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. 2021 Jan 19;21(2):665. doi: 10.3390/s21020665

Table 2.

The system notations and abbreviations.

Notations	Descriptions
BSs	Base Stations
CA	Certificate Authority
ECC	Elliptic Curve Cryptography
$L p i d_{i}$	a list of pseudo identities $(p i d_{i}_{1}, p i d_{i}_{2}, p i d_{i}_{3} \dots, p i d_{i}_{n})$
$L s k_{i}$	list of secret keys $(s k_{i}_{1}, s k_{i}_{2}, s k_{i}_{3}, \dots, s k_{i}_{n})$
$L p k_{i}$	a list of the corresponding public keys $(p k_{i}_{1}, p k_{i}_{2}, p k_{i}_{3}, \dots, p k_{i}_{n})$
$L_{Z I D}$	Zone of ID list
$s k_{i}_{Z I D}$	A random integer number represents a secret key of vehicle $v_{i}$ in each zone
$F_{p}$	Finite field of elements in the range ${1 a n d p - 1}$ .
$G_{1}, G_{2}$	two cyclic additive groups of prime order $p$ based on the elliptic curve $E$ over the finite field $F_{p}$ where $G_{1} \times G_{2}$ → $G_{T}$
$G_{T}$	Acyclic multiplicative group containing the bilinear pairing result of the two groups $G_{1}, G_{2}$ .
$e$	The bilinear pairing function that maps elements from group $G_{1}$ and group $G_{2}$ to group $G_{T}$
$g_{2}_{Z I D}$	Represents the generator point of the group $G_{2}$ for each zone
$p k_{i}_{Z I D}$	An ECC point represents the public key of vehicle $v_{i}$ in each zone: $s k_{i}_{Z I D} . g_{2}_{Z I D}$
$H$	A cryptographic hash function that maps a message to a point in the group $G_{1}$
$p i d_{i}$	Represents the pseudo-identity of $v_{i}$ to hide it’s real identity and allow vehicle to communicate anonymously
$m_{i}$	Represents message payload transmitted from vehicle $v_{i}$
$T_{s}$	Is a timestamp to ensure message freshness
$n$	The number of base stations
$N$	The number of zones
H(m)	Hashed message to a point over the elliptic curve group $G_{1}$
$σ_{i}$	Generate a signatureover message m using the secret key of each vehicle
$\| \|$	Represents the concatenation operation of two elements
$g_{2}_{a g g r}$	Represents the aggregation of zones generators ( $g_{2}_{Z I D})$ to generate a new value $\in G_{2} : {g_{2}}_{1} + {g_{2}}_{2} + \dots + {g_{2}}_{N}$
$s k_{a g g r}$	Represents the aggregation of vehicle $v_{i}$ secret keys for different zone destinations $s k_{1} + s k_{2} + \dots + s k_{N}$ $\in F_{p}$
$p k_{a g g r}$	Represents the aggregated public key of vehicle $v_{i} : s k_{a g g r} . g_{2}_{a g g r},$ where $p k_{a g g r}$ $\in G_{2}$
$σ_{a g g r}$	The aggregated signature over message m using the aggregated secret key $s k_{a g g r} : s k_{a g g r} . H (m)$ , where $σ_{a g g r} \in G_{1}$
$σ_{C}$	The concatenated signatures that consists of $N$ signatures generated by vehicle $v_{i}$ for different zones destinations: $(σ_{1} \| \| σ_{2}, \dots \| \| σ_{N})$
$p k_{C}$	The concatenated public keys that consists of $N$ public keys of vehicle $v_{i}$ for different zones destinations: $(p k_{1} \| \| p k_{2}, \dots \| \| p k_{N})$