# Chebyshev Polynomial-Based Fog Computing Scheme Supporting Pseudonym Revocation for 5G-Enabled Vehicular Networks

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## Abstract

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## 1. Introduction

- We present a fog computing technique for 5G-enabled vehicle networks, which is based on the Chebyshev polynomial and allows for the revocation of pseudonyms.
- We adapt fog computing in the proposed scheme to generate security parameters and check the validity of vehicles.
- We adapt the 5G technology to increase the communication range among vehicles and the system, which avoid the expensive RSU used.
- In order to stop an insider attack, we use fog computing. As soon as the fog server detects that a pseudonym-timestamp ID is about to expire, it will not renew the signature key.
- Our solutions not only pass all privacy and security tests but are also resistant to a wide range of common security assaults.
- In our study, we use the Chebyshev polynomial to sign messages and verify the signature, which results in lower overall performance costs.

## 2. Related Work

## 3. Background

#### 3.1. Architecture

- TA: TA is in charge of providing a valid computation and storage capacity for the main parameter of fog servers and OBUs within its authority. If the system contains incorrect or malicious information, the TA can trace and revoke the pseudonym-ID of the information source. In the vehicular network, all entities hold TA in high regard, and it is impossible to compromise TA.
- 5G-BS: A 5G-BS is a piece of roadway infrastructure that is permanently installed. Through the 5G protocol and secure wired connections, the 5G-BS can communicate with the vehicle’s OBU and the TA within a large range of communication.
- Fog Server: The fog server is in charge of providing the signature key of participating vehicles during joining through 5G-BS. The fog server saves the system’s master key to validate and authenticate the vehicle. We trust the fog server implicitly to help TA reveal the signers’ identities.
- OBU: The vehicle is equipped with an OBU that supports the dedicated short-range communication (DSRC) protocol and 5G protocol. The OBU sends a traffic-related message to the other OBUs or fog servers on a regular basis, informing them of traffic statuses, such as speed, location, and danger warnings.

#### 3.2. Design Goals

- Authentication and Integrity: The authenticity of the owner and validity of the message must be fulfilled to avoid any modification from the attacker.
- Identity Privacy: The true identity of the vehicle should be hidden in pseudonym-ID of the message.
- Traceability: The TA can trace any insider attacker by revealing the true identity of the vehicle.
- Pseudonym Revocation: The TA has the ability to revoke insider attackers to use services.
- Unlinkability: The attacker cannot link several signatures sent by the same owner.
- Resistance to Security Attacks: Our proposal must resist the security attacks, such as replay, forgery, modify, and man-in-the-middle attacks.
- Efficient: Our proposal must lower performances costs in terms of communication and computational overheads.

#### 3.3. Proposed Framework

- System Initialization (Phase 1): In our work, the TA is in charge of setting up the security parameters and registering vehicles and fog servers.
- Vehicle Joining to Fog Server (Phase 2): When a vehicle joins the covered area of the fog server through 5G-BS, it should be validated and authenticated to that fog server to share messages according to the parameters of the fog server. When a vehicle enters the covered area of the new fog server or its pseudonym-ID is expired, it should be joined to the fog server area and obtains the signature key and pseudonym-ID from the fog server.
- Message Signing (Phase 3): Before messages can be transmitted between vehicles on 5G-enabled networks, the linked vehicle must sign them using its signature key.
- Signature Verification (Phase 4): Before proceeding with the message ${M}_{i}$, the receiver-enrolled vehicle must verify the authenticity of the final-tuple {$PI{D}_{{v}_{i}}$, ${M}_{i}$, ${T}_{i}$, ${T}_{1}$, $V{T}_{i}$, ${\delta}_{{m}_{i}}$}.
- Pseudonym Revocation (Phase 5): Prior to the adversary broadcasting counterfeit messages to disrupt the 5G-enabled vehicular networks, we are able to track down the fraudulently registered car and revoke its pseudonym-ID during the pseudonym revocation phase.

