# Dynamic S-Box Construction Using Mordell Elliptic Curves over Galois Field and Its Applications in Image Encryption

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## Abstract

**:**

## 1. Introduction

- The elliptic curves provide great resistance against linear and differential cryptanalysis due to their nonlinear nature.
- Compact S-box designs can be achieved by representing elliptic curves with smaller key sizes than conventional mathematical structures. In terms of efficiency, this can be helpful, particularly in settings with few resources.
- Hardware and software may both effectively implement elliptic curve operations. For real-world applications, such as embedded systems or gadgets with constrained processing power, this efficiency is essential.
- A further degree of protection is provided by the mathematical hardness of elliptic curve problems like the elliptic curve discrete logarithm problem. The cryptographic strength of elliptic curve-based designs is predicated on the difficulty of solving these complex mathematical problems.
- Due to the strong properties of elliptic curves and a highly nonlinear permutation of the Galois field, the proposed strategy for S-boxes and encryption has a greater ability to resist cryptanalysis.

- The generated S-boxes have nonlinearity greater than 105 with four optimal boxes of nonlinearity 112.
- As the degree of irreducible polynomials increases, the number of irreducible polynomials increases quickly, and we can produce millions of S-boxes with the proposed work in a short time.
- The entropy of the proposed cipher image is close to 8, confirming the efficacy of the effectiveness of the method.

## 2. Preliminaries

#### 2.1. Irreducible Polynomial

#### 2.2. Maximal Ideal

#### 2.3. Galois Field

#### 2.4. Elliptic Curve

#### 2.5. Mordell Elliptic Curve

## 3. Proposed Algorithm for the Construction of S-boxes

#### 3.1. S-Boxes Using Mordell Elliptic Curve over $GF\left({2}^{n}\right),n=8,10,12$

- (1)
- Choose any irreducible polynomial of degree $8,10,12$ over the binary field.
- (2)
- Choose the Mordell elliptic curve ${y}^{2}={x}^{3}+b,0\ne b\in GF\left({2}^{n}\right)$.
- (3)
- Choose x-coordinates of points $(x,y)$ satisfying the Mordell curve.
- (4)
- Apply the multiplicative inverse of each non-zero element corresponding to a given irreducible polynomial.
- (5)
- For $GF\left({2}^{n}\right),n=10,12$, apply modulo 256 and choose the 1st 256 unique values.
- (6)
- Reshape into $16\times 16$ matrix.

#### 3.2. S-Boxes Using Mordell Elliptic Curves over $GF\left({2}^{n}\right),n=9,11$

- (1)
- Choose any irreducible polynomial of degree $9,11$ over the binary field.
- (2)
- Choose the Mordell elliptic curve ${y}^{2}={x}^{3}+b,0\ne b\in GF\left({2}^{n}\right),n=9,11$.
- (3)
- Choose y-coordinates of points $(x,y)$ satisfying the Mordell curve.
- (4)
- Apply modulo 256 on y-coordinates to obtain answers in 0–255.
- (5)
- Select the 1st 256 unique values.
- (6)
- Reshape into a $16\times 16$ matrix.

## 4. Security Analysis of S-Boxes

#### 4.1. Nonlinearity (NL)

#### 4.2. Strict Avalanche Criteria (SAC)

#### 4.3. Bit Independence Criteria (BIC)

#### 4.4. Linear Approximation Probability (LAP)

#### 4.5. Differential Approximation Probability (DP)

#### 4.6. Discussion

- Large nonlinearity is required for the S-box to fend off linear attacks. Table 11 shows that there are four S-boxes with optimal nonlinearity, while the remaining also have considerable scores.
- The strict avalanche criterion is deemed to be met rather effectively by the SAC score that is close to the optimal value of 0.50. Table 11 shows that, in comparison to most recently created S-boxes with the avalanche effect, our best SAC score of $0.4998$ is quite near to the ideal value. As a result, the suggested S-box successfully satisfies the strict avalanche criteria.
- Under the bits independence requirement, the pair-wise disjoint boolean functions have demonstrated strong performance for both SAC and nonlinearity scores. Each of our proposed S-boxes has a sound score of nonlinearity and SAC.
- A lower DU score is indicative of a secure S-box. Among all generated S-boxes, none of the S-boxes has a score of DU greater than 10.
- The resistance of the S-box against linear cryptanalysis is likewise correlated with the likelihood of linear approximation. It is claimed that an S-box with a lower LAP score is more resistant to linear cryptanalysis. The LP values of our S-boxes are lower than many of the proposed S-boxes as shown in Table 11.

