Recent cryptographic approaches for private distributed learning, e.g., [119], [42], not only have limited ML functionalities, i.e., regularized or generalized linear models, but also employ traditional encryption schemes that make them vulnerable to post-quantum attacks. This should be cautiously considered, as recent advances in quantum computing [47], [87], [105], [116], increase the need for deploying quantum-resilient cryptographic schemes that eliminate Abstract-In this paper, we address the problem of privacypreserving training and evaluation of neural networks in an N-party, federated learning setting. We propose a novel system, POSEIDON, the first of its kind in the regime of privacy-preserving neural network training. It employs multiparty lattice-based cryptography to preserve the confidentiality of the training data, the model, and the evaluation data, under a passive-adversary model and collusions between up to N − 1 parties. To efficiently execute the secure backpropagation algorithm for training neural networks, we provide a generic packing approach that enables Single Instruction, Multiple Data (SIMD) operations on encrypted data. We also introduce arbitrary linear transformations within the cryptographic bootstrapping operation, optimizing the costly cryptographic computations over the parties, and we define a constrained optimization problem for choosing the cryptographic parameters. Our experimental results show that POSEIDON achieves accuracy similar to centralized or decentralized non-private approaches and that its computation and communication overhead scales linearly with the number of parties. POSEIDON trains a 3-layer neural network on the MNIST dataset with 784 features and 60K samples distributed among 10 parties in less than 2 hours.