# A Conditional Privacy Preserving Generalized Ring Signcryption Scheme for Micro Aerial Vehicles

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## Abstract

**:**

## 1. Introduction

- We propose a conditional privacy-preserving generalized ring signcryption scheme for MAVs using the ECC operation.
- The proposed scheme is conditional privacy-preserving, meaning each entity encrypts its real identity using a common secret key between entity and PKG in the key generation process.
- The proposed scheme enables encryption and digital signature simultaneously as well as independently using generalized signcryption. In ring configurations mode, this scheme guarantees anonymity, spontaneity, flexibility, and equal membership.
- We conducted a formal security study using the Random Oracle Model (ROM) and found that the proposed scheme is secure against a wide range of cyber-attacks.
- Finally, the proposed scheme’s efficiency is compared to its counterparts, validating its low computation cost, communication cost and memory overhead.

## 2. Preliminaries, Network Model and Syntax of the Proposed Scheme

#### 2.1. Preliminaries

#### 2.1.1. Elliptic Curve Cryptography (ECC)

#### 2.1.2. Elliptic Curve Decisional Diffie-Hellman Problem (ECDDHP)

#### 2.1.3. Elliptic Curve Discrete Logarithm Problem (ECDLP)

_{ECC}with prime order q, and given (θ.ξ,ξ,K ∈ G

_{ECC}), extracting θ from K = θ. ξ is called ECDLP.

#### 2.2. Syntax

- Initialization: The ground core network (GCN) can play the role private key generator (PKG), in which he/she can sets ${\mathrm{\xdf}}_{GCN}$ as his/her secret key, ${\mathrm{\delta}}_{GCN}$ as his/her public key, and generates a public parameter set $\mathrm{\u0416}$.
- Key Generation: The device that participates in a network as a legal user will send ($EI{d}_{i},{\Omega}_{i}$) to GCN by using open channel. Based on ($EI{d}_{i},{\Omega}_{i}$), GCN first compute ${\gamma}_{i}$ and recover the real identity $I{d}_{i}$. Then, GCN computes ${\lambda}_{i},$ ${\Phi}_{i}$ and send (${\Phi}_{i},{\lambda}_{i}$) to the legitimate user by using secure channel.
- Generalized Ring Signcryption: This algorithm will run by Micro Aerial Vehicle (MAV), in which the MAV take input that are (${\mathit{EId}}_{\mathit{MAV}},m,{\lambda}_{X},{\pounds}_{X},{\delta}_{GCN}$) and produce the tuple ($\kappa ,\mathrm{\u041b},\Gamma $).
- Generalized Ring Signcryption Verifications: Given the tuple ($EI{d}_{X},{\lambda}_{MAV},{\pounds}_{MAV},{\delta}_{GCN},\kappa ,\mathrm{\u041b},\Gamma ,{\Phi}_{X}.$), a user can verify ($\kappa ,\mathrm{\u041b},\Gamma $).

#### 2.3. Network Model

## 3. Proposed Scheme Construction

- MDN choose ${\chi}_{\mathit{MAV}}\in {f}_{q}$ and compute $\mathrm{\u041b}={\chi}_{\mathit{MAV}}\xb7\xi $.
- Compute $\Psi ={\chi}_{\mathit{MAV}}\left({\lambda}_{X}+{\pounds}_{X}\xb7{\delta}_{GCN}\right)$ and $\Gamma ={\mathrm{\u0426}}_{2}\left(\Psi \right)\oplus \left(m,EI{d}_{MAV}\right).$
- Compute $\omega ={\mathrm{\u0426}}_{3}\left(EI{d}_{MAV},{\lambda}_{MAV},{\lambda}_{X},\mathrm{\u041b},\Gamma \right)$ and $\kappa ={\chi}_{\mathit{MAV}}+\omega \xb7{\Phi}_{MAV}$.
- MAV send ($\omega ,\mathrm{\u041b},\Gamma $) to everything (X).

