Federated learning is an innovative distributed machine learning method that allows for the construction of effective and secure shared models through cooperative learning of data between devices while ensuring data privacy protection. However, current federated learning methods often require uploading some key parameters to a centralized server for model merging, a process that may lead to the risk of privacy leakage. Recently, blockchain technology provides new solutions for the security of federated learning. This paper introduces a lightweight authentication scheme in a blockchain-based federated learning system. Through extensive experimental verification, our method can increase the system performance when the computing capabilities of central nodes and local nodes are unbalanced.