# Using Adaptive Zero-Knowledge Authentication Protocol in VANET Automotive Network

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## Abstract

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## 1. Introduction

- Vehicle-to-vehicle (V2V): When OBUs are interacting objects. This type of interaction is the main one in the absence of infrastructure base stations. It allows organizing the data exchange between road vehicles to enhance road safety;
- Vehicle-to-infrastructure (V2I): When information is transferred from the OBU to the RSU. It is used to accumulate information in control centers in order to organize a control system for monitoring and managing traffic streams;
- Infrastructure-to-infrastructure (I2I): When RSUs interact with each other. This type of interface allows data exchanging both via wired communication channels and wireless ones;
- Vehicle-to-X (V2X) is a universal interface type that allows organizing the V2V and/or V2I type of interaction.

- Taking into account the above requirements for authentication in the VANET, an adaptive authentication protocol was developed, built on zero-knowledge proof. This protocol provides a high degree of anonymity when performing OBU-RSU authentication, RSU-OBU authentication, and OBU-OBU authentication without using encryption methods. In this case, the trusted authorities will not be able to track the vehicle route using the data received by RSUs or OBUs during the authentication process. At the same time, this protocol requires minimal time spent on vehicle authentication. This is achievable by a reduction in the number of execution stages compared to previously known challenge–response protocols.
- A scheme that allows adapting the protocol’s parameters depending on the vehicle traffic intensity. When there is little traffic on the road, the OBU selects a high confidentiality level (level 3). As the traffic congestion increases, the user can lower the confidentiality level (to the level 2). A decrease in this level will reduce the time required for vehicle authentication. A further decrease in confidentiality to the first level ensures minimal time spent on vehicle authentication. It allows RSUs to operate efficiently under high-traffic conditions.
- A protocol for verifying the authority of the vehicle’s owner for requested services. As a result of this protocol implementation, the VRS calculates the public key of the vehicle without using encryption and provides necessary services.

## 2. Comparative Analysis of VANET Authentication Methods

## 3. Analysis of Zero-Knowledge Protocols

- The verifier chooses a random number A and encrypts it using the prover’s public key $C={E}_{{K}_{public}^{P}}^{}(A)$. Then, the verifier calculates a value of the hash function at the number A, that is, ${H}_{A}^{}=H(A)$, where H is the hash function. A pair of the obtained results $(C,{H}_{A}^{})$ is transmitted to the prover.
- The prover, having received $(C,{H}_{A}^{})$, decrypts the message C with his secret key and obtains ${A}_{}^{*}={D}_{{K}_{private}^{P}}^{}(C)$. Then, the prover calculates a value of the hash function ${H}_{{A}_{}^{*}}^{}=H({A}_{}^{*})$. If an equality ${H}_{{A}_{}^{*}}^{}={H}_{A}^{}$ is satisfied, the decrypted number ${A}_{}^{*}$ is returned to the verifier.
- The verifier receives a number ${A}_{}^{*}$ and compares it with his chosen number A. If equality $A={A}_{}^{*}$ is satisfied, the verifier reaches a conclusion about the authenticity of the prover.

## 4. Zero-Knowledge Authentication Protocol with Minimal Authentication Time

- A prime number $S$, a modulo in which the calculations in the developed protocol are performed.
- The number $m$ is the number generating the multiplicative group ${Z}_{S}$.
- ${k}_{\mathit{private}}^{OBU}$, ${k}_{\mathit{private}}^{RSU}$ are images of the OBU and RSU secret keys, where$${k}_{\mathit{private}}^{OBU}={K}_{\mathit{private}}^{OBU}\mathit{mod}S,\text{}{k}_{\mathit{private}}^{RSU}={K}_{\mathit{private}}^{RSU}\mathit{mod}S,$$
- $D$,$G$ are the numbers for obtaining OBU and RSU session keys where $\left\{D,G\right\}S-1$.
- $L$ is the number used by the OBU in the protocol for verifying the vehicle user’s credentials for the services provided where $L<S-1$.
- $T$ is the number used by RSU to get the timestamp where $T<S-1$.