- Compare if $\kappa \xb7\xi =\mathrm{\u041b}+\omega \xb7\left({\lambda}_{MAV}+{\pounds}_{MAV}\xb7{\delta}_{GCN}\right)$ is holds, where $\omega ={\mathrm{\u0426}}_{3}\left(EI{d}_{MAV},{\lambda}_{MAV},{\lambda}_{X},\mathrm{\u041b},\Gamma \right)$.
- Compute $\Psi ={\Phi}_{X}\xb7\mathrm{\u041b}$ and $\left(m,EI{d}_{MAV}\right)=\Gamma \oplus {\mathrm{\u0426}}_{2}\left(\Psi \right)$.

#### Correctness Analysis

## 4. Security Analysis

**Theorem**

**1.**

**Confidentiality:**The proposed generalized ring signcryption is indistinguishable against intruder INT under the ROM, if ECDDHP is hard.

**Proof.**

- ${\mathrm{\u0426}}_{1}$
- Query: $INT$ send a request for ${\mathrm{\u0426}}_{1}$ Query with identity $(I{d}_{i}){C}_{ECDDHP}$ check for a tuple $\left(I{d}_{i},{\lambda}_{i},{\pounds}_{i}\right)$ in the list ${L}_{{\mathrm{\u0426}}_{1}}$, if $\left(I{d}_{i},{\lambda}_{i},{\pounds}_{i}\right)$ is found, ${C}_{ECDDHP}$ returns ${\pounds}_{i}$ to $INT$. Otherwise, ${C}_{ECDDHP}$ choose the value for ${\pounds}_{i}$ randomly and returns it to $INT$.
- ${\mathrm{\u0426}}_{2}$
- Query: $INT$ send a request for ${\mathrm{\u0426}}_{2}$ Query with identity $(I{d}_{i}){C}_{ECDDHP}$ check for a tuple $\left({\Psi}_{i},{\pounds}_{1i}\right)$ in the list ${L}_{{\mathrm{\u0426}}_{2}}$, if $\left({\Psi}_{i},{\pounds}_{1i}\right)$ is found, ${C}_{ECDDHP}$ returns ${\pounds}_{1i}$ to $INT$. Otherwise, ${C}_{ECDDHP}$ choose the value for ${\pounds}_{1\mathrm{i}}$ randomly and returns it to $INT$.
- ${\mathrm{\u0426}}_{3}$
- Query: $INT$ send a request for ${\mathrm{\u0426}}_{3}$ Query with identity $(I{d}_{i}){C}_{ECDDHP}$ check for a tuple $\left(EI{d}_{i},{\lambda}_{i},{\Gamma}_{i},{\mathrm{\u041b}}_{i},{\omega}_{i}\right)$ in the list ${L}_{{\mathrm{\u0426}}_{3}}$, if $\left(EI{d}_{i},{\lambda}_{i},{\Gamma}_{i},{\mathrm{\u041b}}_{i},{\omega}_{i}\right)$ is found, ${C}_{ECDDHP}$ returns ${\omega}_{i}$ to $INT$. Otherwise, ${\mathrm{C}}_{\mathit{ECDDHP}}$ choose the value for ${\omega}_{i}$ randomly and returns it to $INT$.

- At $jth$ query, if $i=j$, ${C}_{ECDDHP}$ set ${\lambda}_{i}=\Omega \xb7\xi $.
- Else, compute ${\lambda}_{i}={\eta}_{i}\xb7\xi $, where it selects ${\eta}_{i}$ randomly.
- At the end, ${C}_{ECDDHP}$ returns ${\lambda}_{i}$ to $INT.$

