Vehicular ad-hoc networks (VANETs) are wireless-equipped vehicles that form networks along the road. The security of this network has been a major challenge. The identity-based cryptosystem (IBC) previously used to secure the networks suffers from membership authentication security features. This paper focuses on improving the detection of intruders in VANETs with a modified identity-based cryptosystem (MIBC). The MIBC is developed using a non-singular elliptic curve with Lagrange interpolation. The public key of vehicles and roadside units on the network are derived from number plates and location identification numbers, respectively. Pseudo-identities are used to mask the real identity of users to preserve their privacy. The membership authentication mechanism ensures that only valid and authenticated members of the network are allowed to join the network. The performance of the MIBC is evaluated using intrusion detection ratio (IDR) and computation time (CT) and then validated with the existing IBC. The result obtained shows that the MIBC recorded an IDR of 99.3% against 94.3% obtained for the existing identity-based cryptosystem (EIBC) for 140 unregistered vehicles attempting to intrude on the network. The MIBC shows lower CT values of 1.17 ms against 1.70 ms for EIBC. The MIBC can be used to improve the security of VANETs.