# S-box Construction Based on Linear Fractional Transformation and Permutation Function

^{1}

^{2}

^{*}

## Abstract

**:**

^{8}). Finally, the produced matrix is permuted to add randomness to the S-box. The strength of the S-box is measured by calculating its potency to create confusion. To analyze the security properties of the S-box, some well-known and commonly used algebraic attacks are used. The proposed S-box is analyzed by nonlinearity test, algebraic degree, differential uniformity, and strict avalanche criterion which are the avalanche effect test, completeness test, and strong S-box test. S-box analysis is done before and after the application of the permutation function and the analysis result shows that the S-box with permutation function has reached the optimal properties as a secure S-box.

## 1. Introduction

## 2. Related Work

## 3. Review on S-Box Properties and Analysis

#### 3.1. Nonlinearity

#### 3.2. Algebraic Degree

#### 3.3. Differential Uniformity

#### 3.4. Strict Avalanche Criterion

#### 3.4.1. Avalanche Effect

#### 3.4.2. Completeness

#### 3.4.3. Strong SBox

## 4. Linear Fractional Transformation and Permutation Function

#### 4.1. Linear Fraction Transform

#### 4.2. Permutation Function

## 5. Constructions of S-Box

## 6. Results and Discussion

#### 6.1. Nonlinearity Test

#### 6.2. Algebraic Degree

#### 6.3. Differential Uniformity

#### 6.4. Strict Avalanche Criterion

#### 6.4.1. Avalanche Effect

#### 6.4.2. Completeness

#### 6.4.3. Strong S-Box

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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1 | ${x}^{8}+{x}^{7}+{x}^{6}+{x}^{5}+{x}^{4}+{x}^{2}+1$ |

2 | ${x}^{8}+{x}^{7}+{x}^{6}+{x}^{5}+{x}^{2}+x+1$ |

3 | ${x}^{8}+{x}^{7}+{x}^{6}+{x}^{3}+{x}^{2}+x+1$ |

4 | ${x}^{8}+{x}^{7}+{x}^{6}+{x}^{3}+{x}^{2}+x+1$ |

5 | ${x}^{8}+{x}^{7}+{x}^{5}+{x}^{3}+1$ |

6 | ${x}^{8}+{x}^{7}+{x}^{3}+{x}^{2}+1$ |

7 | ${x}^{8}+{x}^{7}+{x}^{2}+x+1$ |

8 | ${x}^{8}+{x}^{6}+{x}^{5}+{x}^{4}+1$ |

9 | ${x}^{8}+{x}^{6}+{x}^{5}+{x}^{3}+1$ |

10 | ${x}^{8}+{x}^{6}+{x}^{5}+{x}^{2}+1$ |

11 | ${x}^{8}+{x}^{6}+{x}^{5}+x+1$ |

12 | ${x}^{8}+{x}^{6}+{x}^{4}+{x}^{3}+{x}^{2}+x+1$ |

13 | ${x}^{8}+{x}^{6}+{x}^{3}+{x}^{2}+1$ |

14 | ${x}^{8}+{x}^{5}+{x}^{3}+{x}^{2}+1$ |

15 | ${x}^{8}+{x}^{5}+{x}^{3}+x+1$ |

16 | $\begin{array}{c}{x}^{8}+{x}^{4}+{x}^{3}+{x}^{2}+1\end{array}$ |

$\mathbf{G}\mathbf{F}\left({2}^{8}\right)$ | f(z) = (35z + 15)/(9z + 5) | Matrix Elements |
---|---|---|