- If $EI{d}_{MAV}!=Id$, It choose ${\chi}_{MAV}\in {f}_{q}$ and compute $\mathrm{\u041b}={\chi}_{MAV}\xb7\xi -\omega \left({\lambda}_{MAV}+{\pounds}_{MAV}\xb7{\delta}_{GCN}\right)$.
- Compute $\Psi ={\chi}_{MAV}\left({\lambda}_{X}+{\pounds}_{X}\xb7{\delta}_{GCN}\right)$ and $\Gamma ={\mathrm{\u0426}}_{2}\left(\Psi \right)\oplus \left(m,EI{d}_{MAV}\right).$
- Compute $\omega ={\mathrm{\u0426}}_{3}\left(EI{d}_{MAV},{\lambda}_{MAV},{\lambda}_{X},\mathrm{\u041b},\Gamma \right)$ and $\kappa ={\chi}_{MAV}+y$, where$y$ is randomly selected now here.
- ${C}_{ECDDHP}$ send ($\kappa ,\mathrm{\u041b},\Gamma $) to $INT$.

- It computes $\mathrm{\u041b}=\Omega \xb7\xi $.
- Compute $\Psi =K+{\pounds}_{X}\xb7{\delta}_{GCN}$ and $\Gamma ={\mathrm{\u0426}}_{2}\left(\Psi \right)\oplus \left(m,EI{d}_{MAV}\right).$
- Compute $\omega ={\mathrm{\u0426}}_{3}\left(EI{d}_{MAV},{\lambda}_{MAV},{\lambda}_{X},\mathrm{\u041b},\Gamma \right)$ and $\kappa =\omega \xb7{\Phi}_{MAV}+y+\Omega $, where$y$ is randomly selected now here.
- ${C}_{ECDDHP}$ returns ($\kappa ,\mathrm{\u041b},\Gamma $).

**Phase 2:**In this phase, INT executes ${\mathrm{\u0426}}_{1}$ Query, ${\mathrm{\u0426}}_{2}$ Query, ${\mathrm{\u0426}}_{3}$ Query, User Public Key Query, Generalized Ring Signcryption Query, and Generalized Ring Signcryption Verification Query, respectively. Note that at this stage INT should not perform User Private Key Query on encrypted identity $EI{d}_{X}$ and requested message corresponding to the Generalized ring signcrypted text.

- 1.
- ${\Theta}_{1}:$ ${C}_{ECDDHP}$ succeeded in User Private Key Query.
- 2.
- ${\Theta}_{2}:$ ${\mathrm{C}}_{\mathit{ECDDHP}}$ succeeded in Generalized Ring Signcryption Verification Query.
- 3.
- ${\Theta}_{2}:$ ${\mathrm{C}}_{\mathit{ECDDHP}}$ succeeded in in challenge phase.

**Theorem**

**2.**

**Unforgeability.**Our proposed generalized ring signcryption is indistinguishable against intruder INT under the random oracle model, if ECDLP is hard.

**Proof.**

- $INT$ choose ${\chi}_{INT}\in {f}_{q}$ and compute $\mathrm{\u041b}={\chi}_{INT}\xb7\xi $.
- Compute $\Psi ={\chi}_{INT}\left({\lambda}_{X}+{\pounds}_{X}\xb7{\delta}_{GCN}\right)$ and $\Gamma ={\mathrm{\u0426}}_{2}\left(\Psi \right)\oplus \left(m,EI{d}_{MAV}\right).$
- Compute $\omega ={\mathrm{\u0426}}_{3}\left(EI{d}_{MAV},{\lambda}_{INT},{\lambda}_{X},\mathrm{\u041b},\Gamma \right)$ and $\kappa ={\chi}_{INT}+\omega \xb7{\Phi}_{INT}$.
- Returns ($\omega ,\mathrm{\u041b},\Gamma $).

- 4.
- ${\Theta}_{1}:$ ${C}_{ECDDHP}$ succeeded in User Private Key Query.
- 5.
- ${\Theta}_{2}:$ ${C}_{ECDDHP}$ succeeded in Generalized Ring Signcryption Verification Query.
- 6.
- ${\Theta}_{2}:$ ${C}_{ECDDHP}$ succeeded in in challenge phase.