## 5. Image Encryption

#### 5.1. Entropy

#### 5.2. Correlation

#### 5.3. Contrast

- (1)
- Enhanced Security: Higher contrast can make it more challenging for attackers to analyze or extract meaningful information from the cipher image. Well-defined edges and distinct intensity variations can make it harder to detect patterns or identify specific features within the image.
- (2)
- Robustness Against Attacks: A cipher image with higher contrast can exhibit greater resilience against common attacks, such as statistical analysis, pixel correlation, or known-plaintext attacks. The increased variability in intensity levels can make it more difficult to exploit statistical regularities and effectively break the encryption.
- (3)
- Improved Visual Quality: Although the primary goal of image encryption is security, maintaining a visually appealing and interpretable cipher image is also desirable. Higher contrast often leads to a more visually striking encrypted image, which may enhance the user experience and the overall acceptance of the encryption scheme.

#### 5.4. Homogeneity

#### 5.5. Energy

#### 5.6. Number of Pixel Change Rate (NPCR)

#### 5.7. Unified Average Changing Intensity (UACI)

#### 5.8. Noise Attack Analysis

## 6. Conclusions and Future Study

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Miller, V.S. Use of elliptic curves in cryptography. In Proceedings of the Conference on the Theory and Application of Cryptographic Techniques, Santa Barbara, CA, USA, 18–22 August 1985; pp. 417–426. [Google Scholar]
- Koblitz, N.; Menezes, A.; Vanstone, S. The state of elliptic curve cryptography. Des. Codes Cryptogr.
**2000**, 19, 173–193. [Google Scholar] [CrossRef] - Kodali, R.K.; Patel, K.H.; Sarma, N. Energy efficient elliptic curve point multiplication for WSN applications. In Proceedings of the 2013 National Conference on Communications (NCC), New Delhi, India, 15–17 February 2013; pp. 1–5. [Google Scholar]
- Khalid, I.; Jamal, S.S.; Shah, T.; Shah, D.; Hazzazi, M.M. A novel scheme of image encryption based on elliptic curves isomorphism and substitution boxes. IEEE Access
**2021**, 9, 77798–77810. [Google Scholar] [CrossRef] - Hayat, U.; Azam, N.A.; Asif, M. A method of generating 8 × 8 substitution boxes based on elliptic curves. Wirel. Pers. Commun.
**2018**, 101, 439–451. [Google Scholar] [CrossRef] - Hayat, U.; Azam, N.A. A novel image encryption scheme based on an elliptic curve. Signal Process.
**2019**, 155, 391–402. [Google Scholar] [CrossRef] - Farwa, S.; Sohail, A.; Muhammad, N. A novel application of elliptic curves in the dynamical components of block ciphers. Wirel. Pers. Commun.
**2020**, 115, 1309–1316. [Google Scholar] [CrossRef] - Shah, T.; Aljaedi, A.; Hazzazi, M.M.; Alharbi, A.R. Design of Nonlinear Components Over a Mordell Elliptic Curve on Galois Fields. Comput. Mater. Contin.
**2022**, 71, 1313–1329. [Google Scholar] - Razaq, A.; Yousaf, A.; Shuaib, U.; Siddiqui, N.; Ullah, A.; Waheed, A. A novel construction of substitution box involving coset diagram and a bijective map. Secur. Commun. Netw.
**2017**, 2017, 5101934. [Google Scholar] [CrossRef] - Cheon, J.H.; Chee, S.; Park, C. S-boxes with controllable nonlinearity. In Proceedings of the International Conference on the Theory and Applications of Cryptographic Techniques, EUROCRYPT ’99, Prague, Czech Republiic, 2–6 May 1999; pp. 286–294. [Google Scholar]
- Asghari, P.; Javadi, S.H.H.S. Lightweight Key-Dependent Dynamic S-Boxes based on Hyperelliptic Curve for IoT Devices. arXiv
**2021**, arXiv:2102.13340. [Google Scholar] - Massey, J.; Lai, X. International Data Encryption Algorithm; Eidgenossische Technique Hoehschule (ETH): Zurich, Switzerland, 1991. [Google Scholar]
- Joan, D.; Vincent, R. The Design of Rijndael: AES—The Advanced Encryption Standard; Springer: Berlin/Heidelberg, Germany, 2002. [Google Scholar]
- Shannon, C.E. Communication theory of secrecy systems. Bell Syst. Tech. J.