`→`RSU (verifier).

- The prover (OBU) calculates its true digest value to perform the j-th authentication:$$B(j)={m}_{}^{{k}_{private}^{OBU}(j)}{m}_{}^{D(j)}{m}_{}^{L(j)}\mathit{mod}S.$$The true digest is entered in the OBU’s memory.
- The prover (OBU) chooses random numbers $\left\{\Delta {k}_{private}^{OBU}(j),\Delta D(j),\Delta L(j)\right\}S-2$, thus “distorting” the protocol’s secret parameters:$${\widehat{k}}_{private}^{OBU}(j)=\left({k}_{private}^{OBU}+\Delta {k}_{private}^{OBU}(j)\right)\mathit{mod}\varphi (S),$$$$\widehat{D}(j)=\left(D(j)+\Delta D(j)\right)\mathit{mod}\varphi (S),$$$$\widehat{L}(j)=\left(L(j)+\Delta L(j)\right)\mathit{mod}\varphi (S),$$The OBU then calculates the “distorted” digest:$$\widehat{B}(j)={m}_{}^{{\widehat{k}}_{private}^{OBU}(j)}{m}_{}^{\widehat{D}(j)}{m}_{}^{\widehat{L}(j)}\mathit{mod}S.$$The “distorted” digest is entered into the OBU’s memory.When the OBU (prover) comes into the range of the RSU (verifier), the following actions are performed:
- The verifier (RSU) chooses a random number $M(j)<S-2$, which is a challenge. This number is sent to the OBU.
- The prover (OBU) computes the responses:$${Y}_{1}^{}(j)=({\widehat{k}}_{private}^{OBU}(j)-M(j){k}_{private}^{OBU}(j))\mathit{mod}\varphi (S),$$$${Y}_{2}^{}(j)=(\widehat{D}(j)-M(j)D(j))\mathit{mod}\varphi (S),$$$${Y}_{3}^{}(j)=(\widehat{L}(j)-M(j)L(j))\mathit{mod}\varphi (S).$$
- The prover (OBU) sends a signal to the RSU:$$(B(j)||\widehat{B}(j)||{Y}_{1}^{}(j)||{Y}_{2}^{}(j)||{Y}_{3}^{}(j)).$$
- The RSU (verifier) checks the correctness of the response by calculating$$A(j)={B}_{}^{M(j)}(j){m}_{}^{{Y}_{1}^{}(j)}{m}_{}^{{Y}_{2}^{}(j)}{m}_{}^{{Y}_{3}^{}(j)}.$$

`→`OBU (verifier).

- Prover (RSU):$$H(j)={m}_{}^{{k}_{private}^{RSU}(j)}{m}_{}^{G(j)}{m}_{}^{T(j)}\mathit{mod}S.$$Prover (RSU):$$RSU\stackrel{}{\to}RA{M}_{RSU}^{}:H(j).$$
- Prover (RSU):$$\left\{\Delta {k}_{private}^{RSU}(j),\Delta G(j),\Delta T(j)\right\}S-2,\phantom{\rule{0ex}{0ex}}{\widehat{k}}_{private}^{RSU}(j)=\left({k}_{private}^{RSU}+\Delta {k}_{private}^{RSU}(j)\right)\mathit{mod}\varphi (S),\phantom{\rule{0ex}{0ex}}\widehat{G}(j)=\left(G(j)+\Delta G(j)\right)\mathit{mod}\varphi (S),\phantom{\rule{0ex}{0ex}}\widehat{T}(j)=\left(T(j)+\Delta T(j)\right)\mathit{mod}\varphi (S).$$Prover (RSU):$$\widehat{H}(j)={m}_{}^{{\widehat{k}}_{private}^{RSU}(j)}{m}_{}^{\widehat{G}(j)}{m}_{}^{\widehat{T}(j)}\mathit{mod}S.$$Prover (RSU):$$RSU\stackrel{}{\to}RA{M}_{RSU}^{}:\widehat{H}(j).$$
- Verifier (OBU):$$E(j)<S-2,\text{}OBU\stackrel{}{\to}RSU:E(j).$$
- Prover (RSU):$${C}_{1}^{}(j)=({\widehat{k}}_{private}^{RSU}(j)-E(j){k}_{private}^{RSU}(j))\mathit{mod}\varphi (S),$$$${C}_{2}^{}(j)=(\widehat{G}(j)-E(j)G(j))\mathit{mod}\varphi (S),$$$${C}_{3}^{}(j)=(\widehat{T}(j)-E(j)T(j))\mathit{mod}\varphi (S).$$
- Prover (RSU):$$RSU\stackrel{}{\to}OBU:(H(j)||\widehat{H}(j)||{C}_{1}^{}(j)||{C}_{2}^{}(j)||{C}_{3}^{}(j)).$$
- Verifier (OBU):$$Q(j)={H}_{}^{E(j)}(j){m}_{}^{{C}_{1}^{}(j)}{m}_{}^{{C}_{2}^{}(j)}{m}_{}^{{C}_{3}^{}(j)}=\widehat{H}(j).$$

