encryption Techniques to Encrypt Private Parts of an Image. In Proceedings of the Advances in Electrical and Computer Technologies; Sengodan, T., Murugappan, M., Misra, S., Eds.; Springer Nature: Singapore, 2022; pp. 535–557. [Google Scholar] [CrossRef]
- Chanda, S.; Luhach, A.K.; Alnumay, W.; Sengupta, I.; Sinha Roy, D. A lightweight device-level Public Key Infrastructure with DRAM based Physical Unclonable Function (PUF) for secure cyber physical systems. Comput. Commun.
**2022**, 190, 87–98. [Google Scholar] [CrossRef] - Yuvaraj, N.; Praghash, K.; Karthikeyan, T. Data Privacy Preservation and Trade-off Balance Between Privacy and Utility Using Deep Adaptive Clustering and Elliptic Curve Digital Signature Algorithm. Wirel. Pers. Commun.
**2022**, 124, 655–670. [Google Scholar] [CrossRef] - Chai, X.; Wu, H.; Gan, Z.; Han, D.; Zhang, Y.; Chen, Y. An efficient approach for encrypting double color images into a visually meaningful cipher image using 2D compressive sensing. Inf. Sci.
**2021**, 556, 305–340. [Google Scholar] [CrossRef] - Zhang, Y.; Chen, A.; Chen, B. A unified improvement of the AES algorithm. Multimed. Tools Appl.
**2022**, 81, 18875–18895. [Google Scholar] [CrossRef] - Feixiang, Z.; Mingzhe, L.; Kun, W.; Hong, Z. Color image encryption via Hénon-zigzag map and chaotic restricted Boltzmann machine over Blockchain. Opt. Laser Technol.
**2021**, 135, 106610. [Google Scholar] [CrossRef] - Xingyuan, W.; Junjian, Z.; Guanghui, C. An image encryption algorithm based on ZigZag transform and LL compound chaotic system. Opt. Laser Technol.
**2019**, 119, 105581. [Google Scholar] [CrossRef] - Yaghouti Niyat, A.; Moattar, M.H.; Niazi Torshiz, M. Color image encryption based on hybrid hyper-chaotic system and cellular automata. Opt. Lasers Eng.
**2017**, 90, 225–237. [Google Scholar] [CrossRef] - Liu, X.; Tong, X.; Wang, Z.; Zhang, M. Uniform non-degeneracy discrete chaotic system and its application in image encryption. Nonlinear Dyn.
**2022**, 108, 653–682. [Google Scholar] [CrossRef] - Lin, C.H.; Hu, G.H.; Chan, C.Y.; Yan, J.J. Chaos-Based Synchronized Dynamic Keys and Their Application to Image Encryption with an Improved AES Algorithm. Appl. Sci.
**2021**, 11, 1329. [Google Scholar] [CrossRef] - Yarom, Y.; Genkin, D.; Heninger, N. CacheBleed: A timing attack on OpenSSL constant-time RSA. J. Cryptogr. Eng.
**2017**, 7, 99–112. [Google Scholar] [CrossRef] - K.U., S.; Mohamed, A. A novel image encryption scheme using both pixel level and bit level permutation with chaotic map. Appl. Soft Comput.
**2020**, 90, 106162. [Google Scholar] [CrossRef]

**Figure 4.**Flat image of Barbara (

**a**). Decrypted image of Barbara when a multiplicative noise of size 40% is applied to a figure encrypted with AES–CBC (

**b**). Decrypted image of Barbara when a multiplicative noise of size 40% is applied to a figure encrypted with ICLEBF (

**c**).

**Figure 5.**Image (

**a**) is the Vicky cat image ciphered of Figure 1c using AES–CBC with 40% occlusion damage and (

**b**) is the deciphered image.

**Figure 6.**Vicky cat image (

**a**) ciphered using ICLEBF with 40% occlusion damage and (

**b**) is the deciphered image.

**Figure 7.**(

**a**) The Vicky cat image deciphered with 40% occlusion noise, and (

**b**) deciphered with noise and then median filter is applied.

$({x}_{1}-2,{x}_{2}+2)$ | $({x}_{1}-1,{x}_{2}+2)$ | $({x}_{1},{x}_{2}+2)$ | $({x}_{1}+1,{x}_{2}+2)$ | $({x}_{1}+2,{x}_{2}+2)$ |

$({x}_{1}-2,{x}_{2}+1)$ | $({x}_{1}-1,{x}_{2}+1)$ | $({x}_{1},{x}_{2}+1)$ | $({x}_{1}+1,{x}_{2}+1)$ | $({x}_{1}+2,{x}_{2}+1)$ |

$({x}_{1}-2,{x}_{2})$ | $({x}_{1}-1,{x}_{2})$ | $({x}_{1},{x}_{2})$ | $({x}_{1}+1,{x}_{2})$ | $({x}_{1}+2,{x}_{2})$ |