**1949**, 28, 656–715. [Google Scholar] [CrossRef] - Gan, Z.; Chai, X.; Yuan, K.; Lu, Y. A novel image encryption algorithm based on LFT based S-boxes and chaos. Multimed. Tools Appl.
**2018**, 77, 8759–8783. [Google Scholar] [CrossRef] - Hussain, I.; Shah, T.; Gondal, M.A.; Khan, W.A.; Mahmood, H. A group theoretic approach to construct cryptographically strong substitution boxes. Neural Comput. Appl.
**2013**, 23, 97–104. [Google Scholar] [CrossRef] - Hussain, I.; Shah, T.; Gondal, M.A.; Khan, M.; Khan, W.A. Construction of new S-box using a linear fractional transformation. World Appl. Sci. J.
**2011**, 14, 1779–1785. [Google Scholar] - Younas, I.; Khan, M. A new efficient digital image encryption based on inverse left almost semi group and Lorenz chaotic system. Entropy
**2018**, 20, 913. [Google Scholar] [CrossRef] - Razaq, A.; Al-Olayan, H.A.; Ullah, A.; Riaz, A.; Waheed, A. A Novel Technique for the Construction of Safe Substitution Boxes Based on Cyclic and Symmetric Groups. Secur. Commun. Netw.
**2018**, 2018, 4987021. [Google Scholar] [CrossRef] - Hussain, I.; Shah, T.; Gondal, M.A.; Mahmood, H. An efficient approach for the construction of LFT S-boxes using chaotic logistic map. Nonlinear Dyn.
**2013**, 71, 133–140. [Google Scholar] [CrossRef] - Siddiqui, N.; Afsar, U.; Shah, T.; Qureshi, A. A Novel Construction of S16 AES S-boxes. Int. J. Comput. Sci. Inf. Secur. (IJCSIS)
**2016**, 14, 810–818. [Google Scholar] - Mahmood, S.; Farwa, S.; Rafiq, M.; Riaz, S.M.J.; Shah, T.; Jamal, S.S. To study the effect of the generating polynomial on the quality of nonlinear components in block ciphers. Secur. Commun. Netw.
**2018**, 2018, 5823230. [Google Scholar] [CrossRef] - Attaullah; Jamal, S.S.; Shah, T. A Novel Algebraic Technique for the Construction of Strong Substitution Box. Wirel. Pers. Commun.
**2018**, 99, 213–226. [Google Scholar] [CrossRef] - Naseer, Y.; Shah, T.; Shah, D.; Hussain, S. A novel algorithm of constructing highly nonlinear Sp-boxes. Cryptography
**2019**, 3, 6. [Google Scholar] [CrossRef] - Zhang, T.; Chen, C.P.; Chen, L.; Xu, X.; Hu, B. Design of highly nonlinear substitution boxes based on I-Ching operators. IEEE Trans. Cybern.
**2018**, 48, 3349–3358. [Google Scholar] [CrossRef] - Zahid, A.H.; Arshad, M.J.; Ahmad, M. A novel construction of efficient substitution-boxes using cubic fractional transformation. Entropy
**2019**, 21, 245. [Google Scholar] [CrossRef] - Bin Faheem, Z.; Ali, A.; Khan, M.A.; Ul-Haq, M.E.; Ahmad, W. Highly dispersive substitution box (S-box) design using chaos. ETRI J.
**2020**, 42, 619–632. [Google Scholar] [CrossRef] - Shahzad, I.; Mushtaq, Q.; Razaq, A. Construction of new S-box using action of quotient of the modular group for multimedia security. Secur. Commun. Netw.
**2019**, 2019, 2847801. [Google Scholar] [CrossRef] - Tian, Y.; Lu, Z. Chaotic S-box: Intertwining logistic map and bacterial foraging optimization. Math. Probl. Eng.
**2017**, 2017, 6969312. [Google Scholar] [CrossRef] - Biham, E.; Shamir, A. Differential cryptanalysis of DES-like cryptosystems. J. Cryptol.
**1991**, 4, 3–72. [Google Scholar] [CrossRef] - Yucel, M.; Vergili, I. Avalanche and Bit Independence Properties for the Ensembles of Randomly Chosen nxn S-boxes. Turk. J. Electr. Eng. Comput. Sci.
**2001**, 9, 3. [Google Scholar] - Seberry, J.; Zhang, X.M.; Zheng, Y. Systematic generation of cryptographically robust S-boxes. In Proceedings of the 1st ACM Conference on Computer and Communications Security, Fairfax, VA, USA, 3–5 August 1993; pp. 171–182. [Google Scholar]
- Cipher, D. Linear Cryptanalysis Method for. In Proceedings of the Advances in Cryptology–EUROCRYPT’93: Workshop on the Theory and Application of Cryptographic Techniques, Lofthus, Norway, 23–27 May 1993; Springer: Berlin/Heidelberg, Germany, 2003; Volume 765, p. 386. [Google Scholar]
- Pieprzyk, J.; Finkelstein, G. Towards effective nonlinear cryptosystem design. IEE Proc.-Comput. Digit. Tech.
