# S-Box on Subgroup of Galois Field

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## Abstract

**:**

## 1. Introduction

## 2. Construction of S-box on Subgroup of the Galois Field

**S-1**Define an inversion function $p:{K}_{15}\cup \left\{0\right\}\to {K}_{15}\cup \left\{0\right\}$ by$$p\left(y\right)=\{\begin{array}{c}{y}^{-1},ify\in {K}_{15}\\ 0,ify=0\end{array}$$**S-2**Define a linear scalar multiple function $q:{K}_{15}\cup \left\{0\right\}\to {K}_{15}\cup \left\{0\right\}$ by$$q\left(y\right)=uy,\forall y\in {K}_{15}\cup \left\{0\right\},u\in {K}_{15}\mathrm{is}\text{}\mathrm{fixed}.$$**S-3**Take the composition of $p$ and $q$ and get an $8\times 8$-bit S-box.

## 3. Analyses

#### 3.1. Balance Property

#### 3.2. Nonlinearity Analysis

#### 3.3. Strict Avalanche Criterion

#### 3.4. Linear Approximation Probability Analysis

#### 3.5. Differential Approximation Probability Analysis

#### 3.6. Majority Logic Criterion

- Use LSB’s of the input pixel of the image to select an 8-bit S-box value.
- LSB’s of the S-box value become MSB’s of the output pixel and MSB’s of the input pixel become LSB’s of the output pixel.

## 4. Application of Proposed S-box in Image Watermarking

#### Experimental Results and Discussion

## 5. Conclusions and Future Work

## Author Contributions

## Funding

## Conflicts of Interest

## References

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$\mathit{y}\in {\mathit{K}}_{15}\cup \left\{0\right\}$ | $\mathit{q}\left(\mathit{y}\right)=\mathit{u}\mathit{y}$ $\left(\mathit{u}=10011000\right)$ | $\mathit{p}\left(\mathit{q}\left(\mathit{y}\right)\right)={\left(\mathit{u}\mathit{y}\right)}^{-1}$ |
---|---|---|

00000000 | 00000000 | 00000000 |

10011000 | 01001110 | 01000101 |

01001110 | 00001010 | 11011101 |

00001010 | 10011001 | 11011100 |

10011001 | 11010110 | 11010111 |

11010110 | 01000100 | 10010010 |

01000100 | 10010011 | 01001111 |

10010011 | 01001111 | 10010011 |

01001111 | 10010010 | 01000100 |

10010010 | 11010111 | 11010110 |

11010111 | 11011100 | 10011001 |

11011100 | 11011101 | 00001010 |

11011101 | 01000101 | 01001110 |

01000101 | 00001011 | 10011000 |

00001011 | 00000001 | 00000001 |

00000001 | 10011000 | 00001011 |

LSB’s | 00 | 01 | 10 | 11 |
---|---|---|---|---|

00 | 00000000 | 00001011 | 11010110 | 10010011 |

01 | 01001111 | 10011000 | 10010010 | 10011001 |

10 | 01000101 | 11010111 | 11011100 | 00000001 |

11 | 00001010 | 01001110 | 11011101 | 01000100 |

LSB’s | 0 | 1 | 2 | 3 |
---|---|---|---|---|

0 | 0 | 11 | 214 | 147 |

1 | 79 | 152 | 146 | 153 |

2 | 69 | 215 | 220 | 1 |

3 | 10 | 78 | 221 | 68 |

Boolean function | ${g}_{7}$ | ${g}_{6}$ | ${g}_{5}$ | ${g}_{4}$ | ${g}_{3}$ | ${g}_{2}$ | ${g}_{1}$ | ${g}_{0}$ | Average |

Nonlinearity | 4 | 4 | 0 | 4 | 4 | 4 | 4 | 4 | 3.5 |

Boolean function | ${g}_{7}$ | ${g}_{6}$ | ${g}_{5}$ | ${g}_{4}$ | ${g}_{3}$ | ${g}_{2}$ | ${g}_{1}$ | ${g}_{0}$ | Average |

SAC | 0.5 | 0.5 | 0 | 0.5 | 0.5 | 0 | 0.5 | 0.75 | 0.4688 |

0 | 1 | 2 | 3 |
---|---|---|---|

--- | 0.25 | 0.25 | 0.25 |

0.25 | 0.25 | 0.25 | 0.25 |

0.25 | 0.25 | 0.25 | 0.25 |

0.25 | 0.25 | 0.25 | 0.25 |

Attribute | Ref. [14] | Proposed S-Box | |||
---|---|---|---|---|---|

Lena | Baboon | Pepper | Airplane | ||

Contrast | 3.3220 | 10.4474 | 10.5164 | 10.6092 | 9.8911 |

Correlation | 0.0879 | 0.0127 | −0.0015 | 0.0004 | 0.0664 |

Energy | 0.0244 | 0.0159 | 0.0156 | 0.0157 | 0.0172 |

Entropy | 4.7301 | 7.4451 | 7.3583 | 7.5937 | 6.7025 |

Homogeneity | 0.4835 | 0.4045 | 0.3900 | 0.3949 | 0.4376 |

MAD | 36.3631 | 32.3756 | 31.8342 | 32.4054 | 32.2973 |

Grayscale-Images/Analysis | Proposed S-Box | Ref. [14] | |||
---|---|---|---|---|---|

Airplane | Baboon | Lena | Pepper | ||

MSE | 16.0383 | 15.9333 | 16.2827 | 15.8937 | 11.7755 |

PSNR | 83.1537 | 83.2194 | 83.0025 | 83.2443 | 86.1651 |

SSIM | 0.8280 | 0.9317 | 0.8198 | 0.8250 | 0.9145 |

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**MDPI and ACS Style**

Shah, T.; Qureshi, A.
S-Box on Subgroup of Galois Field. *Cryptography* **2019**, *3*, 13.
https://doi.org/10.3390/cryptography3020013

**AMA Style**

Shah T, Qureshi A.
S-Box on Subgroup of Galois Field. *Cryptography*. 2019; 3(2):13.
https://doi.org/10.3390/cryptography3020013

**Chicago/Turabian Style**

Shah, Tariq, and Ayesha Qureshi.
2019. "S-Box on Subgroup of Galois Field" *Cryptography* 3, no. 2: 13.
https://doi.org/10.3390/cryptography3020013