## 5. The Protocol Parameters’ Adaptation Depending on Traffic Density

`→`RSU (verifier).

- Prover (OBU):$$B(j)={m}_{}^{{k}_{private}^{OBU}(j)}{m}_{}^{D(j)}\mathit{mod}S.$$Prover (OBU):$$OBU\stackrel{}{\to}RA{M}_{OBU}^{}:B(j).$$
- Prover (OBU):$$\left\{\Delta {k}_{private}^{OBU}(j),\Delta D(j)\right\}S-2.$$Prover (OBU):$${\widehat{k}}_{private}^{OBU}(j)=\left({k}_{private}^{OBU}+\Delta {k}_{private}^{OBU}(j)\right)\mathit{mod}\varphi (S),\phantom{\rule{0ex}{0ex}}\widehat{D}(j)=\left(D(j)+\Delta D(j)\right)\mathit{mod}\varphi (S).$$Prover (OBU):$$\widehat{B}(j)={m}_{}^{{\widehat{k}}_{private}^{OBU}(j)}{m}_{}^{\widehat{D}(j)}\mathit{mod}S.$$Prover (OBU):$$OBU\stackrel{}{\to}RA{M}_{OBU}^{}:\widehat{B}(j).$$
- Verifier (RSU):$$M(j)<S-2,\text{}RSU\stackrel{}{\to}OBU:M(j).$$
- Prover (OBU):$${Y}_{1}^{}(j)=({\widehat{k}}_{private}^{OBU}(j)-M(j){k}_{private}^{OBU}(j))\mathit{mod}\varphi (S),\phantom{\rule{0ex}{0ex}}{Y}_{2}^{}(j)=(\widehat{D}(j)-M(j)D(j))\mathit{mod}\varphi (S).$$
- Prover (OBU):$$OBU\stackrel{}{\to}RSU:(B(j)||\widehat{B}(j)||{Y}_{1}^{}(j)||{Y}_{2}^{}(j)).$$
- Verifier (RSU):$$A(j)={B}_{}^{M(j)}(j){m}_{}^{{Y}_{1}^{}(j)}{m}_{}^{{Y}_{2}^{}(j)}=\widehat{B}(j).$$

`→`RSU (verifier).

- Prover (OBU):$$B(j)={m}_{}^{{k}_{private}^{OBU}(j)}\mathit{mod}S.$$Prover (OBU):$$OBU\stackrel{}{\to}RA{M}_{OBU}^{}:B(j).$$
- Prover (OBU):$$\left\{\Delta {k}_{private}^{OBU}(j)\right\}<S-2.$$Prover (OBU):$${\widehat{k}}_{private}^{OBU}(j)=\left({k}_{private}^{OBU}+\Delta {k}_{private}^{OBU}(j)\right)\mathit{mod}\varphi (S).$$Prover (OBU):$$\widehat{B}(j)={m}_{}^{{\widehat{k}}_{private}^{OBU}(j)}\mathit{mod}S.$$Prover (OBU):$$OBU\stackrel{}{\to}RA{M}_{OBU}^{}:\widehat{B}(j).$$
- Verifier (RSU):$$M(j)<S-2,\text{}RSU\stackrel{}{\to}OBU:M(j).$$
- Prover (OBU):$${Y}_{1}^{}(j)=({\widehat{k}}_{private}^{OBU}(j)-M(j){k}_{private}^{OBU}(j))\mathit{mod}\varphi (S).$$
- Prover (OBU):$$OBU\stackrel{}{\to}RSU:(B(j)||\widehat{B}(j)||{Y}_{1}^{}(j)).$$
- Verifier (RSU):$$A(j)={B}_{}^{M(j)}(j){m}_{}^{{Y}_{1}^{}(j)}\mathit{mod}S=\widehat{B}(j).$$