**1988**, 135, 325–335. [Google Scholar] [CrossRef] - Webster, A.F.; Tavares, S.E. On the design of S-boxes. In Proceedings of the Conference on the Theory and Application of Cryptographic Techniques, CRYPTO’85, Santa Barbara, CA, USA, 18–22 August 1985; pp. 523–534. [Google Scholar]
- Lu, Q.; Zhu, C.; Deng, X. An efficient image encryption scheme based on the LSS chaotic map and single S-box. IEEE Access
**2020**, 8, 25664–25678. [Google Scholar] [CrossRef] - Alzaidi, A.A.; Ahmad, M.; Doja, M.N.; Al Solami, E.; Beg, M.S. A new 1D chaotic map and β-hill climbing for generating substitution-boxes. IEEE Access
**2018**, 6, 55405–55418. [Google Scholar] [CrossRef] - Yong, W.; Peng, L. An improved method to obtaining S-box based on chaos and genetic algorithm. HKIE Trans.
**2012**, 19, 53–58. [Google Scholar] [CrossRef] - Lambić, D. A novel method of S-box design based on chaotic map and composition method. Chaos Solitons Fractals
**2014**, 58, 16–21. [Google Scholar] [CrossRef] - Nizam Chew, L.C.; Ismail, E.S. S-box construction based on linear fractional transformation and permutation function. Symmetry
**2020**, 12, 826. [Google Scholar] [CrossRef] - Arshad, B.; Siddiqui, N. Construction of highly nonlinear substitution boxes (S-boxes) based on connected regular graphs. Int. J. Comput. Sci. Inf. Secur. (IJCSIS)
**2020**, 18, 105–122. [Google Scholar] - Siddiqui, N.; Yousaf, F.; Murtaza, F.; Ehatisham-ul Haq, M.; Ashraf, M.U.; Alghamdi, A.M.; Alfakeeh, A.S. A highly nonlinear substitution-box (S-box) design using action of modular group on a projective line over a finite field. PLoS ONE
**2020**, 15, e0241890. [Google Scholar] [CrossRef] [PubMed] - Pali, I.A.; Soomro, M.A.; Memon, M.; Maitlo, A.A.; Dehraj, S.; Umrani, N.A. Construction of an s-box using suppersingular elliptic curve over finite field. J. Hunan Univ. Nat. Sci.
**2023**, 50. [Google Scholar] [CrossRef] - Razaq, A.; Ahmad, M.; El-Latif, A.A.A. A novel algebraic construction of strong S-boxes over double GF (27) structures and image protection. Comput. Appl. Math.
**2023**, 42, 90. [Google Scholar] [CrossRef] - Feng, W.; Wang, Q.; Liu, H.; Ren, Y.; Zhang, J.; Zhang, S.; Qian, K.; Wen, H. Exploiting newly designed fractional-order 3D Lorenz chaotic system and 2D discrete polynomial hyper-chaotic map for high-performance multi-image encryption. Fractal Fract.
**2023**, 7, 887. [Google Scholar] [CrossRef] - Alexan, W.; Elkandoz, M.; Mashaly, M.; Azab, E.; Aboshousha, A. Color image encryption through chaos and kaa map. IEEE Access
**2023**, 11, 11541–11554. [Google Scholar] [CrossRef] - Lavanya, M.; Sundar, K.; Saravanan, S. Simplified Image Encryption Algorithm (SIEA) to enhance image security in cloud storage. Multimed. Tools Appl.
**2024**, 1–33. [Google Scholar] [CrossRef] - Yi, G.; Cao, Z. An Algorithm of Image Encryption based on AES & Rossler Hyperchaotic Modeling. Mob. Netw. Appl.
**2023**, 1–9. [Google Scholar] [CrossRef] - Ali, R.; Jamil, M.K.; Alali, A.S.; Ali, J.; Afzal, G. A robust S box design using cyclic groups and image encryption. IEEE Access
**2023**, 11, 135880–135890. [Google Scholar] [CrossRef] - Ali, J.; Jamil, M.K.; Alali, A.S.; Ali, R.; Guiraiz. A medical image encryption scheme based on Mobius transformation and Galois field. Heliyon
**2024**, 10, e23652. [Google Scholar] [CrossRef] [PubMed] - Wen, H.; Lin, Y. Cryptanalysis of an image encryption algorithm using quantum chaotic map and DNA coding. Expert Syst. Appl.
**2024**, 237, 121514. [Google Scholar] [CrossRef] - Chen, X.; Yu, S.; Wang, Q.; Guyeux, C.; Wang, M. On the cryptanalysis of an image encryption algorithm with quantum chaotic map and DNA coding. Multimed. Tools Appl.
**2023**, 82, 42717–42737. [Google Scholar] [CrossRef] - Hussain, I.; Shah, T.; Mahmood, H.; Gondal, M.A. A projective general linear group based algorithm for the construction of substitution box for block ciphers. Neural Comput. Appl.
**2013**, 22, 1085–1093. [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] - Khan, M.; Shah, T.; Mahmood, H.; Gondal, M.A.; Hussain, I. A novel technique for the construction of strong S-boxes based on chaotic Lorenz systems. Nonlinear Dyn.
**2012**, 70, 2303–2311. [Google Scholar] [CrossRef]

