Latent fingerprint identification is one of the most challenging tasks in fingerprint recognition due to poor image quality, small fingerprint areas, and significant deformations. State-of-the-art approaches typically treat the problem as a minutia classification task, requiring the establishment of weak minutiae correspondence labels and relying on deep minutiae descriptors trained on large private datasets. This paper proposes a novel method to develop efficient minutiae descriptors for latent fingerprint identification without using private datasets. Our approach utilizes self-supervised (contrastive) learning to design a similarity function using a Siamese network between pairs of aligned minutiae patches. We generate synthetic latent fingerprints from high-quality rolled and plain images in the public NIST SD 302 database to address the scarcity of real latent fingerprint databases. During the training, we employ the NTXent loss function to maximize the cosine similarity between the embeddings from a minutia patch and those from its synthetic augmented version. This removes the necessity for establishing weak minutiae correspondence labels. Experiments on the NIST SD27 database demonstrate that our method significantly improves identification performance, achieving a 6.59% increase in the hit rate compared to commercial software. This work highlights the potential of self-supervised learning and data augmentation techniques in advancing latent fingerprint identification systems.