The privacy and security vulnerabilities in fifth-generation (5G)-enabled vehicular networks are often required to cope with schemes based on either bilinear pair cryptography (BPC) or elliptic curve cryptography (ECC). Nevertheless, these schemes suffer from massively inefficient performance related to signing and verifying messages in areas of the high-density traffic stream. Meanwhile, adversaries could launch side-channel attacks to obtain sensitive data protected in a tamper-proof device (TPD) to destroy the system. This paper proposes a Chebyshev polynomial-based scheme for resisting side-channel attacks in 5G-enabled vehicular networks. Our work could achieve both important properties of the Chebyshev polynomial in terms of chaotic and semi-group. Our work consists of five phases: system initialization, enrollment, signing, verification, and pseudonym renew. Moreover, to resist side-channel attacks, our work renews periodically and frequently the vehicle’s information in the TPD. Security analysis shows that our work archives the privacy (pseudonym identity and unlikability) and security (authentication, integrity, and traceability) in 5G-enabled vehicular networks. Finally, our work does not employ the BPC or the ECC; its efficiency performance outperforms other existing recent works, making it suitable for use in vehicular networks.