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108 | 35 | 198 | 3 | 88 | 122 | 212 | 249 | 86 | 234 | 187 | 22 | 76 | 95 | 121 | 134 |

51 | 97 | 37 | 178 | 70 | 162 | 166 | 230 | 228 | 34 | 113 | 186 | 218 | 69 | 195 | 106 |

116 | 94 | 172 | 27 | 251 | 157 | 41 | 66 | 211 | 223 | 255 | 163 | 6 | 190 | 135 | 229 |

79 | 58 | 128 | 59 | 244 | 168 | 161 | 140 | 92 | 84 | 227 | 62 | 109 | 48 | 80 | 30 |

125 | 55 | 73 | 133 | 217 | 54 | 100 | 145 | 21 | 64 | 89 | 40 | 237 | 252 | 87 | 148 |

78 | 242 | 110 | 117 | 189 | 93 | 49 | 143 | 68 | 194 | 175 | 17 | 174 | 111 | 192 | 45 |

0 | 47 | 6 | 203 | 58 | 2 | 79 | 32 | 45 | 174 | 113 | 236 | 115 | 237 | 111 | 83 |

1 | 93 | 179 | 112 | 238 | 180 | 8 | 29 | 204 | 44 | 230 | 66 | 241 | 209 | 169 | 186 |

145 | 89 | 63 | 166 | 16 | 119 | 73 | 247 | 172 | 187 | 182 | 206 | 20 | 253 | 21 | 151 |

30 | 201 | 87 | 55 | 35 | 123 | 67 | 14 | 95 | 51 | 105 | 31 | 183 | 250 | 92 | 242 |

217 | 211 | 189 | 54 | 161 | 244 | 25 | 118 | 127 | 122 | 181 | 234 | 249 | 76 | 106 | 210 |

10 | 61 | 70 | 114 | 171 | 84 | 108 | 156 | 190 | 101 | 215 | 140 | 7 | 136 | 243 | 128 |

143 | 3 | 117 | 192 | 138 | 56 | 34 | 193 | 9 | 107 | 15 | 135 | 202 | 157 | 219 | 245 |

214 | 99 | 64 | 224 | 252 | 33 | 100 | 53 | 124 | 154 | 75 | 74 | 22 | 78 | 167 | 126 |

125 | 43 | 248 | 177 | 17 | 168 | 80 | 68 | 218 | 103 | 23 | 11 | 102 | 130 | 52 | 85 |

178 | 207 | 69 | 50 | 90 | 27 | 220 | 197 | 232 | 240 | 41 | 163 | 62 | 48 | 212 | 109 |

133 | 195 | 144 | 251 | 88 | 194 | 120 | 226 | 152 | 159 | 188 | 49 | 155 | 38 | 196 | 139 |

37 | 110 | 59 | 131 | 221 | 185 | 36 | 97 | 229 | 225 | 153 | 24 | 46 | 223 | 147 | 28 |

86 | 228 | 160 | 184 | 200 | 233 | 173 | 165 | 129 | 222 | 146 | 116 | 13 | 60 | 57 | 175 |

134 | 227 | 4 | 72 | 18 | 96 | 199 | 254 | 81 | 176 | 71 | 162 | 149 | 26 | 77 | 150 |

