# Construction of S-Box Based on Chaotic Map and Algebraic Structures

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. General Linear Group

#### 2.2. Logistic Map

## 3. Propose S-Box

#### 3.1. Step 1: Application of General Linear Group

#### 3.2. Step 2: Applying Logistic Chaotic Map

**a**) Initial vector of step 1 as a basic seed, which is shown in Table 2. Let the initial seed be represented as K with size 1 × 256.

**b**) Define the two chaotic logistic map sequences as defined in Equation (3) with appropriate initial conditions.

**c**) Initial parameters for first logistic map are as follows $p=3.99234589$ and ${y}_{0}=0.5$. The first logistic map sequence is represented as ${f}_{1}$. The length of this sequence is 256.

**d**) Initial parameters for second logistic map are as follows $p=3.99777777$ and ${y}_{0}=0.6$. The second logistic map sequence is represented as y. The length of this sequence is also 256.

**e**) Define ${f}_{2}={y}_{155}$, that is ${f}_{2}$ has a single value of second chaotic sequence which is placed at 155th position.

**f**) Define a function $pos\_min$ as shown in Algorithm 1. It consists of two steps; in step 1, “ones(1,256)” will give us a row vector with 256 values, all with a value of ${f}_{2}$. In step 2, the position at where the minimum difference lies will be set as an output.

**g**) Use the initial seed of S-box generated with the help of general linear group shown in Table 2 and the logistic map, we will get the vector ${K}_{1}$. The whole process of getting the output of this step is depicted in Algorithm 2 and the output of this step is shown in Table 3.

Algorithm 1 To find the positions at where the minimum difference lies |

Inputs: Two distinct logistic chaotic map sequences, ${f}_{1}$, y and ${f}_{2}$.Output: Position location, $pos\_min$.1: $\mathbf{function}\phantom{\rule{4pt}{0ex}}[pos\_min]=find\_pos\_min({f}_{1},{f}_{2})$ 2: $\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}f={f}_{2}\ast ones(1,256)$ 3: $\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}diff=abs\left(minus({f}_{1},f)\right)$ 4: $\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}pos\_min=find(diff==min(diff\left)\right)$ 5: $\mathbf{End}$ |

Algorithm 2 To generate S-box with the input seeds of logistic map and general linear group |

Inputs: Initial vector K of Table 1 (which is a substitution box), functions for logistic map and $pos\_min$.Outputs: Substitution box, ${K}_{1}$.1: $i=1$ 2: while $i<257$ do3: $[pos\_min]=find\_pos\_min({f}_{1},y))$ 4: $p=0$ 5: for $j\leftarrow 1:length\left(V\right)$ do6: if $pos\_min==V\left(j\right)$ then7: $p=1$ 8: end if9: end for10: if $p==0$ then11: $V\left(i\right)=pos\_min$ 12: ${K}_{1}\left(i\right)=K(pos\_min)$ 13: ${x}_{1}=0.9\times {x}_{1}+0.1\times K(pos\_min)/255$ 14: $y=logistic({k}_{1},{x}_{1})$ 15: ${f}_{2}=y\left(155\right)$ 16: $i=i+1$ 17: else18: ${x}_{0}=0.9\times {x}_{0}+0.1\times K(pos-min)/255$ 19: ${f}_{1}=logistic({k}_{0},{x}_{0})$ 20: end if21: end while |

#### 3.3. Step 3: Application of Permutation to Get S-Box

## 4. Analyses for Evaluating the Strength of S-Box

#### 4.1. Nonlinearity

**Definition**

**1.**

#### 4.2. Strict Avalanche Criterion

#### 4.3. Bit Independent Criterion

#### 4.4. Differential Approximation Probability

#### 4.5. Linear Approximation Probability

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Illustration of sensitivity to initial conditions of logistic map. (

**a**) values of the first 100 iterations against the initial parameters of $p=3.7$ and ${y}_{0}=0.5$, (

**b**) comparison between the iteration values generated from two slightly different initial conditions of p, (

**c**) comparison between the iteration values generated from two slightly different initial conditions of ${y}_{0}$.

z | $\mathit{f}\left(\mathit{z}\right)=\frac{(\mathit{az}+\mathit{b})}{(\mathit{cz}+\mathit{d})}$ | Here We Are Taking ${\mathit{\zeta}}^{{\mathit{i}}_{\mathit{i}}}$ and ${\mathit{\zeta}}^{{\mathit{j}}_{\mathit{j}}}$ from Table 2 | S-Box Elements |
---|---|---|---|