#### 3.4. Mathematics Used

- Elliptic Curve: Number theorists place a premium on elliptic curves, and this topic is currently a hotspot for study, for instance, Andrew Wiles’s demonstration of Fermat’s Last Theorem relied on elliptic curves. Elliptic curve cryptography (ECC) and integer factorization are two other areas where they are useful. Some studies [29,30] based on elliptic curve.
- Bilinear Pair: Ingenious systems for one-round three-party key negotiation, identity-based encryption, and aggregate signatures have all been developed using bilinear pairings. For certain selected elliptic curves, the Tate pairing can be used to generate appropriate bilinear pairings. Some studies [31,32,33] based on bilinear pair.

## 4. Proposed Scheme

#### 4.1. System Initialization (Phase 1)

- Let P and ${k}_{S}^{TA},x$ be the large prime and generated values regarding Chebyshev polynomial, respectively.
- TA chooses a random number ${S}_{TA}\in {Z}_{q}^{*}$ as its master secret key.
- Let $h:$${[-0,1]}^{*}\to $${[-0,1]}^{l}$ be one-way hash function according to Chebyshev polynomial.
- Let $\psi =$ {${k}_{S}TA$, x, P, h} be the security parameters.
- TA chooses a random number $e{v}_{i}\in {Z}_{q}^{*}$ as secret parameter.
- Let $A{C}_{{v}_{i}}$ = $h\left({S}_{TA}\right|\left|e{v}_{i}\right)$ be an authentication code.
- TA preloads an authentication code $A{C}_{{v}_{i}}$ and the security parameters $\psi =$ { ${k}_{S}^{TA}$, x, P, h} in each vehicle.
- TA preloads the master key ${S}_{TA}$ and the security parameters $\psi =$ { ${k}_{S}^{TA}$, x, P, h} in each fog server.

#### 4.2. Vehicle Joining to Fog Server (Phase 2)

- The vehicle chooses random number $r\in {Z}_{q}^{*}$ and computes parameter $\alpha ={\mathcal{T}}_{r}\left(x\right)$ mod P and calculates pseudonym-ID $PI{D}_{{v}_{i}}=TI{D}_{{v}_{i}}\oplus h(A{C}_{{v}_{i}}\left|\right|{T}_{1})$, where $TI{D}_{{v}_{i}}$ is a vehicle’s true identity and ${T}_{1}$ is a newness timestamp.
- The vehicle sends the first-tuple $\{\alpha ,PI{D}_{{v}_{i}},{T}_{1}{\delta}_{V-TA}\}$ to TA through 5G-BS by assisting fog server, where ${\delta}_{V-TA}=h\left(\alpha \right||PI{D}_{{v}_{i}}\left|\right|{T}_{1}\left|\right|A{C}_{{v}_{i}})$.
- Upon receiving the first-tuple $\{\alpha ,PI{D}_{{v}_{i}},{T}_{1}{\delta}_{V-TA}\}$, the TA tests the latest timestamp ${T}_{1}$ as Equation (1). If Equation (1) holds, ${T}_{1}$ is freshness. Otherwise, the first-tuple $\{\alpha ,PI{D}_{{v}_{i}},{T}_{1}{\delta}_{V-TA}\}$ is rejected.$${T}_{\u25bf}>{T}_{r}-{T}_{1}$$
- The TA then reveals vehicle’s true identity $TI{D}_{{v}_{i}}$ by computing $TI{D}_{{v}_{i}}=PI{D}_{{v}_{i}}\oplus h(A{C}_{{v}_{i}}\left|\right|{T}_{1})$, where $A{C}_{{v}_{i}}$ is an authentication code.
- Once checking the validity and authenticity of the vehicle’s true identity $TI{D}_{{v}_{i}}$, the TA sends $\{TI{D}_{{v}_{i}},A{C}_{{v}_{i}}\}$ to the fog server via the secure channel.
- The fog server selects the signature key $S{k}_{{v}_{i}}$ as the following Equation.$$S{k}_{{v}_{i}}={\mathcal{T}}_{PI{D}_{{v}_{i}}.{S}_{TA}}\left(x\right)modP$$
- The fog server encrypts the signature key $S{k}_{{v}_{i}}$ as the following Equation.$$S{k}_{{v}_{i}}^{enc}=S{k}_{{v}_{i}}\oplus h(A{C}_{{v}_{i}}\left|\right|{T}_{1})$$
- The fog server sends the second-tuple $\{S{k}_{{v}_{i}}^{enc},V{T}_{i},{T}_{2},{\delta}_{f2v}\}$ to vehicle, where ${\delta}_{f2v}=h(V{T}_{i}\left|\right|S{k}_{{v}_{i}}^{enc}\left|\right|PI{D}_{{v}_{i}}\left|\right|{T}_{2})$ and $V{T}_{i}$ is a valid timestamp obtained from fog server.
- After checking the freshness of timestamp ${T}_{2}$, the vehicle decrypts the signature key $S{k}_{{v}_{i}}$ as the following Equation.$$S{k}_{{v}_{i}}=S{k}_{{v}_{i}}^{enc}\oplus h(A{C}_{{v}_{i}}\left|\right|{T}_{1})$$