235 | 98 | 205 | 121 | 12 | 164 | 158 | 40 | 104 | 191 | 5 | 39 | 148 | 94 | 65 | 137 |

231 | 213 | 246 | 142 | 208 | 239 | 132 | 141 | 42 | 82 | 216 | 170 | 91 | 19 | 255 | 198 |

0 | 174 | 7 | 200 | 185 | 119 | 106 | 56 | 16 | 123 | 13 | 248 | 157 | 142 | 181 | 171 |

1 | 165 | 172 | 15 | 178 | 152 | 226 | 173 | 4 | 132 | 44 | 55 | 175 | 81 | 189 | 65 |

126 | 240 | 100 | 111 | 182 | 2 | 224 | 235 | 208 | 155 | 113 | 99 | 244 | 228 | 28 | 229 |

84 | 76 | 120 | 251 | 188 | 12 | 159 | 74 | 207 | 219 | 225 | 186 | 245 | 78 | 109 | 103 |

191 | 115 | 18 | 168 | 68 | 252 | 20 | 34 | 210 | 146 | 49 | 195 | 51 | 190 | 30 | 82 |

204 | 158 | 166 | 196 | 137 | 144 | 150 | 217 | 220 | 197 | 83 | 206 | 98 | 80 | 88 | 101 |

42 | 127 | 205 | 62 | 234 | 37 | 61 | 91 | 69 | 243 | 10 | 70 | 238 | 38 | 17 | 35 |

218 | 66 | 71 | 213 | 211 | 179 | 117 | 60 | 45 | 96 | 255 | 86 | 8 | 167 | 5 | 147 |

161 | 52 | 193 | 27 | 162 | 54 | 125 | 94 | 97 | 3 | 138 | 64 | 43 | 92 | 183 | 222 |

29 | 214 | 79 | 122 | 24 | 75 | 247 | 36 | 129 | 77 | 85 | 156 | 130 | 216 | 41 | 233 |

230 | 227 | 25 | 39 | 221 | 6 | 110 | 58 | 141 | 163 | 139 | 40 | 232 | 19 | 199 | 241 |

254 | 11 | 249 | 250 | 93 | 121 | 116 | 21 | 134 | 26 | 203 | 145 | 153 | 246 | 14 | 73 |

149 | 136 | 32 | 33 | 143 | 187 | 148 | 23 | 215 | 53 | 46 | 31 | 176 | 124 | 104 | 231 |

59 | 170 | 154 | 180 | 89 | 201 | 135 | 102 | 194 | 63 | 118 | 47 | 105 | 67 | 212 | 90 |

237 | 9 | 108 | 253 | 184 | 50 | 128 | 209 | 72 | 140 | 192 | 160 | 223 | 22 | 198 | 202 |

239 | 87 | 107 | 112 | 114 | 131 | 57 | 169 | 236 | 151 | 133 | 95 | 164 | 48 | 177 | 242 |

149 | 197 | 136 | 96 | 221 | 215 | 123 | 6 | 30 | 144 | 158 | 41 | 2 | 173 | 139 | 32 |

102 | 168 | 153 | 192 | 195 | 69 | 244 | 66 | 92 | 45 | 62 | 224 | 234 | 61 | 225 | 246 |

187 | 27 | 86 | 112 | 227 | 176 | 154 | 200 | 138 | 210 | 209 | 199 | 80 | 126 | 152 | 184 |

0 | 38 | 140 | 169 | 67 | 109 | 125 | 205 | 56 | 29 | 59 | 145 | 193 | 211 | 170 | 203 |

46 | 147 | 37 | 8 | 55 | 76 | 103 | 242 | 130 | 240 | 232 | 53 | 7 | 186 | 71 | 17 |

243 | 88 | 156 | 190 | 117 | 208 | 74 | 70 | 159 | 11 | 124 | 150 | 100 | 22 | 3 | 78 |

39 | 58 | 91 | 110 | 161 | 72 | 229 | 36 | 180 | 105 | 34 | 118 | 194 | 19 | 155 | 33 |

134 | 28 | 23 | 183 | 13 | 218 | 241 | 15 | 116 | 196 | 175 | 207 | 188 | 77 | 137 | 148 |

182 | 254 | 171 | 247 | 115 | 12 | 89 | 83 | 111 | 129 | 44 | 68 | 177 | 49 | 230 | 60 |

217 | 189 | 35 | 172 | 179 | 213 | 132 | 42 | 220 | 47 | 113 | 223 | 107 | 245 | 127 | 253 |

228 | 181 | 21 | 248 | 135 | 87 | 97 | 157 | 235 | 90 | 40 | 255 | 212 | 128 | 25 | 108 |

