A Novel Dynamic S-Box Generation Scheme Based on Quantum Random Walks Controlled by a Hyper-Chaotic Map
Abstract
:1. Introduction
- 1.
- A new compound chaotic system based on a two-dimensional hyper-chaotic map and quantum random walks is proposed, which has a larger key space and better chaotic performance and is more suitable for practical applications;
- 2.
- A simple and efficient dynamic S-Box generation algorithm is proposed;
- 3.
- A comprehensive and detailed security analysis of the generated S-Box is made to evaluate it against cryptographic landscapes. The analytical results demonstrate that the S-Box can well meet multiple cryptographic criteria.
2. A Compound Chaotic System Based on a Two-Dimensional Hyper-Chaotic Map and Quantum Random Walks
2.1. Two-Dimensional Hyper-Chaotic Map Controlling Quantum Random Walks
2.1.1. Trajectory Diagram
2.1.2. Lyapunov Exponents
2.1.3. Bifurcation Diagram
2.2. A Scheme for a Sequence Generated by Quantum Random Walks on a Cycle Graph
Algorithm 1 Random number generation algorithm based on ring graph |
Input: Output: (1) Init: , , , ; (2) , , , . (3) |
2.3. A New Compound Chaotic System Based on Quantum Random Walks Controlled by a Hyper-Chaotic Map
2.3.1. Degree of Non-Periodicity
2.3.2. Statistical Complexity Measures
3. Dynamic S-Box Generation Algorithm Based on the Compound Chaotic System
3.1. Introduction of the S-Box
3.2. Pseudo-Random Number Generator (PRNG) Based on the Proposed Compound Chaotic System
- Initialization ;
- ;
- Let . Furthermore, the following formula is used to generate and output the random number :
NIST Statistical Test
3.3. Dynamic S-Box Generation Algorithm Based on the Proposed PRNG
Algorithm 2 The algorithm for generating S-Boxes |
Input: and m (number of S-Boxes required) Output: (1) ; (2) %% Initialize the state of an S-Box: %% (3) for Obtain a number j from the proposed PRNG; swap values of and ; end for; (4) %% Check whether the S-Box meets the criteria of high diffusion and low differential uniformity [37]. %% for if %% The operator is used to obtain the differential uniformity value of an S-Box. %% ; ; end if end for; If ; else jump to (2); (5) if jump to (2); |
4. Performance Tests of the Constructed S-Box
4.1. Basic Cryptographic Evaluation Criteria
4.1.1. Bijection
4.1.2. Algebraic Degree
4.1.3. Algebraic Complexity (Univariate Degree)
4.1.4. Nonlinearity
4.1.5. Strict Avalanche Criterion (SAC)
4.1.6. Bit Independence Criterion (BIC)
4.1.7. Differential Probability
4.1.8. Maximal Degree of the Product of k Coordinates
4.1.9. Differential Uniformity
4.1.10. Linear Approximation Probability
4.2. Security Analysis
4.2.1. Resistance to Algebraic Attacks
4.2.2. Resistance to Differential Attack
4.2.3. Resistance to Linear Attack
4.2.4. Resistance to Boomerang Attack
4.3. Performance Comparison with Different S-Boxes
5. Conclusions and Discussion
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Abbreviations