#### 4.3. Message Signing (Phase 3)

- The vehicle signs the message ${\sigma}_{{m}_{i}}=h({M}_{i}\left|\right|V{T}_{i}\left|\right|PI{D}_{{v}_{i}}\left|\right|{T}_{i})$, where ${T}_{i}$ is current timestamp.
- A signature of the communication is calculated by the vehicle as ${\delta}_{{m}_{i}}=$${\mathcal{T}}_{{\sigma}_{{m}_{i}}}\left(S{k}_{{v}_{i}}\right)modP$.
- The vehicle then uses V2V communication on 5G-enabled vehicular networks to broadcast the final-tuple {$PI{D}_{{v}_{i}}$, ${M}_{i}$, ${T}_{i}$, ${T}_{1}$, $V{T}_{i}$, ${\delta}_{{m}_{i}}$} to other cars.

#### 4.4. Signature Verification (Phase 4)

- Upon receiving the final-tuple {$PI{D}_{{v}_{i}}$, ${M}_{i}$, ${T}_{i}$, ${T}_{1}$, $V{T}_{i}$, ${\delta}_{{m}_{i}}$}, the checker initially tests the newness of timestamp ${T}_{i}$ as Equation (1).
- The checker then computes the parameter ${\sigma}_{{m}_{i}}^{-}=h({M}_{i}\left|\right|V{T}_{i}\left|\right|PI{D}_{{v}_{i}}\left|\right|{T}_{i})$.
- The checker then uses the signature ${\delta}_{{m}_{i}}$ of the final-tuple {$PI{D}_{{v}_{i}}$, ${M}_{i}$, ${T}_{i}$, ${T}_{1}$, $V{T}_{i}$, ${\delta}_{{m}_{i}}$} to validate the message ${M}_{i}$, where ${\delta}_{{m}_{i}}=$${\mathcal{T}}_{{\sigma}_{{m}_{i}}}\left(S{k}_{{v}_{i}}\right)modP$. The checker accepts the message ${M}_{i}$ when Equation (5) holds. If the message does not meet these requirements, it will be rejected

#### 4.5. Pseudonym Revocation (Phase 5)

- Upon the malicious registered vehicle broadcasting the final-tuple {$PI{D}_{{v}_{i}}$, ${M}_{i}$, ${T}_{i}$, ${T}_{1}$, $V{T}_{i}$, ${\delta}_{{m}_{i}}$}, the fog server computes the vehicle’s true identity $TI{D}_{{v}_{i}}$ as Equation (6).$$TI{D}_{{v}_{i}}=PI{D}_{{v}_{i}}\oplus h(A{C}_{{v}_{i}}\left|\right|{T}_{1})$$
- The fog server then sends $TI{D}_{{v}_{i}}$ to the TA through a secure channel.
- The TA checks the data stored about it in the vehicle registration list and deletes it.
- The TA sends $\{AcknowledgmentMessage\}$ to all fog servers for revoking process.
- Once the valid timestamp $V{T}_{i}$ is close to expiring, the malicious vehicle requests a new signature key from the fog server.
- After checking the $TI{D}_{{v}_{i}}$ in the revocation list, the fog server rejects the request because it is revoked.