## 6. A Mechanism to Verify the Vehicle Owner’s Authority for the Services Provider

- A prime number $S$, a modulo in which the calculations in the developed protocol are performed, where ${\mathrm{log}}_{2}^{}n>512$.
- A number $a$ generating the multiplicative group ${Z}_{S}$.

`→`VRS (verifier).

- Verifier (VRS):$${X}_{1}^{}(i)<n-2,\phantom{\rule{0ex}{0ex}}VRS\stackrel{}{\to}OBU:{X}_{1}^{}(i).$$
- Prover (OBU):$${V}_{1}^{}(i)={K}_{public}^{OBU}{\left({a}_{}^{{(L+i+1)}_{}^{-1}}\right)}_{}^{{X}_{1}^{}}\mathit{mod}n,$$$$OBU\stackrel{}{\to}VRS:{V}_{1}^{}(i).$$
- Verifier (VRS):$${X}_{2}^{}(i)<n-2,\phantom{\rule{0ex}{0ex}}VRS\stackrel{}{\to}OBU:{X}_{2}^{}(i).$$
- Prover (OBU):$${V}_{2}^{}(i)={K}_{public}^{OBU}{\left({a}_{}^{{(L+i+1)}_{}^{-1}}\right)}_{}^{{X}_{2}^{}}\mathit{mod}n,$$$$OBU\stackrel{}{\to}VRS:{V}_{2}^{}(i).$$
- Verifier (VRS):$$K(j)={\left(\frac{{V}_{1}^{{X}_{2}^{}}}{{V}_{2}^{{X}_{1}^{}}}\right)}_{}^{{({X}_{2}^{}-{X}_{1}^{})}_{}^{-1}}\mathit{mod}n={K}_{public}^{OBU}.$$

## 7. Results and Discussion

## 8. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Dependence of the traffic volume available for transmitting information on the number of OBUs per 1 km (1, 2, and 3—confidentiality levels).

**Figure 3.**The average number of packets per OBU at the 3rd confidentiality level, 64-bit authentication protocol.

**Figure 6.**Dependence of the traffic available for transmitting information on the number of OBUs per 1 km (1, 2, and 3—confidentiality levels).

T_{1}, ns | T_{2}, ns | T_{3}, ns | T_{4}, ns | T_{5}, ns | T_{6}, ns |

13,500 | 13,900 | 32,000 | 500 | 96,000 | 27,450 |

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**MDPI and ACS Style**

Kalmykov, I.A.; Olenev, A.A.; Kalmykova, N.I.; Dukhovnyj, D.V.
Using Adaptive Zero-Knowledge Authentication Protocol in VANET Automotive Network. *Information* **2023**, *14*, 27.
https://doi.org/10.3390/info14010027

**AMA Style**

Kalmykov IA, Olenev AA, Kalmykova NI, Dukhovnyj DV.
Using Adaptive Zero-Knowledge Authentication Protocol in VANET Automotive Network. *Information*. 2023; 14(1):27.
https://doi.org/10.3390/info14010027

**Chicago/Turabian Style**

Kalmykov, Igor Anatolyevich, Aleksandr Anatolyevich Olenev, Natalia Igorevna Kalmykova, and Daniil Vyacheslavovich Dukhovnyj.
2023. "Using Adaptive Zero-Knowledge Authentication Protocol in VANET Automotive Network" *Information* 14, no. 1: 27.
https://doi.org/10.3390/info14010027