Notation and their definition.

Fifth-generation (5G)-enabled vehicular fog computing technologies have always been at the forefront of innovation because they support smart transport like the sharing of traffic data and cooperative processing in the urban fabric. Nevertheless, the most important factors limiting progress are concerns over message protection and safety. To cope with these challenges, several scholars have proposed certificateless authentication schemes with pseudonyms and traceability. These schemes avoid complicated management of certificate and escrow of key in the public key infrastructure-based approaches in the identity-based approaches, respectively. Nevertheless, problems such as high communication costs, security holes, and computational complexity still exist. Therefore, this paper proposes an efficient certificateless authentication called the ECA-VFog scheme for fog computing with 5G-assisted vehicular systems. The proposed ECA-VFog scheme applied efficient operations based on elliptic curve cryptography that is supported by a fog server through a 5G-base station. This work conducts a safety analysis of the security designs to analysis the viability and value of the proposed ECA-VFog scheme. In the performance ovulation section, the computation costs for signing and verification process are 2.3539 ms and 1.5752 ms, respectively. While, the communication costs and energy consumption overhead of the ECA-VFog are 124 bytes and 25.610432 mJ, respectively. Moreover, comparing the ECA-VFog scheme to other existing schemes, the performance estimation reveals that it is more cost-effective with regard to computation cost, communication cost, and energy consumption.