# An Innovative Design of Substitution-Boxes Using Cubic Polynomial Mapping

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Related Work

## 3. Proposed Substitution-Box Design

^{n}−1}, mod operation gives the remainder, and both A and B ∈ N – {0} to construct an S-box of size n × n. A cubic polynomial mapping demonstrates a nonlinear behavior and is an inspiration for byte substitution. To ornate the erection of the proposed S-Box by Equation (1), let us have an explicit type of cubic polynomial function as specified in Equation (2). For n = 8, we have N = {0, 1, ……, 2

^{n}−1} = {0, 1, ……, 2

^{8}−1} = {0, 1, ……, 255}. One can choose any values for A and B (A, B ∈ N – {0}) to be used in Equation (1). For the sake of an example here, we have chosen A = 69, and B = 100. CPM function C(t) specified in Equation (2) spawns values N – {31} when t ∈ N – {135}. When t = 135, C(t) calculates to 256 ∉ N. To preserve the function C(t) as bijective one, we explicitly describe the value of C(t) for t=135 as habituated in Equation (2). A CPM function C: N → N to generate 8 × 8 S-box is given as:

## 4. Performance Results

#### 4.1. Bijectiveness

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

#### 4.3. Nonlinearity

_{b}(h) = Walsh spectrum of function b, and it is calculated as:

^{n}and k.h denotes the dot product of k and h, calculated as:

_{1}⊕ h

_{1}) + … + (k

_{n}⊕ h

_{n})

#### 4.4. Bit Independence Criterion (BIC)

#### 4.5. Linear Probability

_{z}and β

_{z}are the corresponding input and output masks and N = {0,1,…, 255}. Maximum value of LP of our S-box is only 0.140, and thus provides good resistance against linear cryptanalysis.

#### 4.6. Differential Probability

#### 4.7. Performance Comparison

- Our S-box has average value of nonlinearity greater than the other S-boxes in Table 7. As a result, proposed S-box provides good resistance against linear cryptanalysis.
- Table 7 validates that SAC value (0.507) of proposed S-box is very near to ideal value of SAC (0.5). We can say that our S-box is gratifying SAC in a respectable manner.
- Differential probability value of proposed S-box is just 0.054. This small value of DP reveals the cryptographic strength of our S-box.
- Proposed S-Box has LP value equal to 0.140. This small value guarantees that our S-box has the potential to confront the linear cryptanalysis.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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100 | 169 | 138 | 164 | 147 | 244 | 98 | 123 | 219 | 29 | 224 | 190 | 84 | 63 | 27 | 133 |

24 | 114 | 46 | 234 | 64 | 207 | 49 | 4 | 229 | 110 | 61 | 239 | 30 | 105 | 107 | 193 |

6 | 217 | 212 | 148 | 182 | 214 | 144 | 129 | 69 | 121 | 185 | 161 | 206 | 220 | 103 | 12 |

104 | 22 | 180 | 221 | 45 | 66 | 184 | 42 | 54 | 120 | 140 | 14 | 156 | 209 | 73 | 162 |

119 | 101 | 8 | 254 | 225 | 78 | 227 | 58 | 242 | 165 | 241 | 113 | 195 | 130 | 75 | 187 |

109 | 255 | 11 | 48 | 9 | 51 | 74 | 235 | 177 | 57 | 32 | 2 | 124 | 41 | 167 | 145 |

132 | 28 | 247 | 175 | 226 | 43 | 40 | 117 | 174 | 111 | 85 | 253 | 1 | 0 | 150 | 94 |

246 | 249 | 3 | 179 | 163 | 112 | 183 | 19 | 34 | 128 | 201 | 153 | 141 | 65 | 82 | 92 |

252 | 205 | 108 | 118 | 135 | 59 | 47 | 31 | 72 | 166 | 181 | 17 | 88 | 37 | 21 | 197 |

208 | 211 | 106 | 50 | 200 | 199 | 204 | 115 | 89 | 26 | 83 | 160 | 157 | 231 | 25 | 210 |

172 | 68 | 55 | 33 | 159 | 76 | 198 | 168 | 143 | 23 | 222 | 126 | 149 | 191 | 152 | 189 |

