A Cross-Layer and Optimized Privacy Method in Vehicular Ad-Hoc Networks

Abstract

Vehicular networks has been recently proposed to connect vehicles and form ad-hoc networks. Many safety and information-entertainment related applications have thus been proposed. These applications, however, potentially have unique privacy challenges. For example, the vehicle’s network identity is strictly linked to the owner’s identity due to the insurance liability. In this paper, we propose cross-layer privacy protection protocols including initialization, joining and exiting protocols. In addition, we also analytically discuss the optimization of the quantity of pseudonyms to save costs. The analytical and numerical results showed the effectiveness of the proposed methods.

Appendix

A list of notations is shown below for better read:

• IA is the identity authority which is the only trusted agents in the system and is the only agent that keeps the real identity of a vehicle

• CA is the certificate authority that can be acted by the elected cell leader which can be partially trusted and be fully monitored by other vehicles

• p is a big prime number

• M is the pseudonym proposal defined by users.

• m is the new message defined by IA, i.e. $m=\{M,x,TTL\}$. To avoid pseudonym collision, a random number x is added. To avoid pseudonym lasts too long time, a time-to-live counter TTL is added.

• x is a random number that IA appends to M.

• TTL is a time-to-live counter that IA appends to M.

• e is the primitive element which is used in our generating function f(x).

• g is the group information which is know by all the group members

• $f(g) = 2^k + 2h(g)+ 1$

• $f^{\prime }(g) = f(g) + \delta$

• $\delta$ is a random number such that $gcd(f^{\prime }(g),p-1) = 1$

• c is the private key of the vehicle i

• k is the maximum length of message g

• d is the public key of the vehicle i and $d=e^c mod p$

• $\{\alpha , \beta , \gamma \}$ are random numbers selected by IA

• $\sigma$ is a random number with condition $gcd(e^\sigma \text{ mod } p,p-1) = 1$

• m is the pseudonym that vehicle i proposes to use

• $a=e^\sigma \text{ mod } p$ is generated by CA and sent to IA.

• $a^{\prime } = a^\alpha d^{\gamma f^{\prime }(g)} e^\beta \text{ mod } p$ is generated by IA and is sent to CA and the public

• $m^{\prime }=\left( \frac{a^{\prime }m-\gamma }{a \alpha }\right) \text{ mod } (p-1)$ is generated by IA and sent to CA

• $b=\left( am^{\prime }cf^{\prime }(g)-k \right) \text{ mod } (p-1)$ is generated by CA and sent to IA

• $b^{\prime } = \frac{\alpha b -\beta }{f^{\prime }(g)} \text{ mod } (p-1)$ is generated by IA and is published to public

Lemma 9

$$\begin{aligned} \alpha b = [b^{\prime } f^{\prime }(g) + \beta ] \text{ mod } (p-1) \end{aligned}$$

Proof

$$\begin{aligned} RHS= &\, [b^{\prime } f^{\prime }(g) + \beta ] \text{ mod } (p-1)\\= &\, \alpha \frac{b^{\prime } f^{\prime }(g) + \beta }{\alpha } \text{ mod } (p-1)\\= &\, \alpha \frac{ \frac{\alpha b -\beta }{f^{\prime }(g)} f^{\prime }(g) + \beta }{\alpha } \text{ mod } (p-1)\\= &\, \alpha b \\= &\, LHS\\ \end{aligned}$$

Therefore, $LHS\equiv RHS$. $\square$

Lemma 10

$$\begin{aligned} \alpha b = \left[ a^{\prime } m f^{\prime }(g) c - \alpha k - c \gamma f^{\prime }(g) \right] \text{ mod } (p-1) \end{aligned}$$