$({x}_{1}-2,{x}_{2}-1)$ | $({x}_{1}-1,{x}_{2}-1)$ | $({x}_{1},{x}_{2}-1)$ | $({x}_{1}+1,{x}_{2}-1)$ | $({x}_{1}+2,{x}_{2}-1)$ |

$({x}_{1}-2,{x}_{2}-2)$ | $({x}_{1}-1,{x}_{2}-2)$ | $({x}_{1},{x}_{2}-2)$ | $({x}_{1}+1,{x}_{2}-2)$ | $({x}_{1}+2,{x}_{2}-2)$ |

**Table 2.**Correlation of the encrypted test images of Figure 1.

Color | Correlation | Lena | Barbara | Vicky | Cameraman |
---|---|---|---|---|---|

Horizontal | $0.0004$ | $-0.0080$ | $-0.0011$ | $-0.0061$ | |

Red | Vertical | $-0.0097$ | $0.0086$ | $0.0034$ | $0.0009$ |

Diagonal | $0.0069$ | $-0.0034$ | $0.0079$ | $0.0133$ | |

Horizontal | $-0.0037$ | $-0.0196$ | $-0.0170$ | $0.0014$ | |

Green | Vertical | $-0.0052$ | $0.0060$ | $0.0006$ | $0.0131$ |

Diagonal | $0.0058$ | $-0.0065$ | $0.0174$ | $0.0062$ | |

Horizontal | $-0.0044$ | $0.0198$ | $-0.0037$ | $0.0030$ | |

Blue | Vertical | $0.0116$ | $-0.0021$ | $-0.0019$ | $-0.0012$ |

Diagonal | $0.0009$ | $-0.0020$ | $0.0042$ | $0.0146$ |

**Table 3.**Entropy of the encrypted test images of Figure 1.

Color | Lena | Barbara | Vicky | Cameraman |
---|---|---|---|---|

Red | $7.99919$ | $7.99935$ | $7.99919$ | $7.99932$ |

Green | $7.99929$ | $7.99925$ | $7.99928$ | $7.99926$ |

Blue | $7.99923$ | $7.99935$ | $7.99927$ | $7.99947$ |

**Table 4.**Entropy analysis of Lena Figure 1a.

Color | ICLEBF System | [48] | [49] | [50] |
---|---|---|---|---|

Red | $7.9992$ | $7.9921$ | $7.9987$ | $7.9972$ |

Green | $7.9993$ | $7.9917$ | $7.9991$ | $7.9973$ |

Blue | $7.9992$ | $7.9972$ | $7.9983$ | $7.9972$ |

Color | Lena | Barbara | Vicky | Cameraman |
---|---|---|---|---|

Red | $99.609$ | $99.606$ | $99.602$ | $99.603$ |

Green | $99.614$ | $99.608$ | $99.598$ | $99.620$ |

Blue | $99.613$ | $99.604$ | $99.608$ | $99.620$ |

Color | Lena | Barbara | Vicky | Cameraman |
---|---|---|---|---|

Red | $33.414$ | $33.437$ | $33.478$ | $33.486$ |

Green | $33.461$ | $33.562$ | $33.483$ | $33.452$ |

Blue | $33.365$ | $33.468$ | $33.421$ | $33.457$ |

Color | Lena | Barbara | Vicky | Cameraman |
---|---|---|---|---|

Red | $49.997$ | $49.994$ | $50.003$ | $49.996$ |

Green | $49.996$ | $50.020$ | $49.985$ | $49.961$ |

Blue | $49.963$ | $49.981$ | $50.014$ | $49.944$ |

**Table 8.**Energy of Figure 1 images after encryption.

Color | Lena | Barbara | Vicky | Cameraman |
---|---|---|---|---|

Red | $0.0156$ | $0.0156$ | $0.0156$ | $0.0156$ |

Green | $0.0156$ | $0.0156$ | $0.0156$ | $0.0155$ |

Blue | $0.0156$ | $0.0156$ | $0.0155$ | $0.0156$ |

**Table 9.**Contrast of Figure 1 images after encryption.

Color | Lena | Barbara | Vicky | Cameraman |
---|---|---|---|---|

Red | $10.501$ | $10.516$ | $10.475$ | $10.472$ |

Green | $10.504$ | $10.450$ | $10.514$ | $10.538$ |

Blue | $10.526$ | $10.494$ | $10.503$ | $10.503$ |

**Table 10.**Homogeneity of Figure 1 images after encryption.

Color | Lena | Barbara | Vicky | Cameraman |
---|---|---|---|---|

Red | $0.388$ | $0.389$ | $0.390$ | $0.389$ |

Green | $0.389$ | $0.389$ | $0.389$ | $0.388$ |

Blue | $0.389$ | $0.388$ | $0.389$ | $0.389$ |

**Table 11.**The randomness measurement using the Discrete Fourier Transform (✓ Accept, x Reject), with $\alpha =0.01$.

Color | Lena | Barbara | Vicky | Cameraman |
---|---|---|---|---|

Red | $0.671/$✓ | $0.978/$✓ | $0.342/$✓ | $0.093/$✓ |

Green | $0.368/$✓ | $0.116/$✓ | $0.737/$✓ | $0.272/$✓ |