202 | 91 | 13 | 125 | 70 | 5 | 87 | 216 | 35 | 215 | 142 | 230 | 122 | 232 | 203 | 192 |

99 | 81 | 38 | 127 | 248 | 44 | 186 | 60 | 80 | 146 | 158 | 16 | 134 | 155 | 236 | 20 |

178 | 96 | 188 | 97 | 237 | 251 | 39 | 15 | 79 | 131 | 71 | 56 | 243 | 18 | 52 | 245 |

240 | 194 | 7 | 93 | 95 | 170 | 218 | 139 | 90 | 228 | 196 | 151 | 250 | 136 | 223 | 154 |

86 | 176 | 67 | 173 | 137 | 116 | 10 | 233 | 171 | 238 | 77 | 102 | 213 | 53 | 36 | 62 |

0.500 | 0.469 | 0.500 | 0.516 | 0.547 | 0.453 | 0.563 | 0.469 |

0.531 | 0.578 | 0.453 | 0.500 | 0.453 | 0.484 | 0.531 | 0.531 |

0.531 | 0.484 | 0.547 | 0.531 | 0.594 | 0.469 | 0.516 | 0.484 |

0.469 | 0.531 | 0.500 | 0.516 | 0.453 | 0.547 | 0.531 | 0.516 |

0.438 | 0.531 | 0.406 | 0.500 | 0.500 | 0.453 | 0.547 | 0.484 |

0.563 | 0.500 | 0.453 | 0.500 | 0.531 | 0.453 | 0.468 | 0.547 |

0.563 | 0.516 | 0.531 | 0.547 | 0.469 | 0.422 | 0.531 | 0.531 |

0.547 | 0.563 | 0.438 | 0.578 | 0.516 | 0.516 | 0.516 | 0.500 |

Boolean Function | b_{1} | b_{2} | b_{3} | b_{4} | b_{5} | b_{6} | b_{7} | b_{8} |
---|---|---|---|---|---|---|---|---|

Nonlinearity | 106 | 104 | 106 | 108 | 108 | 106 | 108 | 108 |

S-box Method | Minimum | Maximum | Average |
---|---|---|---|

[17] | 98 | 108 | 102.5 |

[28] | 96 | 110 | 104.3 |

[30] | 102 | 108 | 105.3 |

[38] | 102 | 108 | 105.3 |

[43] | 102 | 108 | 104.5 |

[44] | 104 | 110 | 106 |

[48] | 98 | 108 | 104 |

[54] | 98 | 108 | 104 |

[55] | 102 | 106 | 104 |

[56] | 102 | 108 | 105.3 |

[57] | 100 | 110 | 105.5 |

[58] | 104 | 106 | 105.3 |

[59] | 100 | 108 | 105.7 |

[60] | 100 | 108 | 104.8 |

[61] | 94 | 104 | 99.5 |

[62] | 96 | 108 | 103.5 |

[63] | 100 | 106 | 103.3 |

[64] | 84 | 106 | 100 |

[65] | 100 | 108 | 104.5 |

Proposed | 104 | 108 | 106.8 |

Boolean Function | b_{1} | b_{2} | b_{3} | b_{4} | b_{5} | b_{6} | b_{7} | b_{8} |
---|---|---|---|---|---|---|---|---|

b_{1} | - | 104 | 106 | 106 | 104 | 104 | 102 | 102 |

b_{2} | 104 | - | 104 | 102 | 108 | 104 | 104 | 100 |

b_{3} | 106 | 104 | - | 104 | 102 | 104 | 108 | 106 |

b_{4} | 106 | 102 | 104 | - | 106 | 106 | 100 | 102 |

b_{5} | 104 | 108 | 102 | 106 | - | 108 | 106 | 100 |

b_{6} | 104 | 104 | 104 | 106 | 108 | - | 98 | 106 |

b_{7} | 102 | 104 | 108 | 100 | 106 | 98 | - | 104 |

b_{8} | 102 | 100 | 106 | 102 | 100 | 106 | 104 | - |

Boolean Function | b_{1} | b_{2} | b_{3} | b_{4} | b_{5} | b_{6} | b_{7} | b_{8} |
---|---|---|---|---|---|---|---|---|

