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Several researchers have proposed secure authentication techniques for addressing privacy and security concerns in the fifth-generation (5G)-enabled vehicle networks. To verify vehicles, however, these conditional privacy-preserving authentication (CPPA) systems required a roadside unit, an expensive component of vehicular networks. Moreover, these CPPA systems incur exceptionally high communication and processing costs. This study proposes a CPPA method based on fog computing (FC), as a solution for these issues in 5G-enabled vehicle networks. In our proposed FC-CPPA method, a fog server is used to establish a set of public anonymity identities and their corresponding signature keys, which are then preloaded into each authentic vehicle. We guarantee the security of the proposed FC-CPPA method in the context of a random oracle. Our solutions are not only compliant with confidentiality and security standards, but also resistant to a variety of threats. The communication costs of the proposal are only 84 bytes, while the computation costs are 0.0031, 2.0185 to sign and verify messages. Comparing our strategy to similar ones reveals that it saves time and money on communication and computing during the performance evaluation phase.