# Novel Authentication Protocols Based on Quadratic Diophantine Equations

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

- A protocol must constitute at least two clients, and everyone associated must be aware of the protocol ahead of time.
- The protocol must be followed by all the concerned clients.
- The steps involved in the protocol must be well defined.
- A protocol must be comprehensive, containing instructions for every possible scenario.

## 2. Mathematical Properties of the Equation ${\mathit{u}}_{\mathbf{1}}^{\mathbf{2}}+{\mathit{u}}_{\mathbf{2}}^{\mathbf{2}}={\mathit{v}}_{\mathbf{1}}^{\mathbf{2}}+{\mathit{v}}_{\mathbf{2}}^{\mathbf{2}}$

**Theorem**

**1.**

**Proof.**

**Note**

**1.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

**Note**

**2.**

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

**Example**

**4.**

**Theorem**

**5.**

**Proof.**

## 3. Existence of Trapdoor Function in Diophantine Cryptography

Algorithm 1 Pseudocode for the authentication protocol |

Input: B, the trapdoor message in a binary form. |

Input: ${x}_{1},{y}_{1}>0$. |

Output: decimal $=0$, base $=1$ |

while
$B\ne 0$ do |

remainder = B% 10 |

decimal = decimal + remainder * base |

B = B / 10 |

base = base * 2 |

end while |

Output: n = decimal |

${x}_{2}\leftarrow n*({x}_{1}+{y}_{1})$; |

${y}_{2}\leftarrow sqrt\{{x}_{1}^{2}+{x}_{2}^{2}-{y}_{1}^{2}\}$; |

$sum1\leftarrow {x}_{1}^{2}+{x}_{2}^{2}$; |

$sum2\leftarrow {y}_{1}^{2}+{y}_{2}^{2}$; |

if
$sum1==sum2$ then |

Trapdoor is possible using Diophantine equation; |

else |

No solution |

end if |

## 4. Cost and Effort Evaluation Using COCOMO Equations

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Table 1.**Coefficients and exponent values as in [25].

Software Project | $\mathit{\alpha}$ | $\mathit{\beta}$ | $\mathit{\gamma}$ | $\mathit{\delta}$ |
---|---|---|---|---|

Organic | 2.4 | 1.05 | 2.5 | 0.38 |

Semi-detached | 3.0 | 1.12 | 2.5 | 0.35 |

Embedded | 3.6 | 1.20 | 2.5 | 0.32 |

Code Length (Lines in Thousands) | Effort | Time Period | Manpower |
---|---|---|---|

50 | 0.10 | 1.04 | 0.09 |

150 | 0.33 | 1.64 | 0.20 |

550 | 1.28 | 2.75 | 0.47 |

1200 | 2.91 | 3.75 | 0.78 |

1400 | 3.42 | 3.99 | 0.86 |

2000 | 4.96 | 4.59 | 1.08 |

5000 | 13.00 | 6.62 | 1.96 |

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**MDPI and ACS Style**

Vijayarangan, A.; Narayanan, V.; Natarajan, V.; Raghavendran, S. Novel Authentication Protocols Based on Quadratic Diophantine Equations. *Mathematics* **2022**, *10*, 3136.
https://doi.org/10.3390/math10173136

**AMA Style**

Vijayarangan A, Narayanan V, Natarajan V, Raghavendran S. Novel Authentication Protocols Based on Quadratic Diophantine Equations. *Mathematics*. 2022; 10(17):3136.
https://doi.org/10.3390/math10173136

**Chicago/Turabian Style**

Vijayarangan, Avinash, Veena Narayanan, Vijayarangan Natarajan, and Srikanth Raghavendran. 2022. "Novel Authentication Protocols Based on Quadratic Diophantine Equations" *Mathematics* 10, no. 17: 3136.
https://doi.org/10.3390/math10173136