## 3 Preliminaries  
Now we will formalize the background knowledge of the building blocks for the proposed scheme. The notations used in the designed algorithm are given and described in Table 1. ECC is a public key cryptosystem based on elliptic curve theory and has an advantage for being a structure for faster and more efficient cryptosystems with robust security. ECC cryptosystems have low computational requirement hence its viable for securing resource constrained network systems that require seamless and real-time operations like the IoT and SG systems [48].  
_Elliptic curve:_ Given a prime number \(q\), equation \(y^{3}=x^{2}+ax+bmodp\) defines an elliptic curve over a prime field \(E(F_{p})\), where \(p>3,a,b\in F_{q}\) and satisfies \(\triangle=4a^{3}+27b^{2}\
eq 0 modp\). The points on \(F_{p}\) together with the point at infinity \(\mathcal{O}\) form an additive cyclic group  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Symbols** & **Meanings of Symbols in the Scheme** \\ \hline \(V_{i}\) & \(i^{th}\) vehicle \\ \(p,\ q\) & Two large primes \\ \(E\) & Is the chosen elliptic curve, \\  & \(y^{3}=x^{2}+ax+bmodp\) where \(a,b\in Z_{q}^{*}\) \\ \(E(F_{p})\) & Is the prime field of an elliptic curve \(E\) order \(p\) \\ \(p\) & Is the generator of \(E(F_{p})\) with large prime order \(q\) \\ \(G\) & A cyclic group generated by a point \(P\) on a non-singular \\  & elliptic curve \(E\) \\ \(1D_{i}\) & A pseudo-identity of \(V_{i}\) such that \(ID=(PID_{1},PID_{2},T_{i})\) \\ \(psk_{i}\) & Partial private key for a vehicle, \(V_{i}\) \\ \((x_{i},x_{i}p)\) & Secret key and public key for \(V_{i}\) \\ \(sk_{i}\) & Full private key for \(V_{i}\) \\ \(T_{i}\) & Validity period for the pseudo-identity \(ID_{i}\) for \(V_{i}\) \\ \(RID_{i}\) & A real identity for the vehicle \(V_{i}\) \\ \((P_{pub},a)\) & KGC’s public key and master key respectively \\ \((T_{pub},\beta)\) & TRA’s public key and master key respectively \\ \(M_{i}\) & Traffic-related message generated by \(V_{i}\) \\ \(t_{i}\) & Current timestamp \\ \(H_{1},\ H_{2},\ H_{3}\) & Hash function: \(H_{1},\ H_{2}:\{0,1\}^{*}\rightarrow\ Z_{q}^{*}\) \\ \(\oplus\) & Exclusive-OR operation \\ \(||\) & concatenation \\ \hline \hline \end{tabular}
\end{table}
Table 1: Notations Used in the Proposed Scheme.  
\(G\). Let \(P\) be the generator point of order \(n\), the scalar multiple operation is defined as, \(nP=P+P+\cdots+P\), \(n\) times addition, where \(n\in Z_{q}^{*}\), is a positive integer. So, there are a number of intractable problems in an elliptic curve group \(G\) of order \(n\), suitable for cryptographic purposes as there is no polynomial algorithm to solve them efficiently by brute-force within probabilistic polynomial time.  
_Elliptic Discrete Logarithm (ECDL) Problem_: Given an element \(Q\in G\), the ECDL problem is to extract an element \(x\in Z_{q}^{*}\), such that \(Q=xP\).  
_Elliptic Curve Computational Diffie-Hellman (ECCDH) Problem_: Given two elements \(xP,yP\in G\), with unknown elements \(x,y\in Z_{q}^{*}\), the ECCDH problem is to compute \(Q=xyP\).  
_Elliptic Curve Decisional Diffie-Hellman (ECDDH) Problem_: No any probabilistic polynomial time algorithm can distinguish between the tuples \((P_{1},xP_{1},yP_{1},T)\) and \((P_{1},xP_{1},yP_{1},xyP_{1})\) where \(P_{1},T\in G\), with unknown elements \(x,y\in Z_{q}^{*}\).