# Construction Method and Performance Analysis of Chaotic S-Box Based on a Memorable Simulated Annealing Algorithm

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## Abstract

**:**

## 1. Introduction

## 2. Chaotic System

## 3. Optimization Process of MSAA

## 4. Construction Method of Chaotic S-Box

## 5. Testing and Analysis of Performance

#### 5.1. Nonlinearity

#### 5.2. Difference Uniformity

#### 5.3. Strict Avalanche Criterion

#### 5.4. Bit Independence Criterion

#### 5.5. Bijectivity

#### 5.6. Implementation Efficiency

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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96 | 87 | 213 | 3 | 159 | 215 | 185 | 225 | 14 | 94 | 175 | 164 | 219 | 127 | 211 | 253 |

160 | 40 | 100 | 177 | 187 | 68 | 220 | 83 | 108 | 135 | 128 | 183 | 53 | 138 | 224 | 232 |

43 | 134 | 133 | 201 | 63 | 151 | 32 | 248 | 205 | 30 | 158 | 144 | 247 | 196 | 155 | 191 |

89 | 114 | 60 | 214 | 84 | 146 | 161 | 91 | 143 | 157 | 124 | 231 | 78 | 95 | 131 | 189 |

69 | 130 | 93 | 148 | 36 | 106 | 12 | 16 | 218 | 167 | 85 | 58 | 65 | 90 | 33 | 217 |

44 | 226 | 156 | 104 | 80 | 71 | 136 | 239 | 49 | 10 | 129 | 27 | 48 | 182 | 39 | 70 |

241 | 139 | 59 | 115 | 153 | 184 | 11 | 45 | 47 | 210 | 31 | 173 | 204 | 25 | 72 | 140 |

152 | 75 | 145 | 250 | 172 | 202 | 99 | 195 | 237 | 110 | 207 | 208 | 216 | 67 | 20 | 125 |

82 | 222 | 64 | 198 | 23 | 118 | 37 | 186 | 46 | 238 | 209 | 28 | 79 | 35 | 255 | 141 |

73 | 50 | 77 | 111 | 163 | 107 | 19 | 244 | 199 | 21 | 234 | 112 | 119 | 181 | 105 | 98 |

221 | 76 | 246 | 254 | 137 | 229 | 18 | 1 | 212 | 123 | 223 | 101 | 42 | 81 | 9 | 242 |

165 | 121 | 194 | 38 | 56 | 236 | 176 | 88 | 252 | 249 | 179 | 57 | 178 | 174 | 61 | 192 |

54 | 86 | 8 | 251 | 147 | 74 | 26 | 97 | 193 | 243 | 190 | 17 | 169 | 2 | 188 | 206 |

117 | 113 | 230 | 150 | 103 | 7 | 240 | 149 | 24 | 116 | 92 | 15 | 66 | 109 | 4 | 62 |

34 | 122 | 233 | 171 | 132 | 41 | 168 | 170 | 235 | 29 | 142 | 166 | 55 | 22 | 120 | 227 |

162 | 197 | 154 | 200 | 5 | 6 | 228 | 102 | 126 | 0 | 245 | 51 | 180 | 203 | 52 | 13 |

