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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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0.0031</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2.0185</mn></mrow></semantics></math></inline-formula> 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.