**Theorem**

**3.**

**Sender Anonymity.**In the key generation phase, the sender device called MAV will send his encrypted real identity$EI{d}_{MAV}={\gamma}_{MAV}\oplus I{d}_{MAV}$, and${\Omega}_{MAV}={\alpha}_{MAV}\xb7\xi ,$to GCN by using open channel, where${\gamma}_{MAV}={\alpha}_{MAV}\xb7{\delta}_{GCN}$ and${\alpha}_{MAV}\in {f}_{q}$. Based on ($EI{d}_{MAV},{\Omega}_{MAV}$), GCN firs compute${\gamma}_{MAV}={\mathrm{\xdf}}_{GCN}\xb7{\Omega}_{MAV}$ and recover the real identity$I{d}_{MAV}$as$I{d}_{MAV}=EI{d}_{i}\oplus {\gamma}_{MAV}$. Here, if INT wants the real identity$I{d}_{MAV}$of MAV, he will pass the following two cases.

- INT first struggle to access ${\alpha}_{MAV}$ from ${\Omega}_{MAV}={\alpha}_{MAV}\xb7\xi $ to made ${\gamma}_{MAV}={\alpha}_{MAV}\xb7{\delta}_{GCN}$.
- Secondly INT can access ${\mathrm{\xdf}}_{GCN}$ from ${\delta}_{GCN}={\mathrm{\xdf}}_{GCN}\xb7\xi $ to made ${\gamma}_{MAV}={\mathrm{\xdf}}_{GCN}\xb7{\Omega}_{MAV}$.

**Theorem**

**4.**

**Receiver Anonymity.**In the key generation phase, the receiver device called$X$ will send his encrypted real identity$EI{d}_{X}={\gamma}_{X}\oplus I{d}_{X}$, and${\Omega}_{X}={\alpha}_{X}\xb7\xi ,$to GCN by using open channel, where${\gamma}_{X}={\alpha}_{X}\xb7{\delta}_{GCN}$ and${\alpha}_{X}\in {f}_{q}$. Based on ($EI{d}_{X},{\Omega}_{X}$), GCN firs compute${\gamma}_{X}={\mathrm{\xdf}}_{GCN}\xb7{\Omega}_{X}$and recover the real identity$I{d}_{X}$as$I{d}_{X}=EI{d}_{X}\oplus {\gamma}_{X}$. Here, if INT wants the real identity$I{d}_{X}$ of$X$, he will pass the following two cases.

- INT first struggle to access ${\alpha}_{X}$ from ${\Omega}_{X}={\alpha}_{X}\xb7\xi $ to made ${\gamma}_{X}={\alpha}_{X}\xb7{\delta}_{GCN}$.
- Secondly INT can access ${\mathrm{\xdf}}_{GCN}$ from ${\delta}_{GCN}={\mathrm{\xdf}}_{GCN}\xb7\xi $ to made ${\gamma}_{X}={\mathrm{\xdf}}_{GCN}\xb7{\Omega}_{X}$.

## 5. Performance Comparison

#### 5.1. Computation Cost

#### 5.2. Communication Cost

#### 5.3. Memory Overhead

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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S. No | Notation | Descriptions |
---|---|---|

1 | GCN | Ground core network |

2 | PKG | Private key generator |

3 | $\mathrm{\u0416}$ | Public parameter param |

4 | ${\mathrm{\u0426}}_{1}$,${\mathrm{\u0426}}_{2},{\mathrm{\u0426}}_{3}$ | Irreversible and collision resistant hash functions |

