With the development of laws and regulations related to privacy preservation, it has become difficult to collect personal data to perform machine learning. In this context, federated learning, which is distributed learning without sharing personal data, has been proposed. In this paper, we focus on federated learning for user authentication. We show that it is difficult to achieve both privacy preservation and high accuracy with existing methods. To address these challenges, we propose IPFed which is privacy-preserving federated learning using random projection for class embedding. Furthermore, we prove that IPFed is capable of learning equivalent to the state-of-the-art method. Experiments on face image datasets show that IPFed can protect the privacy of personal data while maintaining the accuracy of the state-of-the-art method.