## 5. Result

#### 5.1. Security Analysis

- Authentication and Integrity: The checker in our work can verify the legality and integrity of the final-tuple {$PI{D}_{{v}_{i}}$, ${M}_{i}$, ${T}_{i}$, ${T}_{1}$, $V{T}_{i}$, ${\delta}_{{m}_{i}}$} by verifying whether Equation (5) holds. Thus, the legality and integrity of our work are satisfied.
- Identity Privacy: In a 5G-enabled vehicular network, the vehicle’s true identity of $TI{D}_{{v}_{i}}$ is involved in the pseudonym-ID of the final-tuple {$PI{D}_{{v}_{i}}$, ${M}_{i}$, ${T}_{i}$, ${T}_{1}$, $V{T}_{i}$, ${\delta}_{{m}_{i}}$} generated by the vehicle. Hence, no attacker can obtain the true identity $TI{D}_{{v}_{i}}$ of the vehicle through the $TI{D}_{{v}_{i}}$. Thus, our work satisfies the identity privacy requirement.
- Traceability: The true identity of the vehicle $TI{D}_{{v}_{i}}$ is hidden in the $PI{D}_{{v}_{i}}$ generated by the vehicle, where $PI{D}_{{v}_{i}}=TI{D}_{{v}_{i}}\oplus h(A{C}_{{v}_{i}}\left|\right|{T}_{1})$. By using an authentication code $A{C}_{{v}_{i}}$, the TA or fog server computes $TI{D}_{{v}_{i}}$ by calculating $TI{D}_{{v}_{i}}=PI{D}_{{v}_{i}}\oplus h(A{C}_{{v}_{i}}\left|\right|{T}_{1})$. Hence, our work provides a traceability requirement.
- Pseudonym Revocation: Once the process of traceability is done, the TA has the ability to revoke the pseudonym of the vehicle as shown in Section 4.5. Hence, our work provides a pseudonym revocation requirement.
- Unlinkability: The vehicle uses a pseudonym-ID to create the final-tuple {$PI{D}_{{v}_{i}}$, ${M}_{i}$, ${T}_{i}$, ${T}_{1}$, $V{T}_{i}$, ${\delta}_{{m}_{i}}$}. Once the valid timestamp $V{T}_{i}$ is close to expiring, the vehicle creates a new pseudonym-ID to request a new signature for completing the joining process. Our work also utilizes the freshness timestamp ${T}_{i}$ to compute the signature ${\delta}_{{m}_{i}}$. Any attacker who tries to link multiple final-tuples {$PI{D}_{{v}_{i}}$, ${M}_{i}$, ${T}_{i}$, ${T}_{1}$, $V{T}_{i}$, ${\delta}_{{m}_{i}}$} may not succeed because of changes in their pseudonym-ID and timestamp. Nevertheless, no linkability issue arises in our work.
- Resistance to Replay Attacks: The newness timestamp ${T}_{i}$ is included in the final-tuple {$PI{D}_{{v}_{i}}$, ${M}_{i}$, ${T}_{i}$, ${T}_{1}$, $V{T}_{i}$, ${\delta}_{{m}_{i}}$}. Before accepting the message ${M}_{i}$, the verifier checks whether the inequality $({T}_{\u25bf}>{T}_{r}-{T}_{1})$ holds. If it is valid, the verifier accepts the message ${M}_{i}$ to be checked further; otherwise, the message ${M}_{i}$ is rejected. Therefore, the resistance to replay attacks is satisfied in our work.
- Resistance to Forgery Attacks: The attacker cannot forge a valid final-tuple {$PI{D}_{{v}_{i}}$, ${M}_{i}$, ${T}_{i}$, ${T}_{1}$, $V{T}_{i}$, ${\delta}_{{m}_{i}}$} in our work. This is because the checker can verify the authenticity of the final-tuple {$PI{D}_{{v}_{i}}$, ${M}_{i}$, ${T}_{i}$, ${T}_{1}$, $V{T}_{i}$, ${\delta}_{{m}_{i}}$} by computing whether the equation ${\mathcal{T}}_{PI{D}_{{v}_{i}}.{{\sigma}_{{m}_{i}}}^{-}}\left(S{k}_{{v}_{i}}\right)\stackrel{?}{=}{\mathcal{T}}_{PI{D}_{{v}_{i}}}\left({\delta}_{{m}_{i}}\right)$ holds. If it is valid, the checker accepts the message ${M}_{i}$; otherwise, it is rejected. Therefore, the resistance to forgery attacks is satisfied in our work.
- Resistance to Modify Attacks: The attacker cannot easily modify a valid final-tuple {$PI{D}_{{v}_{i}}$, ${M}_{i}$, ${T}_{i}$, ${T}_{1}$, $V{T}_{i}$, ${\delta}_{{m}_{i}}$}, where ${\delta}_{{m}_{i}}=$${\mathcal{T}}_{{\sigma}_{{m}_{i}}}\left(S{k}_{{v}_{i}}\right)modP$. The checker can check the validity of the final-tuple {$PI{D}_{{v}_{i}}$, ${M}_{i}$, ${T}_{i}$, ${T}_{1}$, $V{T}_{i}$, ${\delta}_{{m}_{i}}$} by computing whether the equation ${\mathcal{T}}_{PI{D}_{{v}_{i}}.{{\sigma}_{{m}_{i}}}^{-}}\left(S{k}_{{v}_{i}}\right)\stackrel{?}{=}{\mathcal{T}}_{PI{D}_{{v}_{i}}}\left({\delta}_{{m}_{i}}\right)$ holds. If it is so, the checker accepts the message ${M}_{i}$; otherwise, it is rejected. Therefore, the resistance to modify attacks is satisfied in our work.
- Resistance to Man-In-The-Middle Attacks: The investigation of the node authenticity and message validity above proves that it is necessary to verify that the relation between the signer and the checker should be verified and that a real message cannot be modified and fabricated. Therefore, the resistance to man-in-the-middle attacks is satisfied in our work.