0 | f(z) = (35(0) + 15)/(9(0) + 5) | 198 |

1 | f(z) = (35(1) + 15)/(9(1) + 5) | 214 |

$\dots $ | $\dots $ | $\dots $ |

$\dots $ | $\dots $ | $\dots $ |

254 | f(z) = (35(254) + 15)/(9(254) + 5) | 6 |

255 | f(z) = (35(255) + 15)/(9(255) + 5) | 76 |

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

0 | C6 | D6 | F1 | A3 | 82 | A5 | D9 | 7F | B3 | 7B | 6F | C5 | 2B | 8D | ED | 03 |

1 | A8 | C9 | 11 | 79 | 8E | 65 | E8 | AE | 0B | F9 | 10 | 9C | 0A | 32 | B7 | 41 |

2 | 48 | B8 | C8 | 84 | 3A | 2F | 1B | 9F | E7 | BD | 08 | 12 | CE | C2 | B1 | 1F |

3 | C1 | 5C | 7A | C0 | 55 | 89 | F3 | 31 | B2 | AA | 24 | 87 | E6 | 5F | 64 | 80 |

4 | 0D | 6D | E3 | 00 | E0 | 90 | D0 | 4E | AD | 20 | 8B | EA | 6B | 52 | AC | 51 |

5 | 33 | E9 | 0C | 9A | 5E | A1 | F4 | 37 | 07 | 22 | FB | E1 | 99 | 5D | FE | 8A |

6 | 66 | F0 | 73 | F2 | 6E | 86 | 7C | 4F | 9D | A0 | 5A | EE | 49 | 35 | A9 | FA |

7 | 88 | 76 | 70 | 30 | 28 | 72 | 16 | F6 | 2E | 83 | 17 | 45 | 34 | EB | F8 | 02 |

8 | 74 | 5B | 75 | 1A | A6 | 19 | DB | 3B | 36 | E5 | 78 | F5 | 59 | B9 | 63 | E2 |

9 | 69 | 2D | 3C | C7 | A4 | BF | E4 | CA | 25 | 68 | 8F | D1 | DC | 93 | 2C | BA |

10 | 91 | 7D | CB | 1D | 26 | 29 | D7 | 6C | 40 | 58 | 77 | 4A | D5 | 60 | D3 | 53 |

11 | DA | 92 | C4 | CD | 43 | 98 | 81 | AF | 54 | 9E | CF | B0 | 50 | 3E | 96 | 56 |

12 | 39 | 9B | C3 | D8 | 4B | 13 | 01 | 57 | 21 | 44 | 47 | EC | EF | FF | 23 | D4 |

13 | 94 | BC | 85 | 0F | CC | BB | 2A | B6 | 61 | 38 | 18 | DD | FC | 1E | 4D | B5 |

14 | 04 | F7 | A7 | 15 | 09 | DE | B4 | BE | 97 | 8C | 27 | AB | 0E | 7E | 42 | FD |

15 | 67 | DF | 46 | 62 | 1C | 14 | 3F | A2 | 3D | 71 | 95 | D2 | 6A | 05 | 06 | 4C |

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

0 | 212 | 139 | 143 | 19 | 31 | 230 | 50 | 172 | 191 | 39 | 174 | 86 | 184 | 156 | 224 | 109 |

1 | 209 | 185 | 176 | 171 | 61 | 23 | 60 | 128 | 220 | 18 | 252 | 85 | 81 | 186 | 237 | 22 |

2 | 98 | 204 | 9 | 102 | 182 | 46 | 126 | 54 | 119 | 248 | 107 | 233 | 4 | 47 | 197 | 190 |

3 | 131 | 17 | 232 | 254 | 6 | 65 | 201 | 243 | 132 | 150 | 30 | 16 | 26 | 3 | 137 | 69 |

4 | 63 | 92 | 111 | 7 | 112 | 68 | 236 | 21 | 40 | 78 | 250 | 104 | 219 | 89 | 0 | 161 |

5 | 113 | 120 | 148 | 251 | 66 | 169 | 175 | 12 | 216 | 145 | 10 | 165 | 214 | 181 | 179 | 189 |

6 | 64 | 166 | 15 | 207 | 75 | 117 | 247 | 215 | 14 | 79 | 44 | 52 | 33 | 108 | 228 | 8 |

7 | 25 | 1 | 115 | 70 | 173 | 123 | 100 | 13 | 211 | 133 | 155 | 67 | 56 | 57 | 223 | 5 |

8 | 127 | 35 | 103 | 29 | 2 | 141 | 180 | 142 | 183 | 87 | 217 | 195 | 151 | 196 | 213 | 125 |

9 | 135 | 110 | 205 | 203 | 229 | 97 | 129 | 27 | 194 | 114 | 208 | 249 | 76 | 59 | 177 | 42 |

10 | 93 | 37 | 225 | 82 | 147 | 168 | 88 | 222 | 124 | 239 | 11 | 48 | 136 | 20 | 178 | 28 |

11 | 122 | 210 | 245 | 158 | 235 | 241 | 91 | 55 | 118 | 72 | 41 | 43 | 188 | 130 | 200 | 53 |

12 | 32 | 94 | 246 | 202 | 80 | 164 | 116 | 221 | 193 | 101 | 198 | 121 | 146 | 162 | 74 | 34 |

13 | 152 | 255 | 140 | 159 | 167 | 62 | 73 | 106 | 24 | 160 | 163 | 138 | 51 | 105 | 71 | 99 |

14 | 226 | 58 | 84 | 192 | 238 | 234 | 170 | 154 | 244 | 242 | 206 | 49 | 240 | 199 | 90 | 231 |

15 | 157 | 96 | 77 | 253 | 218 | 83 | 134 | 149 | 187 | 45 | 227 | 144 | 153 | 95 | 36 | 38 |