0 | $f\left(0\right)=\frac{\left(180\right(0)+144)}{\left(83\right(0)+4)}$ | $f\left(0\right)=\frac{{\Delta}^{i}}{{\Delta}^{j}}$ | 255 |

1 | $f\left(1\right)=\frac{\left(180\right(1)+144)}{\left(83\right(1)+4)}$ | $f\left(1\right)=\frac{{\Delta}^{{i}_{2}}}{{\Delta}^{{j}_{2}}}$ | 125 |

⋮ | ⋮ | ⋮ | |

254 | $f\left(254\right)=\frac{\left(180\right(254)+144)}{\left(83\right(254)+4)}$ | $f\left(254\right)=\frac{{\Delta}^{{i}_{254}}}{{\Delta}^{{j}_{254}}}$ | 106 |

255 | $f\left(255\right)=\frac{\left(180\right(255)+144)}{\left(83\right(255)+4)}$ | $f\left(255\right)=\frac{{\Delta}^{{i}_{255}}}{{\Delta}^{{j}_{255}}}$ | 95 |

**Table 2.**Initial vector corresponding to $a=183$, $b=144$, $c=83$ and $d=4$ of Mobius transformation in matrix form.

255 | 125 | 26 | 23 | 249 | 72 | 44 | 32 | 33 | 34 | 202 | 30 | 191 | 186 | 248 | 211 |

166 | 189 | 195 | 62 | 242 | 150 | 253 | 45 | 165 | 239 | 143 | 169 | 2 | 103 | 183 | 65 |

86 | 130 | 91 | 40 | 219 | 223 | 60 | 210 | 168 | 73 | 115 | 139 | 154 | 175 | 187 | 75 |

124 | 39 | 152 | 218 | 131 | 251 | 185 | 81 | 28 | 157 | 109 | 12 | 6 | 13 | 224 | 226 |

135 | 230 | 188 | 164 | 149 | 128 | 116 | 228 | 217 | 173 | 212 | 8 | 84 | 80 | 243 | 93 |

20 | 146 | 194 | 36 | 42 | 35 | 79 | 77 | 179 | 9 | 151 | 85 | 172 | 52 | 118 | 222 |

15 | 14 | 101 | 18 | 112 | 117 | 55 | 92 | 48 | 178 | 22 | 25 | 54 | 231 | 4 | 16 |

138 | 64 | 160 | 134 | 46 | 200 | 120 | 238 | 137 | 114 | 108 | 241 | 61 | 153 | 50 | 3 |

132 | 78 | 163 | 110 | 90 | 201 | 129 | 47 | 236 | 53 | 104 | 246 | 49 | 41 | 100 | 158 |

232 | 68 | 21 | 159 | 141 | 227 | 197 | 208 | 245 | 38 | 215 | 156 | 70 | 133 | 43 | 127 |

198 | 180 | 74 | 190 | 89 | 37 | 69 | 209 | 98 | 136 | 29 | 182 | 87 | 126 | 207 | 237 |

51 | 122 | 155 | 204 | 192 | 247 | 206 | 59 | 24 | 82 | 63 | 83 | 17 | 107 | 184 | 142 |

205 | 167 | 19 | 121 | 216 | 177 | 96 | 66 | 105 | 123 | 229 | 113 | 214 | 11 | 234 | 94 |

0 | 221 | 240 | 220 | 31 | 196 | 119 | 161 | 252 | 181 | 148 | 99 | 111 | 56 | 97 | 244 |

67 | 250 | 199 | 57 | 254 | 7 | 203 | 145 | 171 | 225 | 140 | 193 | 213 | 102 | 174 | 1 |

58 | 10 | 88 | 147 | 233 | 170 | 5 | 176 | 71 | 235 | 27 | 144 | 162 | 76 | 106 | 95 |

Rows/Columns | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0 | 84 | 44 | 60 | 125 | 119 | 238 | 63 | 110 | 207 | 22 | 191 | 187 | 48 | 252 | 202 | 93 |

1 | 181 | 50 | 57 | 118 | 151 | 56 | 92 | 192 | 86 | 229 | 171 | 19 | 113 | 10 | 233 | 24 |

2 | 242 | 161 | 139 | 27 | 13 | 126 | 170 | 21 | 134 | 122 | 17 | 20 | 34 | 216 | 159 | 40 |

