Peer-to-Peer (P2P) energy trading, facilitated by prosumers who both produce and consume energy, provides a new type of for energy trading. Prosumers generate renewable energy in various environments, from industrial to residential. Traditional centralized energy trading methods pose risks, such as single point of failure and security issues. In contrast, de-centralized energy trading methods that use blockchains provide high security and reliability. However, the blockchain technology is not without limitations; in particular, the blockchain-based authentication mechanisms face three limitations, namely, they do not fully protect prosumer privacy due to unencrypted transactions, they are susceptible to multiple security attacks, and their authentication processes demand high computational and communication resources. To address these limitations, this paper proposes a novel Privacy-Preserving Mutual Authentication (PPMA) mechanism for P2P energy trading in smart grid networks. By employing Elliptic Curve Cryptography (ECC), symmetric encryption, and hash functions, the PPMA mechanism provides secure, privacy-preserving, and cost-effective mutual authentication for prosumers in P2P energy trading. When integrated with a permissioned blockchain and smart contract, PPMA aims to facilitate secure and scalable P2P energy trading. The efficacy of the PPMA mechanism is evaluated through comprehensive security and cost analyses.