216 | 219 | 16 | 95 | 63 | 141 | 165 | 85 | 20 | 122 | 131 | 251 | 178 | 185 | 4 | 26 |

252 | 52 | 249 | 5 | 121 | 238 | 10 | 104 | 174 | 9 | 43 | 201 | 133 | 160 | 81 | 120 |

237 | 191 | 214 | 93 | 146 | 50 | 163 | 94 | 106 | 143 | 51 | 79 | 239 | 151 | 75 | 54 |

167 | 202 | 84 | 73 | 114 | 233 | 14 | 18 | 142 | 162 | 119 | 24 | 206 | 1 | 57 | 65 |

98 | 31 | 250 | 231 | 222 | 236 | 198 | 204 | 99 | 82 | 166 | 101 | 164 | 48 | 226 | 64 |

0 | 188 | 251 | 4 | 58 | 82 | 164 | 231 | 117 | 104 | 76 | 237 | 41 | 128 | 78 | 203 |

1 | 12 | 6 | 137 | 191 | 110 | 48 | 88 | 129 | 225 | 163 | 19 | 31 | 245 | 73 | 91 |

224 | 214 | 126 | 178 | 34 | 42 | 89 | 161 | 255 | 193 | 32 | 172 | 238 | 186 | 147 | 62 |

64 | 90 | 235 | 55 | 21 | 194 | 201 | 11 | 121 | 175 | 95 | 229 | 92 | 119 | 60 | 199 |

240 | 143 | 192 | 254 | 211 | 205 | 57 | 29 | 116 | 35 | 252 | 140 | 25 | 93 | 239 | 226 |

127 | 81 | 45 | 152 | 96 | 135 | 66 | 24 | 18 | 70 | 124 | 79 | 198 | 249 | 227 | 241 |

160 | 14 | 134 | 69 | 246 | 74 | 166 | 158 | 77 | 37 | 33 | 132 | 253 | 159 | 23 | 3 |

118 | 105 | 167 | 141 | 123 | 54 | 180 | 145 | 236 | 234 | 113 | 173 | 242 | 87 | 151 | 244 |

120 | 40 | 72 | 133 | 195 | 52 | 179 | 115 | 59 | 27 | 83 | 53 | 181 | 155 | 208 | 144 |

109 | 107 | 217 | 102 | 184 | 22 | 183 | 106 | 146 | 233 | 185 | 43 | 153 | 38 | 5 | 85 |

223 | 61 | 7 | 207 | 51 | 138 | 17 | 170 | 65 | 47 | 247 | 228 | 68 | 49 | 71 | 112 |

28 | 100 | 220 | 111 | 216 | 149 | 215 | 9 | 243 | 139 | 202 | 200 | 8 | 46 | 171 | 98 |

80 | 75 | 84 | 250 | 36 | 2 | 15 | 248 | 136 | 165 | 162 | 212 | 86 | 13 | 20 | 196 |

122 | 10 | 154 | 157 | 67 | 190 | 125 | 168 | 209 | 197 | 103 | 177 | 206 | 94 | 156 | 114 |

187 | 204 | 148 | 176 | 230 | 63 | 218 | 108 | 219 | 30 | 26 | 182 | 189 | 44 | 210 | 232 |

213 | 222 | 169 | 131 | 99 | 130 | 174 | 221 | 50 | 150 | 39 | 142 | 16 | 101 | 97 | 56 |

S-Boxes | Nonlinearity | SAC | BIC-NL | BIC-SAC | LAP | DAP |
---|---|---|---|---|---|---|

S-box-283 | 112 | 0.5032 | 112 | 0.5059 | 0.0625 | 0.0156 |

S-box-299 | 112 | 0.4998 | 112 | 0.5046 | 0.0625 | 0.0156 |

S-box-313 | 112 | 0.5032 | 112 | 0.5015 | 0.0625 | 0.0156 |

S-box-505 | 112 | 0.5022 | 112 | 0.5020 | 0.0625 | 0.0156 |

S-box-529 | 106 | 0.5020 | 102.5714 | 0.5056 | 0.1094 | 0.0391 |

S-box-787 | 106.25 | 0.5027 | 103.5 | 0.5036 | 0.0859 | 0.0391 |

S-box-1315 | 106 | 0.5039 | 105 | 0.5025 | 0.0859 | 0.0391 |

S-box-1789 | 105.75 | 0.5024 | 103.3571 | 0.5022 | 0.1016 | 0.0469 |

S-box-3441 | 105.75 | 0.4995 | 103.0714 | 0.5018 | 0.0859 | 0.0391 |

S-box-7105 | 105.25 | 0.5066 | 104.2143 | 0.4994 | 0.1016 | 0.0469 |

S-box over $GF\left({2}^{8}\right)$ [8] | 112 | 0.4871 | 112 | - | 0.0625 | 0.0156 |