3 | 138 | 201 | 107 | 100 | 152 | 0 | 81 | 127 | 42 | 213 | 65 | 128 | 251 | 197 | 237 | 172 |

4 | 136 | 95 | 221 | 102 | 165 | 45 | 49 | 135 | 190 | 85 | 99 | 222 | 41 | 162 | 80 | 32 |

5 | 106 | 137 | 164 | 109 | 72 | 53 | 43 | 12 | 89 | 101 | 26 | 38 | 74 | 124 | 71 | 158 |

6 | 16 | 54 | 163 | 147 | 209 | 64 | 120 | 160 | 66 | 186 | 97 | 239 | 166 | 112 | 178 | 8 |

7 | 47 | 37 | 3 | 133 | 2 | 108 | 247 | 223 | 206 | 250 | 114 | 76 | 15 | 211 | 155 | 231 |

8 | 123 | 88 | 248 | 203 | 115 | 208 | 210 | 245 | 1 | 195 | 144 | 154 | 156 | 196 | 230 | 96 |

9 | 79 | 182 | 67 | 117 | 111 | 168 | 183 | 142 | 232 | 175 | 157 | 131 | 193 | 220 | 184 | 189 |

10 | 146 | 148 | 35 | 87 | 91 | 31 | 77 | 61 | 236 | 4 | 167 | 234 | 205 | 33 | 52 | 94 |

11 | 73 | 212 | 9 | 83 | 214 | 227 | 145 | 200 | 51 | 62 | 149 | 30 | 59 | 11 | 103 | 98 |

12 | 70 | 194 | 14 | 243 | 235 | 199 | 169 | 174 | 68 | 5 | 224 | 140 | 218 | 179 | 255 | 246 |

13 | 254 | 215 | 188 | 39 | 75 | 23 | 82 | 253 | 29 | 173 | 78 | 143 | 153 | 249 | 28 | 225 |

14 | 180 | 58 | 150 | 244 | 176 | 217 | 105 | 204 | 116 | 46 | 69 | 185 | 130 | 219 | 177 | 6 |

15 | 198 | 228 | 141 | 132 | 104 | 121 | 18 | 7 | 226 | 240 | 90 | 129 | 241 | 25 | 55 | 36 |

Rows/Columns | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0 | 94 | 30 | 171 | 84 | 96 | 215 | 28 | 246 | 3 | 216 | 245 | 255 | 152 | 86 | 31 | 180 |

1 | 118 | 208 | 184 | 237 | 204 | 112 | 185 | 109 | 183 | 182 | 76 | 159 | 19 | 149 | 44 | 239 |

2 | 123 | 173 | 103 | 12 | 195 | 154 | 63 | 244 | 256 | 188 | 65 | 130 | 18 | 194 | 38 | 72 |

3 | 162 | 45 | 78 | 137 | 119 | 83 | 165 | 98 | 27 | 142 | 249 | 125 | 100 | 238 | 120 | 199 |

4 | 9 | 7 | 20 | 200 | 174 | 42 | 243 | 136 | 91 | 102 | 52 | 139 | 242 | 117 | 213 | 59 |

5 | 60 | 47 | 232 | 43 | 145 | 181 | 114 | 167 | 229 | 8 | 150 | 221 | 172 | 132 | 23 | 210 |

6 | 192 | 231 | 35 | 69 | 22 | 115 | 201 | 151 | 247 | 193 | 222 | 39 | 54 | 178 | 56 | 85 |

7 | 138 | 104 | 214 | 48 | 107 | 175 | 240 | 108 | 16 | 21 | 17 | 141 | 62 | 88 | 74 | 14 |

8 | 61 | 248 | 226 | 144 | 90 | 95 | 71 | 202 | 10 | 81 | 53 | 163 | 110 | 254 | 75 | 32 |

9 | 11 | 224 | 101 | 129 | 177 | 253 | 111 | 37 | 24 | 33 | 140 | 131 | 113 | 2 | 155 | 206 |

10 | 68 | 197 | 66 | 147 | 6 | 79 | 189 | 25 | 187 | 49 | 134 | 5 | 64 | 146 | 241 | 70 |

11 | 217 | 168 | 124 | 205 | 158 | 170 | 143 | 209 | 207 | 191 | 223 | 196 | 15 | 51 | 50 | 169 |

12 | 153 | 73 | 36 | 160 | 127 | 219 | 87 | 122 | 135 | 55 | 97 | 41 | 190 | 233 | 126 | 157 |

