# Sustainable Network by Enhancing Attribute-Based Selection Mechanism Using Lagrange Interpolation

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## Abstract

**:**

## 1. Introduction

## 2. Related Work

## 3. Building Blocks of ABSM

**Definition**

**1:**

_{n}are already secured; hence, validation is required for the request of other relative networks O

_{n}. The demanding node d

_{n}from the O

_{n}requests the sender S

_{n}of C

_{n}. To validate d

_{n}, S

_{n}builds a tree structure by applying the Zig-Zag rule. The S

_{n}takes the d

_{n}as the first node. S

_{n}identifies the nearby nodes of d

_{n}by applying the distance formula.

_{n}, S

_{n}will follow the Right-Left mechanism. The nearest right node (R

_{n}) of the d

_{n}followed by the nearest node of R

_{n}will be the first consideration. The S

_{n}will ask for the share.

_{1}, y

_{1}= f (x

_{1})), (x

_{2}, y

_{2}= (x

_{2})), …, (x

_{n}, y

_{n}= f(x

_{n})), which is given by Lagrange polynomial as

_{k}match the network key. Second, security at the RSU level is added, making the network more secure. The pseudo-code is also supplied to help demonstrate how this security is constructed.In Figure 1a, d

_{n}is requesting for the shared key from S

_{n}and in Figure 1b, a tree type structure is formed to transfer the shared key from left right nodes.

Algorithm 1 Share Verification Pseudo Code |

(1) Order = 2; // order of Interpolation |

(2) My_{VALUE} = [ ] // Empty value be initialized |

(3) For x = 1: 3 // For 3 nodes |

token = 1; |

Current_{1} = Node_{ID}I // Using the first node as a starting point |

(4) for y=1: Nodes; |

Current = Vehicles_{j}; // There would be two Rest Nodes for each interpolation. |

if Current_{1}~ = Current // If nodes are not equal |

Rest(token) = current; |

Token = token + 1; |

End If |

(5) End |

(6) For Deno = 0 |

(7) Deno = Current_{1} − Rest_{1} ∗ Current_{1} − Rest_{2} // The denominator value |

(8) Num = Rest_{1} ∗ Rest_{2} |

(9) My_{value}[x] = Num/ Deno |

(10) Shared_{key} = Share_{Current1} * My_{value}[x] |

(11) End For |

## 4. Curve Fitting

#### 4.1. Fourier Series Fitting (FST)

_{0}/2 + Σ∞ n = 1 an cos nΠa/l + Σ∞ n = 1 bn sin nΠa/l

_{0}/2, while the others are known as periodic terms. The finite summation is used when the Fourier series is applied to a given function. An equation is given below to calculate the partial sum.

#### 4.1.1. Computation Complexity

#### 4.1.2. Randomness

#### 4.2. Moving Average (Simple)

^{n − 1}I = 0 dm − I

#### 4.2.1. Computation Intricacy

#### 4.2.2. Randomness

#### 4.2.3. Setup

Algorithm 2 Communication range prediction (Nodes, m, n, Nodeid) |

CommunicationPrediction = [] |

For node 1 in environment |

For node 2 in environment |

When node 𝜕 = node 1 |

D = √ [(node) − M(node1)]2 + [N(node) − N(node1)]2 |

Predict the coverage (node, node1) = Node_{i}(node1) |

End_For |

End_For |

End Algorithm 1 |

#### 4.2.4. Route Discovery

Algorithm 3 Route Dicovery |

Input: Source at Transmission and Destination at Receiving end |

Send Message = Send (‘Hi’) |

For each responder of Send Message |

Compute Requirements of the route = (); |

Choose Node (Hi); |

If reply comes back then |

Add Route |

EndIF |

Repeat the step until destination dose not found |

EndFor |

End Algorithm |

## 5. Results and Discussion

**Throughput:**Throughput is known as the ratio of total packets received at the destination end per unit time. In mathematical terms, it is written as follows:$$\mathrm{Throughput}=\frac{\mathrm{Total}\mathrm{packets}\mathrm{at}\mathrm{the}\mathrm{destination}}{\mathrm{time}}$$**PDR:**defined as the ratio of the total number of packets received from targets to packets generated by source nodes on the transmission side, the PDR is calculated using the following mathematical formula:$$\mathrm{PDR}=\frac{\mathrm{Data}\mathrm{packet}\mathrm{received}\mathrm{from}\mathrm{the}\mathrm{targets}}{\mathrm{Data}\mathrm{packet}\mathrm{from}\mathrm{the}\mathrm{source}\mathrm{end}}$$

## 6. Limitation and Future Directions

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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Sr. No. | References | Technique Used |
---|---|---|

1 | [14] | RSS |

2 | [15] | RSS |

3 | [16] | two-level execution using CIR and RSS |

4 | [17] | Blocks and a two-level quantifier |

5 | [18] | Key generation ranking mechanisms and interpolation |

6 | [19] | Technique based on the KLT (Karhunen-Loeve Transform) |

7 | [20] | CSI (channel state information) |

8 | [21] | Angle-of-arrival for signature key extraction |

9 | [22] | Chaotic signals |

Number of Iterations | Throughput without LI | Throughput with LI |
---|---|---|

50 | 11,000 | 22,000 |

100 | 11,500 | 22,500 |

150 | 12,000 | 23,000 |

200 | 12,500 | 23,500 |

250 | 13,000 | 23,800 |

Total Number of Iterations | PDR without LI | PDR with LI |
---|---|---|

50 | 0.34 | 0.71 |

100 | 0.36 | 0.72 |

150 | 0.38 | 0.74 |

200 | 0.41 | 0.76 |

250 | 0.4 | 0.79 |

Total Number of Simulations | Noise without LI | Noise with Proposed Methodology Incorporating LI |
---|---|---|

50 | 0.26 | 0.16 |

100 | 0.25 | 0.22 |

150 | 0.31 | 0.24 |

200 | 0.36 | 0.26 |

250 | 0.39 | 0.27 |

300 | 0.4 | 0.27 |

350 | 0.41 | 0.23 |

400 | 0.42 | 0.21 |

450 | 0.42 | 0.21 |

Iterations | Delay without LI (Seconds) | Delay with LI (Seconds) |
---|---|---|

50 | 0.42 | 0.34 |

100 | 0.43 | 0.33 |

150 | 0.44 | 0.33 |

200 | 0.55 | 0.32 |

250 | 0.44 | 0.31 |

300 | 0.5 | 0.3 |

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**MDPI and ACS Style**

Monga, C.; Gupta, D.; Prasad, D.; Juneja, S.; Muhammad, G.; Ali, Z.
Sustainable Network by Enhancing Attribute-Based Selection Mechanism Using Lagrange Interpolation. *Sustainability* **2022**, *14*, 6082.
https://doi.org/10.3390/su14106082

**AMA Style**

Monga C, Gupta D, Prasad D, Juneja S, Muhammad G, Ali Z.
Sustainable Network by Enhancing Attribute-Based Selection Mechanism Using Lagrange Interpolation. *Sustainability*. 2022; 14(10):6082.
https://doi.org/10.3390/su14106082

**Chicago/Turabian Style**

Monga, Chetna, Deepali Gupta, Devendra Prasad, Sapna Juneja, Ghulam Muhammad, and Zulfiqar Ali.
2022. "Sustainable Network by Enhancing Attribute-Based Selection Mechanism Using Lagrange Interpolation" *Sustainability* 14, no. 10: 6082.
https://doi.org/10.3390/su14106082