Blue | $0.191/$✓ | $0.214/$✓ | $0.182/$✓ | $0.469/$✓ |

Color | Lena | Barbara | Vicky | Cameraman |
---|---|---|---|---|

Red | $242.9/$✓ | $256.3/$✓ | $259.3/$✓ | $243.9/$✓ |

Green | $252.7/$✓ | $229.8/$✓ | $254.6/$✓ | $251.9/$✓ |

Blue | $287.5/$✓ | $260.7/$✓ | $264.2/$✓ | $272.3/$✓ |

Parameter | Color | Black Image | White Image |
---|---|---|---|

Red | $99.610$ | $99.588$ | |

NPCR | Green | $99.606$ | $99.605$ |

Blue | $99.603$ | $99.627$ | |

Red | $33.429$ | $33.384$ | |

UACI | Green | $33.428$ | $33.520$ |

Blue | $33.480$ | $33.410$ | |

Red | $50.039$ | $50.005$ | |

AC | Green | $49.950$ | $49.989$ |

Blue | $49.986$ | $50.006$ |

**Table 14.**SP parameter for distinct noise size of the testing images after encryption, utilizing chi square damage.

Color | Size Noise | Lena | Barbara | Vicky | Cameraman |
---|---|---|---|---|---|

20% | $80.17$ | $82.12$ | $77.06$ | $81.44$ | |

Red | $30\%$ | $70.39$ | $73.25$ | $65.49$ | $72.15$ |

40% | $60.26$ | $64.31$ | $54.10$ | $62.75$ | |

20% | $81.66$ | $82.17$ | $76.83$ | $81.45$ | |

Green | 30% | $72.49$ | $73.21$ | $64.86$ | $72.08$ |

40% | $63.35$ | $64.28$ | $53.30$ | $62.64$ | |

20% | $83.43$ | $82.12$ | $76.47$ | $81.34$ | |

Blue | 30% | $75.14$ | $73.15$ | $64.68$ | $72.12$ |

40% | $66.92$ | $64.33$ | $52.89$ | $62.80$ |

**Table 15.**SP after a 5 × 5 median filter was applied to encrypted images with 40% damage from different noise sources.

Color | Noise Type | Lena | Barbara | Vicky | Cameraman |
---|---|---|---|---|---|

Chi square | $92.39$ | $85.76$ | $93.78$ | $92.38$ | |

Additive | $92.36$ | $85.81$ | $93.76$ | $92.49$ | |

Red | Multiplicative | $92.54$ | $85.93$ | $94.11$ | $92.57$ |

Gaussian | $92.43$ | $85.80$ | $93.81$ | $92.41$ | |

Occlusion | $92.59$ | $85.84$ | $93.86$ | $92.47$ | |

Chi square | $90.71$ | $85.85$ | $93.61$ | $92.47$ | |

Additive | $90.72$ | $85.78$ | $93.59$ | $92.51$ | |

Green | Multiplicative | $90.87$ | $85.93$ | $93.91$ | $92.52$ |

Gaussian | $90.76$ | $85.80$ | $93.75$ | $92.40$ | |

Occlusion | $90.80$ | $85.87$ | $93.69$ | $92.44$ | |

Chi square | $91.94$ | $85.85$ | $93.42$ | $92.48$ | |

Additive | $91.96$ | $85.75$ | $93.53$ | $92.45$ | |

Blue | Multiplicative | $92.03$ | $85.94$ | $93.94$ | $92.60$ |

Gaussian | $91.93$ | $85.82$ | $93.63$ | $92.40$ | |

Occlusion | $91.97$ | $85.87$ | $93.57$ | $92.45$ |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Silva-García, V.M.; Flores-Carapia, R.; Cardona-López, M.A.; Villarreal-Cervantes, M.G.
Generation of Boxes and Permutations Using a Bijective Function and the Lorenz Equations: An Application to Color Image Encryption. *Mathematics* **2023**, *11*, 599.
https://doi.org/10.3390/math11030599

**AMA Style**

Silva-García VM, Flores-Carapia R, Cardona-López MA, Villarreal-Cervantes MG.
Generation of Boxes and Permutations Using a Bijective Function and the Lorenz Equations: An Application to Color Image Encryption. *Mathematics*. 2023; 11(3):599.
https://doi.org/10.3390/math11030599

**Chicago/Turabian Style**

Silva-García, Víctor Manuel, Rolando Flores-Carapia, Manuel Alejandro Cardona-López, and Miguel Gabriel Villarreal-Cervantes.
2023. "Generation of Boxes and Permutations Using a Bijective Function and the Lorenz Equations: An Application to Color Image Encryption" *Mathematics* 11, no. 3: 599.
https://doi.org/10.3390/math11030599