13 | 13 | 133 | 121 | 235 | 1 | 252 | 93 | 34 | 251 | 211 | 212 | 179 | 92 | 106 | 67 | 82 |

14 | 29 | 99 | 234 | 148 | 227 | 176 | 225 | 203 | 40 | 198 | 105 | 220 | 58 | 4 | 250 | 166 |

15 | 186 | 26 | 218 | 156 | 161 | 236 | 164 | 57 | 128 | 46 | 89 | 228 | 230 | 77 | 116 | 80 |

Rows/Columns | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0 | 75 | 220 | 207 | 219 | 234 | 91 | 95 | 101 | 136 | 1 | 79 | 27 | 254 | 231 | 59 | 206 |

1 | 114 | 34 | 113 | 221 | 250 | 209 | 71 | 232 | 61 | 228 | 42 | 63 | 180 | 44 | 202 | 96 |

2 | 175 | 253 | 163 | 14 | 142 | 159 | 239 | 116 | 140 | 45 | 109 | 233 | 201 | 240 | 137 | 133 |

3 | 4 | 103 | 11 | 99 | 144 | 166 | 5 | 178 | 7 | 130 | 32 | 106 | 123 | 15 | 170 | 205 |

4 | 17 | 35 | 28 | 146 | 147 | 94 | 210 | 40 | 194 | 155 | 230 | 171 | 25 | 107 | 31 | 36 |

5 | 195 | 225 | 0 | 125 | 8 | 252 | 169 | 211 | 90 | 115 | 190 | 153 | 82 | 84 | 208 | 119 |

6 | 224 | 127 | 58 | 251 | 67 | 85 | 139 | 37 | 69 | 249 | 2 | 223 | 192 | 156 | 183 | 56 |

7 | 193 | 43 | 64 | 55 | 162 | 181 | 152 | 237 | 188 | 174 | 242 | 9 | 128 | 255 | 235 | 226 |

8 | 117 | 20 | 131 | 124 | 215 | 167 | 68 | 135 | 100 | 47 | 222 | 157 | 76 | 213 | 145 | 203 |

9 | 72 | 33 | 87 | 244 | 10 | 26 | 160 | 48 | 70 | 126 | 184 | 132 | 246 | 214 | 19 | 243 |

10 | 104 | 138 | 154 | 18 | 81 | 6 | 12 | 212 | 98 | 227 | 60 | 74 | 161 | 165 | 108 | 217 |

11 | 111 | 112 | 143 | 93 | 53 | 229 | 86 | 57 | 92 | 198 | 236 | 122 | 77 | 218 | 62 | 16 |

12 | 186 | 216 | 13 | 30 | 148 | 46 | 172 | 102 | 120 | 245 | 204 | 151 | 83 | 189 | 51 | 50 |

13 | 200 | 158 | 173 | 78 | 80 | 3 | 238 | 22 | 73 | 141 | 199 | 185 | 38 | 97 | 149 | 182 |

14 | 105 | 248 | 176 | 129 | 89 | 241 | 54 | 164 | 179 | 150 | 39 | 121 | 118 | 197 | 24 | 247 |

15 | 52 | 41 | 49 | 21 | 191 | 110 | 66 | 88 | 65 | 177 | 29 | 23 | 168 | 196 | 187 | 134 |

${\mathit{ftn}}_{0}$ | ${\mathit{ftn}}_{1}$ | ${\mathit{ftn}}_{2}$ | ${\mathit{ftn}}_{3}$ | ${\mathit{ftn}}_{4}$ | ${\mathit{ftn}}_{5}$ | ${\mathit{ftn}}_{6}$ | ${\mathit{ftn}}_{7}$ |
---|---|---|---|---|---|---|---|

112 | 112 | 112 | 112 | 112 | 112 | 112 | 112 |

0.515625 | 0.515625 | 0.453125 | 0.484375 | 0.562500 | 0.500000 | 0.453125 | 0.453125 |

0.468750 | 0.484375 | 0.562500 | 0.453125 | 0.5000 | 0.531250 | 0.500000 | 0.484375 |

0.515625 | 0.515625 | 0.500000 | 0.500000 | 0.4608750 | 0.500000 | 0.531250 | 0.562500 |

0.531250 | 0.531250 | 0.468750 | 0.531250 | 0.453125 | 0.546875 | 0.500000 | 0.500000 |