b_{1} | - | 0.502 | 0.510 | 0.506 | 0.500 | 0.504 | 0.484 | 0.477 |

b_{2} | 0.502 | - | 0.512 | 0.479 | 0.510 | 0.488 | 0.512 | 0.518 |

b_{3} | 0.510 | 0.512 | - | 0.479 | 0.520 | 0.492 | 0.461 | 0.500 |

b_{4} | 0.506 | 0.479 | 0.479 | - | 0.504 | 0.518 | 0.520 | 0.467 |

b_{5} | 0.500 | 0.510 | 0.520 | 0.504 | - | 0.521 | 0.498 | 0.510 |

b_{6} | 0.504 | 0.488 | 0.492 | 0.518 | 0.521 | - | 0.488 | 0.512 |

b_{7} | 0.484 | 0.512 | 0.461 | 0.520 | 0.498 | 0.488 | - | 0.504 |

b_{8} | 0.477 | 0.518 | 0.500 | 0.467 | 0.510 | 0.512 | 0.504 | - |

S-box Method | Nonlinearity Min. Max. Average | SAC | BIC-NL | LP | DP | ||
---|---|---|---|---|---|---|---|

[17] | 98 | 108 | 102.5 | 0.492 | 103.3 | 0.141 | 0.062 |

[28] | 96 | 110 | 104.3 | 0.497 | 103.4 | 0.133 | 0.047 |

[30] | 102 | 108 | 105.3 | 0.491 | 103.6 | 0.133 | 0.039 |

[38] | 102 | 108 | 105.3 | 0.496 | 103.8 | 0.156 | 0.039 |

[43] | 102 | 108 | 104.5 | 0.498 | 104.6 | 0.125 | 0.047 |

[44] | 104 | 110 | 106 | 0.520 | 104.2 | 0.132 | 0.039 |

[48] | 98 | 108 | 104 | 0.505 | 103.4 | 0.133 | 0.250 |

[54] | 98 | 108 | 104 | 0.507 | 102.9 | 0.086 | 0.047 |

[55] | 102 | 106 | 104 | 0.498 | 102.9 | 0.148 | 0.039 |

[56] | 102 | 108 | 105.3 | 0.502 | 103.7 | 0.125 | 0.047 |

[57] | 100 | 110 | 105.5 | 0.499 | 106 | 0.133 | 0.125 |

[58] | 104 | 106 | 105.3 | 0.504 | 104.6 | 0.133 | 0.039 |

[59] | 100 | 108 | 105.7 | 0.498 | 104.3 | 0.109 | 0.047 |

[60] | 100 | 108 | 104.8 | 0.501 | 105.1 | 0.125 | 0.125 |

[61] | 94 | 104 | 99.5 | 0.516 | 101.7 | 0.132 | 0.281 |

[62] | 96 | 108 | 103.5 | 0.494 | 103.6 | 0.152 | 0.039 |

[63] | 100 | 106 | 103.3 | 0.505 | 103.7 | 0.133 | 0.039 |

[64] | 84 | 106 | 100 | 0.481 | 101.9 | 0.180 | 0.063 |

[65] | 100 | 108 | 104.5 | 0.498 | 103.6 | 0.141 | 0.047 |

Proposed | 104 | 108 | 106.8 | 0.507 | 103.9 | 0.140 | 0.054 |

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**MDPI and ACS Style**

Zahid, A.H.; Arshad, M.J.
An Innovative Design of Substitution-Boxes Using Cubic Polynomial Mapping. *Symmetry* **2019**, *11*, 437.
https://doi.org/10.3390/sym11030437

**AMA Style**

Zahid AH, Arshad MJ.
An Innovative Design of Substitution-Boxes Using Cubic Polynomial Mapping. *Symmetry*. 2019; 11(3):437.
https://doi.org/10.3390/sym11030437

**Chicago/Turabian Style**

Zahid, Amjad Hussain, and Muhammad Junaid Arshad.
2019. "An Innovative Design of Substitution-Boxes Using Cubic Polynomial Mapping" *Symmetry* 11, no. 3: 437.
https://doi.org/10.3390/sym11030437