#### 5.2. Performance Evaluation

#### 5.2.1. Overhead of Computational

#### 5.2.2. Overhead of Communication

## 6. Conclusions and Future Work

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Table 1.**Assessment of Present Authentication Systems’ Security. Look to Section 3.2 for explanation of the seven criteria of comparison.

Bayat et al. [22] | Li et al. [23] | Cui et al. [24] | Al-Shareeda et al. [25] | Our Work | |
---|---|---|---|---|---|

Authentication and Integrity | Yes | Yes | Yes | Yes | Yes |

Identity Privacy | Yes | Yes | Yes | Yes | Yes |

Traceability | Yes | Yes | Yes | Yes | Yes |

Unlinkability | Yes | Yes | Yes | Yes | Yes |

Resistance to Attacks | Yes | Yes | Yes | Yes | Yes |

Pseudonyms Revocation | NO | NO | NO | NO | Yes |

Efficient | NO | NO | NO | NO | Yes |

Math Symbol | Definition |
---|---|

TA | Trusted Authority |

5G-BS | Fifth-Generation Base Station |

$P,{k}_{S}^{TA},x$ | Chebyshev Polynomial Parameters |

$PI{D}_{{v}_{i}}$ | Pseudonym-IDs of Vehicle |

$S{k}_{{v}_{i}}$ | Signature Key |

h | One-Way Hash Function |

${S}_{TA}$ | Master’s Private Key |

$\psi $ | Security Parameter |

$A{C}_{{v}_{i}}$ | Authentication Code |

$TI{D}_{{v}_{i}}$ | Vehicle’s True Identity |

${T}_{i}$ | Current Timestamp |

|| | Operations of Concatenation |

⊕ | Operation of X-OR |

Operations | Definition | Times (ms) |
---|---|---|

${T}_{pair}^{bp}$ | The duration of the bilinear pairing cryptography (BPC) operation’s runtime. | 1.537 |

${T}_{mul}^{bp}$ | The scale multiplication operation’s runtime for the BPC. | 0.137 |