0.453125 | 0.500000 | 0.453125 | 0.500000 | 0.515625 | 0.531250 | 0.546875 | 0.500000 |

0.453125 | 0.515625 | 0.515625 | 0.546875 | 0.468750 | 0.531250 | 0.531250 | 0.468750 |

0.531250 | 0.531250 | 0.468750 | 0.531250 | 0.515625 | 0.484375 | 0.531250 | 0.468750 |

0.515625 | 0.562500 | 0.515625 | 0.531250 | 0.531250 | 0.515625 | 0.484375 | 0.484375 |

—— | 0.515625 | 0.486328 | 0.517578 | 0.500000 | 0.515625 | 0.509766 | 0.494141 |

0.515625 | —— | 519531 | 0.490234 | 0.511719 | 0.480469 | 0.501953 | 0.496094 |

0.486328 | 0.519531 | —— | 0.496094 | 0.525391 | 0.490234 | 0.507813 | 0.507813 |

0.517578 | 0.490234 | 0.496094 | —— | 0.494141 | 0.513672 | 0.505859 | 0.511719 |

0.500000 | 0.511719 | 0.525391 | 0.494141 | —— | 0.505859 | 0.494141 | 0.517578 |

0.515625 | 0.480469 | 0.490234 | 0.513672 | 0.505859 | —— | 0.509766 | 0.515625 |

0.509766 | 0.501953 | 0.507813 | 0.505859 | 0.494141 | 0.509766 | —— | 0.494141 |

0.494141 | 0.496094 | 0.507813 | 0.511719 | 0.517578 | 0.515625 | 0.494141 | —— |

0 | 112 | 112 | 112 | 112 | 112 | 112 | 112 |

112 | 0 | 112 | 112 | 112 | 112 | 112 | 112 |

112 | 112 | 0 | 112 | 112 | 112 | 112 | 112 |

112 | 112 | 112 | 0 | 112 | 112 | 112 | 112 |

112 | 112 | 112 | 112 | 0 | 112 | 112 | 112 |

112 | 112 | 112 | 112 | 112 | 0 | 112 | 112 |

112 | 112 | 112 | 112 | 112 | 112 | 0 | 112 |

112 | 112 | 112 | 112 | 112 | 112 | 112 | 0 |

Rows/Columns | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 |

1 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 |

2 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 |

3 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 |

4 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 |

5 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 |

6 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 |

7 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 |

8 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 |

9 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 |

10 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 |

11 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 |

12 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 |

13 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 |

14 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 |

15 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 |

S-Boxes/Analyses | Minimum Nonlinearity | SAC Offset | Minimum BIC-Nonlinearity | DP | LP |
---|---|---|---|---|---|

Ref [2] | 112 | 0.02637 | 112 | 0.015625 | 0.0625 |

Ref [9], S-box1 | 112 | 0.02579 | 112 | 0.015625 | 0.0625 |

Ref [12] | 112 | 0.02502 | 112 | 0.015625 | 0.0625 |

Ref [13] | 100 | 0.03125 | 100 | 0.0290525 | 0.070557 |

Ref [14] | 104 | 0.02007 | 96 | 0.0390625 | 0.148438 |

Ref [25] | 108 | 0.01833 | 104 | 0.03125 | 0.09375 |

Ref [9], S-box2 | 107.5 | 0.4971 | 103.85 | NA | NA |

Ref [9], S-box3 | 104 | 0.4531 | 112 | NA | NA |

Proposed | 112 | 0.01567 | 112 | 0.01562 | 0.0625 |

© 2019 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 (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Hussain, I.; Anees, A.; Al-Maadeed, T.A.; Mustafa, M.T.
Construction of S-Box Based on Chaotic Map and Algebraic Structures. *Symmetry* **2019**, *11*, 351.
https://doi.org/10.3390/sym11030351

**AMA Style**

Hussain I, Anees A, Al-Maadeed TA, Mustafa MT.
Construction of S-Box Based on Chaotic Map and Algebraic Structures. *Symmetry*. 2019; 11(3):351.
https://doi.org/10.3390/sym11030351

**Chicago/Turabian Style**

Hussain, Iqtadar, Amir Anees, Temadher Alassiry Al-Maadeed, and Muhammad Tahir Mustafa.
2019. "Construction of S-Box Based on Chaotic Map and Algebraic Structures" *Symmetry* 11, no. 3: 351.
https://doi.org/10.3390/sym11030351