5 | ${\delta}_{GCN}$ | Master secret key of ground core network |

6 | ${\delta}_{GCN}$ | Master public key of ground core network |

7 | $\xi $ | Generator of group ${G}_{ECC}$ |

8 | ${G}_{ECC}$ | Finite cyclic group on the elliptic curve ${E}_{ECC}$ |

9 | ${E}_{ECC}$ | The elliptic curve defined on ${V}^{2}={U}^{3}+sU+t$ |

10 | $EI{d}_{MAV}$ | Encrypted identity of $MAV$ |

11 | $MAV$ | It represents a Micro Aerial Vehicle ($MAV$) |

12 | $EI{d}_{X}$ | Encrypted identity of everything ($X$) |

13 | $I{d}_{MAV}$ | Real identity of $MAV$ |

14 | $I{d}_{X}$ | Real identity of everything ($X$) |

15 | ${f}_{q}$ | Finite field on the elliptic curve ${E}_{ECC}$ of order $q$ |

16 | ${\Phi}_{MAV}$ | Private key of $MAV$ |

17 | ${\Phi}_{X}$ | Private key of everything ($X$) |

18 | ${\lambda}_{X}$ | Public key of everything ($X$) |

19 | ${\lambda}_{MAV}$ | Public key of $MAV$ |

20 | $\mathsf{\Delta}$ | Identities of ring group $\{EI{d}_{MAV1}$, $EI{d}_{\mathit{MAV}2},EI{d}_{\mathit{MAV}3},\dots \dots ,EI{d}_{\mathit{MAV}n}\}$ |

21 | ${\gamma}_{MAV}$ | Encryption and decryption key for real identity of $MAV$ |

22 | ${\gamma}_{X}$ | Encryption and decryption key for real identity of everything ($X$) |

23 | $\Psi $ | Encryption and decryption key for message $MAV$ and everything ($X$) |

24 | $\oplus $ | Used for Encryption and decryption |

Schemes | Sender | Receiver | Total |
---|---|---|---|

Zhou et al. [14] | $7B{P}_{BM}+1M{D}_{EXP}+1B{P}_{OP}$ | $1B{P}_{BM}+3B{P}_{OP}$ | $8B{P}_{BM}+1M{D}_{EXP}+4B{P}_{OP}$ |

Zhou et al. [15] | $10B{P}_{BM}+3M{D}_{EXP}+2B{P}_{OP}$ | $3B{P}_{BM}+4B{P}_{OP}$ | $13B{P}_{BM}+3M{D}_{EXP}+6B{P}_{OP}$ |

Luo and Zhou [16] | $7B{P}_{BM}+2M{D}_{EXP}$ | $1B{P}_{BM}+1M{D}_{EXP}+2B{P}_{OP}$ | $8B{P}_{BM}+3M{D}_{EXP}+2B{P}_{OP}$ |

Proposed Scheme | $4EC{C}_{PM}$ | $4EC{C}_{PM}$ | $8EC{C}_{PM}$ |

Schemes | Sender | Receiver | Total |
---|---|---|---|

Zhou et al. [14] | $7\times 4.31+1\times 1.25+1\times 14.9=46.32$ | $1\times 4.31+3\times 14.90=49.01$ | $8\times 4.31+1\times 1.25+4\times 14.90=95.33$ |

Zhou et al. [15] | $10\times 4.31+3\times 1.25+2\times 14.90=76.65$ | $3\times 4.31+4\times 14.90=72.53$ | $13\times 4.31+3\times 1.25+6\times 14.90=149.18$ |

Luo and Zhou [16] | $7\times 4.31+2\times 1.25=32.67$ | $1\times 4.31+1\times 1.25+2\times 14.90=35.36$ | $8\times 4.31+3\times 1.25+2\times 14.90=68.03$ |

Proposed Scheme | $4\times 0.97=3.88$ | $4\times 0.97=3.88$ | $8\times 0.97=7.76$ |

Schemes | Communication Cost | Communication Cost in Bits |
---|---|---|

Zhou et al. [14] | $\left|m\right|+3\left|B{P}_{G}\right|$ | $\left|1024\right|+3\times \left|1024\right|=4096$ |

Zhou et al. [15] | $\left|m\right|+3\left|B{P}_{G}\right|$ | $\left|1024\right|+3\times \left|1024\right|=4096$ |

Luo and Zhou [16] | $\left|m\right|+5\left|B{P}_{G}\right|$ | $\left|1024\right|+5\times \left|1024\right|=6144$ |