S-Box | Maximum | Minimum | Average |
---|---|---|---|

${S}_{1}$-box | 110 | 104 | 108 |

${S}_{2}$-box | 110 | 104 | 108 |

${S}_{3}$-box | 110 | 104 | 107.5 |

${S}_{4}$-box | 116 | 100 | 103 |

${S}_{5}$-box | 110 | 104 | 108 |

${S}_{6}$-box | 108 | 100 | 106.5 |

Ref. [15] | 110 | 104 | 107.25 |

Ref. [16] | 108 | 106 | 107.5 |

Ref. [26] | 108 | 106 | 107 |

Ref. [27] | 108 | 104 | 106.5 |

- | 8 | 6 | 6 | 6 | 10 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 |

6 | 10 | 6 | 8 | 6 | 6 | 6 | 6 | 8 | 8 | 6 | 8 | 6 | 8 | 6 | 6 |

10 | 6 | 10 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 6 |

8 | 6 | 8 | 6 | 6 | 6 | 10 | 8 | 6 | 6 | 6 | 8 | 6 | 6 | 8 | 6 |

8 | 6 | 6 | 6 | 4 | 8 | 6 | 6 | 6 | 8 | 8 | 6 | 6 | 6 | 6 | 6 |

6 | 8 | 8 | 8 | 8 | 6 | 6 | 6 | 8 | 6 | 6 | 8 | 8 | 6 | 8 | 8 |

6 | 6 | 6 | 6 | 6 | 6 | 8 | 6 | 8 | 6 | 6 | 6 | 8 | 6 | 6 | 6 |

10 | 6 | 6 | 6 | 8 | 6 | 8 | 6 | 8 | 6 | 8 | 6 | 6 | 8 | 8 | 6 |

6 | 4 | 6 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 6 | 6 | 6 | 6 | 8 | 6 |

10 | 8 | 8 | 8 | 6 | 6 | 6 | 8 | 6 | 6 | 6 | 6 | 8 | 6 | 6 | 6 |

6 | 6 | 8 | 6 | 6 | 6 | 10 | 6 | 8 | 6 | 6 | 6 | 6 | 6 | 6 | 8 |

6 | 6 | 6 | 8 | 8 | 8 | 6 | 8 | 6 | 10 | 6 | 8 | 6 | 8 | 6 | 6 |

6 | 10 | 10 | 6 | 6 | 6 | 8 | 8 | 6 | 6 | 8 | 6 | 6 | 6 | 6 | 8 |

8 | 6 | 8 | 8 | 8 | 8 | 8 | 6 | 8 | 6 | 6 | 6 | 6 | 6 | 8 | 6 |

6 | 6 | 6 | 4 | 8 | 6 | 6 | 8 | 10 | 8 | 4 | 6 | 10 | 8 | 6 | 8 |

8 | 6 | 8 | 8 | 6 | 4 | 10 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 6 | 6 |