Proof

$$\begin{aligned} RHS= &\, \left[ a^{\prime } m f^{\prime }(g) c - \alpha k - c \gamma f^{\prime }(g) \right] \text{ mod } (p-1)\\= &\, \alpha \frac{a^{\prime } m f^{\prime }(g) c - \alpha k - c \gamma f^{\prime }(g)}{\alpha } \text{ mod } (p-1)\\= &\, \left[ \alpha \left[ \frac{a^{\prime } m f^{\prime }(g) c - c \gamma f^{\prime }(g)}{\alpha } \right] -k\right] \text{ mod } (p-1)\\= &\, \left[ \alpha \left[ \frac{(a^{\prime } m - \gamma )f^{\prime }(g) c}{\alpha } \right] -k\right] \text{ mod } (p-1)\\= &\, \left[ \alpha \left[ \frac{a(a^{\prime } m - \gamma )f^{\prime }(g) c}{a \alpha } \right] -k\right] \text{ mod } (p-1)\\= &\, \alpha \left[ a m^{\prime } c f^{\prime }(g) -k\right] \text{ mod } (p-1)\\= &\, \alpha b \\= &\, LHS \end{aligned}$$

$\square$

Lemma 11

$$\begin{aligned} a^{\prime } m f^{\prime }(g) c = \left[ b^{\prime } f^{\prime }(g) + \beta + \alpha k + c \gamma f^{\prime }(g) \right] mod (p-1) \end{aligned}$$

Proof

To prove this lemma, we refer lemma 9 and 10. We shuffle them in a different order. We write:

$$\begin{aligned} a^{\prime } m f^{\prime }(g) c = \left[ b^{\prime } f^{\prime }(g) + \beta + \alpha k + c \gamma f^{\prime }(g) \right] \text{ mod } (p-1). \end{aligned}$$

$\square$

Lemma 12

$\{X(t)~|~ t \ge 0\}$ is a Poisson process with parameter

$$\begin{aligned} \varLambda (t) = \lambda \int _{0}^t p(t- u) \left[ 1- F_G(u) \right] \,{\hbox{d}}u. \end{aligned}$$

(5)

Proof

We begin by determining the probability $\gamma (t)$ that an arbitrary arriving car is marked as it enters the cell and that it will still be resident in the cell at time t. For this purpose, assume that $n,\ (n \ge 0),$ cars have arrived in (0, t). It is well known that, with this assumption, the individual arrival times of cars are uniformly distributed in (0, t). Now, consider a car arriving at time $\tau$:

• with probability $p(\tau )$ the car is marked;

• with probability $\Pr [\{G > t- \tau \}] = 1- F_G(t- \tau )$ the car is still resident in the cell at time t.

Since the event that a car is marked is independent of whether or not the car will be resident in the cell at time t, it is clear that the probability that a generic car arriving at time $\tau$ is both marked and resident in the cell at time t is

$$\begin{aligned} p(\tau ) \left[ 1- F_G(t- \tau ) \right] . \end{aligned}$$

Now, letting U denote the uniform random variable on (0, t), the Law of Total Probability guarantees that

$$\begin{aligned} \gamma (t)= & \int _{0}^t p(\tau ) \left[ 1- F_G(t- \tau ) \right] \,{\hbox{d}}F_U(\tau ) \\= & \int _{0}^t p(\tau ) \left[ 1- F_G(t- \tau ) \right] \frac{{\hbox{d}}\tau }{t} \\= & \frac{1}{t} \int _{0}^t p(t- u) \left[ 1- F_G(u) \right] \,{\hbox{d}}u. \end{aligned}$$

(6)

We now have all the ingredients necessary to evaluate the probability $P_k(t) = \Pr [\{X(t) =k\}]$.

$$\begin{aligned} P_k(t)= & \sum _{n \ge 0} \Pr [\{X(t) =k\} ~|arrive(n)] \\&*&\Pr [\{\{n\ {\hbox{cars have arrived in}}\ (0,t)\}] \\= & \sum _{n \ge 0} {n \atopwithdelims ()k} \left[ \gamma (t) \right] ^k \left[ 1- \gamma (t) \right] ^{n-k} \frac{\left[ \lambda t\right] ^n}{n!} e^{- \lambda t} \\= & \frac{\left[ \varLambda (t)\right] ^k}{k!} e^{- \varLambda (t)} \end{aligned}$$

(7)

where $\varLambda (t)$ has been defined in (5), arrive(n) is $~\{n\ {\hbox{cars have arrived in}}\ (0,t)\}$.

Observe that (7) tells us that the process $\{X(t)~|~ t \ge 0\}$ is a Poisson process with parameter $\varLambda (t)$. This completes the proof of the lemma. $\square$