Construction of S-Box Based on Chaotic Map and Algebraic Structures
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
Algorithm 1 To find the positions at where the minimum difference lies |
Inputs: Two distinct logistic chaotic map sequences, , y and . Output: Position location, . 1: 2: 3: 4: 5: |
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 . Outputs: Substitution box, . 1: 2: while do 3: 4: 5: for do 6: if then 7: 8: end if 9: end for 10: if then 11: 12: 13: 14: 15: 16: 17: else 18: 19: 20: end if 21: 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
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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z | Here We Are Taking and from Table 2 | S-Box Elements | |
---|---|---|---|
0 | 255 | ||
1 | 125 | ||
⋮ | ⋮ | ⋮ | |
254 | 106 | ||
255 | 95 |
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 |
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 |
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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
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 StyleHussain, 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