S-Box | DP |
---|---|

${S}_{1}$-box | 3.9062% |

${S}_{2}$-box | 3.9062% |

${S}_{3}$-box | 3.9062% |

${S}_{4}$-box | 4.6875% |

${S}_{5}$-box | 3.9062% |

${S}_{6}$-box | 3.9062% |

Ref. [16] | 3.9062% |

Ref. [17] | 3.9062% |

Ref. [26] | 3.9062% |

Ref. [27] | 4.2960% |

0.4688 | 0.5938 | 0.4688 | 0.4844 | 0.5469 | 0.5000 | 0.5156 | 0.4375 |

0.4531 | 0.4844 | 0.4844 | 0.5156 | 0.5000 | 0.4844 | 0.5000 | 0.5156 |

0.4688 | 0.5000 | 0.4219 | 0.4688 | 0.5156 | 0.4844 | 0.5313 | 0.6094 |

0.5156 | 0.5313 | 0.5313 | 0.5469 | 0.5625 | 0.4688 | 0.4844 | 0.4531 |

0.5000 | 0.4844 | 0.5156 | 0.5156 | 0.5156 | 0.5313 | 0.4688 | 0.5156 |

0.5000 | 0.4531 | 0.5625 | 0.5313 | 0.5000 | 0.5000 | 0.5625 | 0.5156 |

0.4844 | 0.5156 | 0.5313 | 0.4844 | 0.5781 | 0.4688 | 0.4844 | 0.4844 |

0.5313 | 0.5000 | 0.5313 | 0.5000 | 0.4531 | 0.4844 | 0.4688 | 0.5469 |

S-Box | Average |
---|---|

${S}_{1}$-box | 0.5007 |

${S}_{2}$-box | 0.5007 |

${S}_{3}$-box | 0.5008 |

${S}_{4}$-box | 0.4836 |

${S}_{5}$-box | 0.5010 |

${S}_{6}$-box | 0.5048 |

Ref. [15] | 0.5046 |

Ref. [16] | 0.4943 |

Ref. [17] | 0.4953 |

Ref. [26] | 0.5015 |

Ref. [27] | 0.4990 |

- | 108 | 106 | 108 | 102 | 108 | 102 | 104 |

108 | - | 106 | 102 | 102 | 104 | 106 | 106 |

106 | 106 | - | 106 | 102 | 108 | 104 | 106 |

108 | 102 | 106 | - | 102 | 104 | 108 | 102 |

102 | 102 | 102 | 102 | - | 100 | 104 | 104 |

108 | 104 | 104 | 104 | 100 | - | 104 | 100 |

102 | 106 | 104 | 108 | 104 | 104 | - | 104 |

104 | 106 | 106 | 102 | 104 | 100 | 104 | - |

- | 0.5020 | 0.4902 | 0.5000 | 0.4883 | 0.5195 | 0.5098 | 0.4980 |

0.5020 | - | 0.4844 | 0.4961 | 0.5059 | 0.5039 | 0.5098 | 0.5056 |

0.4902 | 0.4844 | - | 0.5052 | 0.5007 | 0.4717 | 0.5015 | 0.5093 |

0.5000 | 0.4961 | 0.5052 | - | 0.5059 | 0.5017 | 0.5010 | 0.5005 |

0.4883 | 0.5059 | 0.5017 | 0.5059 | - | 0.4941 | 0.5056 | 0.5056 |

0.5195 | 0.5039 | 0.4707 | 0.5017 | 0.4941 | - | 0.5056 | 0.5059 |

0.5098 | 0.5098 | 0.5015 | 0.5010 | 0.5056 | 0.5056 | - | 0.5059 |

0.4980 | 0.5056 | 0.5093 | 0.5005 | 0.5056 | 0.5059 | 0.5059 | - |

S-Box | BIC-Nonlinearity Average | BIC-SAC Average |
---|---|---|

${S}_{1}$-box | 104.21 | 0.5012 |

${S}_{2}$-box | 104.21 | 0.5012 |

${S}_{3}$-box | 104.20 | 0.5011 |

${S}_{4}$-box | 101.90 | 0.4945 |

${S}_{5}$-box | 104.21 | 0.5016 |

${S}_{6}$-box | 103.53 | 0.5038 |

Ref. [15] | 103.86 | 0.5034 |

Ref. [16] | 104.35 | 0.4982 |

Ref. [17] | 104.07 | 0.5021 |

Ref. [26] | 104.21 | 0.5016 |

Ref. [27] | 103.18 | 0.4992 |

The Number of LEs | Highest Frequency | |
---|---|---|

Proposed S-box | 73 | 192.93 MHz |

logic circuits of the AES SubBytes | 87 | 47.55 MHz |

LUT-based of AES S-box | 237 | 183.82 MHz |

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**MDPI and ACS Style**

Wang, J.; Zhu, Y.; Zhou, C.; Qi, Z.
Construction Method and Performance Analysis of Chaotic S-Box Based on a Memorable Simulated Annealing Algorithm. *Symmetry* **2020**, *12*, 2115.
https://doi.org/10.3390/sym12122115

**AMA Style**

Wang J, Zhu Y, Zhou C, Qi Z.
Construction Method and Performance Analysis of Chaotic S-Box Based on a Memorable Simulated Annealing Algorithm. *Symmetry*. 2020; 12(12):2115.
https://doi.org/10.3390/sym12122115

**Chicago/Turabian Style**

Wang, Juan, Yangqing Zhu, Chao Zhou, and Zhiming Qi.
2020. "Construction Method and Performance Analysis of Chaotic S-Box Based on a Memorable Simulated Annealing Algorithm" *Symmetry* 12, no. 12: 2115.
https://doi.org/10.3390/sym12122115