LAP | Linear approximation probability |
SAC | Strict avalanche criterion |
CWT | Continuous wavelet transform |
DP | Differential approximation probability |
SCM | Statistical complexity measure |
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Test Name | p-Value | Pass Rate | Result |
---|---|---|---|
Frequency | 0.816537 | 99/100 | Pass |
Block Frequency (m = 128) | 0.249284 | 98/100 | Pass |
Cumulative Sums (Forward) | 0.289667 | 99/100 | Pass |
Cumulative Sums (Reverse) | 0.319084 | 99/100 | Pass |
Runs | 0.935716 | 98/100 | Pass |
Longest Run of Ones | 0.419021 | 99/100 | Pass |
Rank | 0.955835 | 100/100 | Pass |
FFT | 0.202268 | 99/100 | Pass |
Non-Overlapping Templates | 0.964295 | 100/100 | Pass |
(m = 9, B = 000000001) | |||
Overlapping Templates (m = 9) | 0.595549 | 100/100 | Pass |
Universal | 0.494392 | 99/100 | Pass |
Approximate Entropy (m = 10) | 0.191687 | 98/100 | Pass |
Random-Excursions (data1) | 0.848588 | 63/63 | Pass |
Random-Excursions Variant Serial (data7) | 0.788728 | 63/63 | Pass |
Serial Test 1 (m = 16) | 0.383827 | 99/100 | Pass |
Serial Test 2 (m = 16) | 0.319084 | 100/100 | Pass |
Linear complexity (M = 500) | 0.595549 | 100/100 | Pass |
i/j | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | 145 | 129 | 70 | 86 | 165 | 18 | 91 | 84 | 29 | 3 | 88 | 132 | 47 | 20 | 17 | 49 |
2 | 217 | 158 | 107 | 219 | 147 | 10 | 39 | 130 | 60 | 179 | 202 | 34 | 190 | 185 | 128 | 63 |
3 | 79 | 213 | 41 | 114 | 97 | 74 | 166 | 169 | 230 | 118 | 199 | 135 | 142 | 31 | 75 | 124 |
4 | 73 | 93 | 6 | 83 | 151 | 99 | 120 | 203 | 19 | 122 | 64 | 248 | 106 | 4 | 183 | 250 |
5 | 187 | 148 | 116 | 143 | 164 | 125 | 81 | 35 | 138 | 161 | 9 | 111 | 223 | 159 | 176 | 108 |
6 | 157 | 2 | 50 | 45 | 155 | 178 | 113 | 7 | 14 | 181 | 194 | 174 | 244 | 239 | 54 | 0 |
7 | 180 | 229 | 76 | 149 | 72 | 141 | 117 | 207 | 51 | 27 | 231 | 154 | 16 | 110 | 126 | 236 |
8 | 28 | 242 | 197 | 121 | 25 | 104 | 172 | 240 | 37 | 221 | 13 | 24 | 94 | 103 | 198 | 205 |
9 | 243 | 33 | 175 | 171 | 80 | 193 | 153 | 245 | 152 | 11 | 5 | 53 | 208 | 137 | 90 | 234 |
10 | 38 | 42 | 177 | 67 | 85 | 26 | 101 | 201 | 215 | 186 | 127 | 78 | 150 | 216 | 96 | 218 |
11 | 252 | 65 | 62 | 43 | 184 | 220 | 112 | 52 | 225 | 66 | 224 | 182 | 36 | 115 | 237 | 98 |
12 | 77 | 191 | 57 | 123 | 95 | 8 | 167 | 55 | 227 | 56 | 131 | 30 | 222 | 200 | 136 | 235 |
13 | 69 | 206 | 23 | 255 | 102 | 22 | 196 | 253 | 92 | 211 | 170 | 254 | 133 | 40 | 249 | 15 |
14 | 160 | 251 | 68 | 214 | 192 | 109 | 87 | 48 | 189 | 195 | 188 | 228 | 12 | 139 | 105 | 163 |
15 | 61 | 32 | 58 | 246 | 241 | 46 | 209 | 134 | 1 | 146 | 204 | 168 | 226 | 173 | 238 | 89 |
16 | 21 | 144 | 119 | 82 | 233 | 156 | 140 | 162 | 44 | 59 | 232 | 212 | 100 | 247 | 210 | 71 |
7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | 0 | 1 | 2 | 4 | 8 | 16 | 32 | 64 | 128 | 29 | 58 | 116 | 232 | 205 | 135 | 19 |
2 | 38 | 76 | 152 | 45 | 90 | 180 | 117 | 234 | 201 | 143 | 3 | 6 | 12 | 24 | 48 | 96 |
