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Training face recognition models requires a large amount of identity-labeled face images, which are often collected by crawling the web, and therefore have ethical and privacy concerns. Recently, generating synthetic face datasets and training face recognition models using synthetic datasets has emerged to be a viable solution. This paper presents BIFFace, a new framework to generate synthetic face recognition datasets. We use the Brownian identity diffusion to generate synthetic identities, and then build synthetic face recognition datasets by generating different samples per each identity using a foundation model. In our experiments, we use the generated face datasets to train face recognition models and evaluate them on several real benchmarking dataset. Our experimental results show that face recognition models trained with BIF-Face achieve competitive performance with face recognition models trained on state-of-the-art synthetic face recognition datasets.