S-box over $GF\left({2}^{9}\right)$ [8] | 106.25 | 0.4992 | 103.8 | - | 0.1328 | 0.0391 |

[49] | 112 | 0.5034 | 112 | 0.5066 | 0.0625 | 0.0156 |

[50] | 112 | 0.4988 | 112 | 0.5008 | 0.0625 | 0.0156 |

[53] | 105.5 | 0.507 | 106 | 0.462 | 0.140 | 0.0242 |

[9] | 106.75 | 0.5032 | 103.6429 | 0.5074 | 0.1484 | 0.0469 |

[54] | 106 | 0.5051 | 98 | - | 0.148 | 0.039 |

Skipjack | 105.75 | 0.503 | 104.14 | 0.499 | 0.109 | 0.0468 |

Residue prime | 99.5 | 0.515 | 101.71 | 0.502 | 0.132 | 0.281 |

[16] | 104.87 | 0.493 | 99 | 0.504 | 0.105 | 0.0390 |

[55] | 96 | 0.4900 | 92 | 0.5100 | 0.23 | 0.050 |

S-Boxes | Entropy | Correlation | Contrast | Energy | Homogeneity | NPCR | UACI |
---|---|---|---|---|---|---|---|

S-box-283 | 7.9995 | −0.0049 | 10.5706 | 0.0156 | 0.3882 | 99.61 | 33.52 |

S-box-299 | 7.9994 | −0.0042 | 10.5556 | 0.0156 | 0.3884 | 99.59 | 33.48 |

S-box-505 | 7.9995 | −0.0079 | 10.6160 | 0.0156 | 0.3882 | 99.61 | 33.43 |

S-box-313 | 7.9994 | −0.0039 | 10.5683 | 0.0156 | 0.3885 | 99.62 | 33.53 |

S-box-529 | 7.9994 | −0.0028 | 10.5213 | 0.0156 | 0.3917 | 99.64 | 33.39 |

S-box-787 | 7.9994 | −0.0078 | 10.6421 | 0.0156 | 0.3885 | 99.62 | 33.62 |

S-box-1315 | 7.9993 | −0.0061 | 10.5935 | 0.0156 | 0.3876 | 99.58 | 33.48 |

S-box-1789 | 7.9994 | −0.0096 | 10.6143 | 0.0156 | 0.3881 | 99.63 | 33.46 |

S-box-3441 | 7.9994 | −0.0001 | 10.5683 | 0.0156 | 0.3890 | 99.58 | 33.54 |

S-box-7105 | 7.9994 | −0.0045 | 10.5679 | 0.0156 | 0.3888 | 99.61 | 33.49 |

[8] | 7.9479 | 0.0036 | 9.9955 | 0.0158 | 0.3948 | 99.42 | 33.21 |

[49] | 7.9994 | −0.0079 | 10.6137 | 0.0156 | 0.3879 | 99.59 | 33.35 |

Intensity of Salt and Pepper Attack | PSNR |
---|---|

$0.001$ | 36.29 |

$0.1$ | 33.64 |

$0.3$ | 32.90 |

$0.5$ | 32.05 |

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## Share and Cite

**MDPI and ACS Style**

Alali, A.S.; Ali, R.; Jamil, M.K.; Ali, J.; Gulraiz.
Dynamic S-Box Construction Using Mordell Elliptic Curves over Galois Field and Its Applications in Image Encryption. *Mathematics* **2024**, *12*, 587.
https://doi.org/10.3390/math12040587

**AMA Style**

Alali AS, Ali R, Jamil MK, Ali J, Gulraiz.
Dynamic S-Box Construction Using Mordell Elliptic Curves over Galois Field and Its Applications in Image Encryption. *Mathematics*. 2024; 12(4):587.
https://doi.org/10.3390/math12040587

**Chicago/Turabian Style**

Alali, Amal S., Rashad Ali, Muhammad Kamran Jamil, Javed Ali, and Gulraiz.
2024. "Dynamic S-Box Construction Using Mordell Elliptic Curves over Galois Field and Its Applications in Image Encryption" *Mathematics* 12, no. 4: 587.
https://doi.org/10.3390/math12040587