3 | 192 | 157 | 39 | 78 | 156 | 37 | 74 | 148 | 53 | 106 | 212 | 181 | 119 | 238 | 193 | 159 |
4 | 35 | 70 | 140 | 5 | 10 | 20 | 40 | 80 | 160 | 93 | 186 | 105 | 210 | 185 | 111 | 222 |
5 | 161 | 95 | 190 | 97 | 194 | 153 | 47 | 94 | 188 | 101 | 202 | 137 | 15 | 30 | 60 | 120 |
6 | 240 | 253 | 231 | 211 | 187 | 107 | 214 | 177 | 127 | 254 | 225 | 223 | 163 | 91 | 182 | 113 |
7 | 226 | 217 | 175 | 67 | 134 | 17 | 34 | 68 | 136 | 13 | 26 | 52 | 104 | 208 | 189 | 103 |
8 | 206 | 129 | 31 | 62 | 124 | 248 | 237 | 199 | 147 | 59 | 118 | 236 | 197 | 151 | 51 | 102 |
9 | 204 | 133 | 23 | 46 | 92 | 184 | 109 | 218 | 169 | 79 | 158 | 33 | 66 | 132 | 21 | 42 |
10 | 84 | 168 | 77 | 154 | 41 | 82 | 164 | 85 | 170 | 73 | 146 | 57 | 114 | 228 | 213 | 183 |
11 | 115 | 230 | 209 | 191 | 99 | 198 | 145 | 63 | 126 | 252 | 229 | 215 | 179 | 123 | 246 | 241 |
12 | 255 | 227 | 219 | 171 | 75 | 150 | 49 | 98 | 196 | 149 | 55 | 110 | 220 | 165 | 87 | 174 |
13 | 65 | 130 | 25 | 50 | 100 | 200 | 141 | 7 | 14 | 28 | 56 | 112 | 224 | 221 | 167 | 83 |
14 | 166 | 81 | 162 | 89 | 178 | 121 | 242 | 249 | 239 | 195 | 155 | 43 | 86 | 172 | 69 | 138 |
15 | 9 | 18 | 36 | 72 | 144 | 61 | 122 | 244 | 245 | 247 | 243 | 251 | 235 | 203 | 139 | 11 |
16 | 22 | 44 | 88 | 176 | 125 | 250 | 233 | 207 | 131 | 27 | 54 | 108 | 216 | 173 | 71 | 142 |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | 145 | 33 | 253 | 168 | 50 | 241 | 221 | 168 | 150 | 208 | 54 | 179 | 157 | 196 | 216 | 21 |
2 | 44 | 151 | 203 | 161 | 238 | 133 | 253 | 107 | 223 | 199 | 184 | 244 | 229 | 154 | 44 | 176 |
3 | 129 | 69 | 236 | 230 | 239 | 250 | 186 | 45 | 158 | 139 | 78 | 198 | 29 | 56 | 99 | 31 |
4 | 36 | 206 | 82 | 115 | 33 | 35 | 1 | 194 | 139 | 133 | 45 | 47 | 163 | 233 | 203 | 184 |
5 | 169 | 112 | 221 | 31 | 45 | 205 | 101 | 169 | 141 | 190 | 234 | 148 | 172 | 100 | 129 | 157 |
6 | 201 | 14 | 91 | 123 | 222 | 143 | 49 | 209 | 221 | 39 | 156 | 206 | 76 | 19 | 7 | 99 |
7 | 37 | 69 | 33 | 126 | 22 | 103 | 140 | 233 | 73 | 72 | 186 | 3 | 168 | 9 | 175 | 49 |
8 | 49 | 174 | 30 | 81 | 115 | 124 | 232 | 221 | 123 | 23 | 75 | 207 | 18 | 139 | 22 | 11 |
9 | 254 | 82 | 194 | 182 | 201 | 173 | 168 | 226 | 90 | 104 | 205 | 190 | 62 | 15 | 40 | 20 |
10 | 193 | 114 | 89 | 84 | 219 | 1 | 98 | 146 | 52 | 106 | 139 | 194 | 69 | 199 | 39 | 170 |
11 | 176 | 185 | 66 | 62 | 245 | 220 | 99 | 94 | 43 | 44 | 43 | 11 | 246 | 199 | 83 | 110 |
12 | 63 | 238 | 98 | 188 | 167 | 30 | 180 | 214 | 95 | 249 | 101 | 5 | 148 | 154 | 219 | 76 |
13 | 115 | 5 | 215 | 170 | 233 | 45 | 112 | 37 | 119 | 253 | 88 | 89 | 223 | 32 | 52 | 243 |
14 | 107 | 171 | 144 | 56 | 148 | 61 | 150 | 130 | 117 | 255 | 55 | 22 | 29 | 143 | 248 | 210 |
15 | 63 | 251 | 113 | 209 | 158 | 134 | 61 | 59 | 164 | 114 | 130 | 147 | 184 | 249 | 143 | 194 |
16 | 106 | 98 | 219 | 204 | 166 | 88 | 116 | 200 | 170 | 54 | 212 | 225 | 23 | 64 | 186 | 0 |
104 | 106 | 106 | 104 | 110 | 106 | 112 | 104 | 106.5 |