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

0 | AC | 76 | A6 | 5F | CE | 02 | 55 | 00 | FA | C8 | FB | 77 | 37 | F6 | 9D | 73 |

1 | 87 | 5C | F9 | A3 | 60 | 4E | 41 | 65 | 61 | 88 | E6 | CA | 53 | 1A | 24 | 82 |

2 | 39 | 49 | D4 | 5B | 06 | 7D | 4C | 7B | AD | CF | BA | B0 | 5A | 71 | 2F | C2 |

3 | 4A | AB | D9 | FC | EE | 56 | 9F | AF | 34 | EB | F7 | 93 | E8 | 8E | BB | 0D |

4 | 66 | 89 | 5E | 45 | 90 | 80 | 30 | 4D | 9E | 2A | 23 | 6E | DC | 46 | 20 | A0 |

5 | 4B | 0A | 1D | 14 | A7 | 9C | C5 | E5 | D7 | 52 | 42 | 81 | 6D | 91 | 9B | 05 |

6 | DF | BF | 48 | B5 | 16 | 44 | 84 | 75 | EA | 1E | B6 | 08 | 35 | 03 | 2D | E3 |

7 | E0 | 33 | 68 | 70 | 01 | 86 | 54 | E7 | E9 | EC | DA | 72 | 40 | E2 | 1B | 74 |

8 | AE | E4 | 3E | C1 | B2 | 83 | 3F | 69 | D5 | 64 | DD | D6 | 85 | 19 | 3B | F1 |

9 | D2 | 22 | EF | 26 | 0C | A2 | AA | 59 | 94 | 6A | BE | 17 | 8D | 67 | CD | 0F |

10 | 38 | 51 | FF | 18 | 13 | E1 | F0 | CC | 29 | A1 | B4 | 79 | 7F | 28 | 6F | F4 |

11 | 11 | 2C | D3 | FE | DB | 5D | 3A | 36 | 2B | C9 | 32 | 3D | 50 | 8A | 1F | B3 |

12 | 15 | 21 | 25 | F5 | B9 | B1 | 47 | 7E | 96 | F3 | D8 | C7 | B8 | 3C | 27 | F2 |

13 | 8F | A8 | 92 | 2E | C6 | 63 | 99 | 4F | 07 | 78 | 1C | 6B | 0B | 57 | 6C | F8 |

14 | ED | CB | 04 | 95 | A9 | A4 | A5 | FD | 7A | 12 | DE | 43 | D0 | B7 | 09 | 58 |

15 | 0E | 98 | 8C | 31 | 97 | C4 | C3 | 7C | BD | D1 | 8B | 9A | 10 | 62 | C0 | BC |

S-Box before Permutation Function | S-box after Permutation Function | |
---|---|---|

Nonlinearity Test | 95 | 112 |

Algebraic degree | 7 | 7 |

Difference uniformity | 8 | 4 |

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |
---|---|---|---|---|---|---|---|---|

Maximum entry (before permutation function) | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 |

Maximum entry (after permutation function) | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |

S-Box before Permutation Function | S-Box after Permutation Function | |
---|---|---|

Avalanche effect | Normal | Normal |

Completeness | Nonuniform | Uniform |

Strong S-box | Nonuniform | Uniform |

Nonlinearity Test | Algebraic Degree | Difference Uniformity | |
---|---|---|---|

Proposed S-box | 112 | 7 | 4 |

AES S-box [31] | 112 | 7 | 4 |

Camellia S-box 1 [32] | 112 | 7 | 4 |

Camellia S-box 2 [32] | 112 | 7 | 4 |

Camellia S-box 3 [32] | 112 | 7 | 4 |

Camellia S-box 4 [32] | 112 | 7 | 4 |

Hierocrypt-Higher Level S-box [33] | 112 | 7 | 4 |

Cui Jie et al. S-box [13] | 112 | 7 | 4 |

APA S-box [3] | 112 | 7 | 4 |

ARIA [34] | 112 | 7 | 4 |

HyRAL [35] | 112 | 7 | 4 |

Hussain et al. S-box [28] | 112 | 7 | 4 |

Yang et al. S-box 1 [21] | 114 | 7 | 4 |

Yang et al. S-box 2 [21] | 110 | 7 | 4 |

Yang et al. S-box 3 [21] | 112 | 7 | 6 |

Yang et al. S-box 4 [21] | 110 | 7 | 6 |

Isa et al. S-box [15] | 108 | 7 | 4 |

Hierocrypt-Lower Level S-box [33] | 106 | 7 | 6 |

Mamadolimov et al. S-box [14] | 102 | 7 | 6 |

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## Share and Cite

**MDPI and ACS Style**

Nizam Chew, L.C.; Ismail, E.S.
S-box Construction Based on Linear Fractional Transformation and Permutation Function. *Symmetry* **2020**, *12*, 826.
https://doi.org/10.3390/sym12050826

**AMA Style**

Nizam Chew LC, Ismail ES.
S-box Construction Based on Linear Fractional Transformation and Permutation Function. *Symmetry*. 2020; 12(5):826.
https://doi.org/10.3390/sym12050826

**Chicago/Turabian Style**

Nizam Chew, Liyana Chew, and Eddie Shahril Ismail.
2020. "S-box Construction Based on Linear Fractional Transformation and Permutation Function" *Symmetry* 12, no. 5: 826.
https://doi.org/10.3390/sym12050826