Proposed Scheme | $\left|m\right|+2\left|EC{C}_{q}\right|$ | $\left|1024\right|+2\times \left|160\right|=1344$ |

Schemes | Sender | Receiver | Total |
---|---|---|---|

Zhou et al. [14] | 9|$B{P}_{G}|+3\left|H\right|$+ $\left|m\right|$ | 3|$B{P}_{G}|+2\left|H\right|$+ $\left|m\right|$ | 12|$B{P}_{G}|+5\left|H\right|$+ $2\left|m\right|$ |

Zhou et al. [15] | 11|$B{P}_{G}|+4\left|H\right|$+ $\left|m\right|$ | 4|$B{P}_{G}|+4\left|H\right|$+ $\left|m\right|$ | 15|$B{P}_{G}|+8\left|H\right|$+ $2\left|m\right|$ |

Luo and Zhou [16] | 11|$B{P}_{G}|+4\left|H\right|$+ $\left|m\right|$ | 5|$B{P}_{G}|+2\left|H\right|$+ $\left|m\right|$ | 16|$B{P}_{G}|+6\left|H\right|$+ $2\left|m\right|$ |

Proposed Scheme | 10|$EC{C}_{q}|+1\left|H\right|$+ $\left|m\right|$ | 8|$EC{C}_{q}|+1\left|H\right|$+ $\left|m\right|$ | 18|$EC{C}_{q}|+2\left|H\right|$+ $2\left|m\right|$ |

Schemes | Sender | Receiver | Total |
---|---|---|---|

Zhou et al. [14] | 9$\left|1024\left|+3\right|256\right|$+ $\left|1024\right|=10996$ | 3|$1024\left|+2\right|256|$+ $\left|1024\right|=4608$ | 12|$1024\left|+5\right|256|$+ $2\left|1024\right|=15604$ |

Zhou et al. [15] | 11|$1024\left|+4\right|256|$+ $\left|1024\right|=13312$ | 4|$1024\left|+4\right|256|$+ $\left|1024\right|=6144$ | 15|$1024\left|+8\right|256|$+ $2\left|1024\right|=19456$ |

Luo and Zhou [16] | 11|$1024\left|+4\right|256|$+ $\left|1024\right|=13312$ | 5|$1024\left|+2\right|256|$+ $\left|1024\right|=6656$ | 16|$1024\left|+6\right|256|$+ $2\left|1024\right|=19968$ |

Proposed Scheme | 10|$160\left|+1\right|256|$+ $\left|1024\right|=2880$ | 8|$160\left|+1\right|256|$+ $\left|1024\right|=2560$ | 18|$160\left|+2\right|256|$+ $2\left|1024\right|=5440$ |

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## Share and Cite

**MDPI and ACS Style**

Ullah, I.; Khan, M.A.; Abdullah, A.M.; Mohsan, S.A.H.; Noor, F.; Algarni, F.; Innab, N.
A Conditional Privacy Preserving Generalized Ring Signcryption Scheme for Micro Aerial Vehicles. *Micromachines* **2022**, *13*, 1926.
https://doi.org/10.3390/mi13111926

**AMA Style**

Ullah I, Khan MA, Abdullah AM, Mohsan SAH, Noor F, Algarni F, Innab N.
A Conditional Privacy Preserving Generalized Ring Signcryption Scheme for Micro Aerial Vehicles. *Micromachines*. 2022; 13(11):1926.
https://doi.org/10.3390/mi13111926

**Chicago/Turabian Style**

Ullah, Insaf, Muhammad Asghar Khan, Ako Muhammad Abdullah, Syed Agha Hassnain Mohsan, Fazal Noor, Fahad Algarni, and Nisreen Innab.
2022. "A Conditional Privacy Preserving Generalized Ring Signcryption Scheme for Micro Aerial Vehicles" *Micromachines* 13, no. 11: 1926.
https://doi.org/10.3390/mi13111926