0.5746 | 0.5517 | 0.5188 | 0.4753 | 0.4772 | 0.5379 | 0.4979 | 0.5276 |
0.4719 | 0.4821 | 0.5756 | 0.4837 | 0.5895 | 0.5004 | 0.4463 | 0.4622 |
0.5407 | 0.4823 | 0.5816 | 0.5187 | 0.4307 | 0.4070 | 0.4942 | 0.4878 |
0.5474 | 0.4812 | 0.5162 | 0.4815 | 0.5114 | 0.5339 | 0.4757 | 0.4535 |
0.5065 | 0.4680 | 0.5365 | 0.5185 | 0.5197 | 0.4902 | 0.5321 | 0.4392 |
0.5367 | 0.5240 | 0.4952 | 0.5165 | 0.5976 | 0.5691 | 0.5347 | 0.5091 |
0.5654 | 0.5288 | 0.5458 | 0.5255 | 0.4708 | 0.4594 | 0.4648 | 0.5513 |
0.5229 | 0.5791 | 0.5664 | 0.5164 | 0.4262 | 0.6032 | 0.3924 | 0.4726 |
- | 0.46875 | 0.54688 | 0.54688 | 0.51562 | 0.54688 | 0.56250 | 0.51562 | |
0.54688 | - | 0.48438 | 0.48438 | 0.46875 | 0.53125 | 0.53125 | 0.57812 | |
0.51562 | 0.57812 | - | 0.51562 | 0.54688 | 0.48438 | 0.54688 | 0.56250 | |
0.46875 | 0.48438 | 0.51562 | - | 0.51562 | 0.51562 | 0.53125 | 0.51562 | |
0.48438 | 0.59375 | 0.42188 | 0.50000 | - | 0.59375 | 0.46875 | 0.42188 | |
0.53125 | 0.50000 | 0.40625 | 0.53125 | 0.48438 | - | 0.45312 | 0.59375 | |
0.50000 | 0.43750 | 0.50000 | 0.46875 | 0.53125 | 0.53125 | - | 0.39062 | |
0.51562 | 0.46875 | 0.48438 | 0.45312 | 0.43750 | 0.51562 | 0.54688 | - |
- | 108 | 108 | 108 | 106 | 104 | 112 | 106 | |
108 | - | 108 | 106 | 106 | 102 | 108 | 108 | |
108 | 108 | - | 102 | 102 | 108 | 100 | 104 | |
108 | 106 | 102 | - | 108 | 102 | 106 | 106 | |
106 | 106 | 102 | 108 | - | 106 | 102 | 104 | |
104 | 102 | 108 | 102 | 106 | - | 106 | 106 | |
112 | 108 | 100 | 106 | 102 | 106 | - | 106 | |
106 | 108 | 104 | 106 | 104 | 106 | 106 | - |
k | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|
7 | 7 | 7 | 7 | 7 | 7 | 7 |
S-Box Method | Nonlinearity | SAC | BIC-NL | LAP | DP | AC | ||
---|---|---|---|---|---|---|---|---|
Min | Max | Average | ||||||
Ref. [1] | 102 | 108 | 104.5 | 0.498 | 104.6 | 0.125 | 0.048 | 254 |
Ref. [16] | 104 | 108 | 106.8 | 0.507 | 103.9 | 0.140 | 0.054 | 254 |
Ref. [20] | 104 | 110 | 106 | 0.499 | 103.8 | 0.125 | 0.039 | 255 |
Ref. [48] | 104 | 110 | 106.5 | 0.495 | 103.8 | 0.141 | 0.039 | 255 |
Ref. [49] | 98 | 110 | 102 | 0.493 | 104.6 | 0.140 | 0.046 | 255 |
Proposed S-Box | 104 | 112 | 106.5 | 0.506 | 105.6 | 0.132 | 0.039 | 255 |
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Zhang, L.; Ma, C.; Zhao, Y.; Zhao, W. A Novel Dynamic S-Box Generation Scheme Based on Quantum Random Walks Controlled by a Hyper-Chaotic Map. Mathematics 2024, 12, 84. https://doi.org/10.3390/math12010084
Zhang L, Ma C, Zhao Y, Zhao W. A Novel Dynamic S-Box Generation Scheme Based on Quantum Random Walks Controlled by a Hyper-Chaotic Map. Mathematics. 2024; 12(1):84. https://doi.org/10.3390/math12010084
Chicago/Turabian StyleZhang, Lijun, Caochuan Ma, Yuxiang Zhao, and Wenbo Zhao. 2024. "A Novel Dynamic S-Box Generation Scheme Based on Quantum Random Walks Controlled by a Hyper-Chaotic Map" Mathematics 12, no. 1: 84. https://doi.org/10.3390/math12010084