${T}_{mul}^{ecc}$ | The scale multiplication operation’s runtime for the elliptic curve cryptography (ECC). | 0.063075 |

${T}_{chev}$ | Chebyshev’s polynomial mapping operation’s runtime. | 0.021025 |

Schemes | Message Signing | Signature Verification |
---|---|---|

Bayat et al. [22] | $5{T}_{mul}^{bp}\approx 0.685$ | $3{T}_{pair}^{bp}+1{T}_{mul}^{bp}\approx 4.748$ |

Li et al. [23] | $1{T}_{mul}^{ecc}\approx 0.063075$ | $4{T}_{mul}^{ecc}\approx 0.2523$ |

Cui et al. [24] | $3{T}_{mul}^{ecc}\approx 0.18921$ | $3{T}_{mul}^{ecc}\approx 0.18921$ |

Al-Shareeda et al. [25] | $1{T}_{mul}^{ecc}\approx 0.06307$ | $2{T}_{mul}^{ecc}\approx 0.12614$ |

Our Proposal | $1{T}_{chev}\approx 0.021025$ | $2{T}_{chev}\approx 0.04205$ |

Schemes | Final-Tuple | Size of Tuple (bytes) | Size of n Tuples (bytes) |
---|---|---|---|

Bayat et al. [22] | $\{PI{D}_{i},M,V,r,{T}_{i1},{T}_{i2},{T}_{i3},t{s}_{i}\}$ | $4+20+5\xb7128\approx 664$ | 664 n |

Li et al. [23] | $\{{M}_{j},RI{D}_{j},{Y}_{j},{W}_{j},{T}_{j},Rsi{g}_{j}\}$ | $64\xb72+4+20\approx 152$ | 152 n |

Cui et al. [24] | $\{D{T}_{ij},{D}_{j},PI{D}_{j},{\delta}_{j},{T}_{j}\}$ | $4+2\xb720+2\xb764\approx 172$ | 172 n |

Al-Shareeda et al. [25] | $\{{M}_{i},{R}_{i},AI{D}_{i},{T}_{i},{\sigma}_{i}\}$ | $4+20+2\xb764\approx 152$ | 152 n |

Our Proposal | {$PI{D}_{{v}_{i}}$, ${M}_{i}$, ${T}_{i}$, ${T}_{1}$, $V{T}_{i}$, ${\delta}_{{m}_{i}}$} | $4\xb73+20+64\approx 96$ | 96 n |

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## Share and Cite

**MDPI and ACS Style**

Al-Mekhlafi, Z.G.; Al-Shareeda, M.A.; Manickam, S.; Mohammed, B.A.; Alreshidi, A.; Alazmi, M.; Alshudukhi, J.S.; Alsaffar, M.; Alsewari, A.
Chebyshev Polynomial-Based Fog Computing Scheme Supporting Pseudonym Revocation for 5G-Enabled Vehicular Networks. *Electronics* **2023**, *12*, 872.
https://doi.org/10.3390/electronics12040872

**AMA Style**

Al-Mekhlafi ZG, Al-Shareeda MA, Manickam S, Mohammed BA, Alreshidi A, Alazmi M, Alshudukhi JS, Alsaffar M, Alsewari A.
Chebyshev Polynomial-Based Fog Computing Scheme Supporting Pseudonym Revocation for 5G-Enabled Vehicular Networks. *Electronics*. 2023; 12(4):872.
https://doi.org/10.3390/electronics12040872

**Chicago/Turabian Style**

Al-Mekhlafi, Zeyad Ghaleb, Mahmood A. Al-Shareeda, Selvakumar Manickam, Badiea Abdulkarem Mohammed, Abdulrahman Alreshidi, Meshari Alazmi, Jalawi Sulaiman Alshudukhi, Mohammad Alsaffar, and Abdulrahman Alsewari.
2023. "Chebyshev Polynomial-Based Fog Computing Scheme Supporting Pseudonym Revocation for 5G-Enabled Vehicular Networks" *Electronics* 12, no. 4: 872.
https://doi.org/10.3390/electronics12040872