Striking a balance between protecting data privacy and enabling collaborative computation is a critical challenge for distributed machine learning. While privacy-preserving techniques for federated learning have been extensively developed, methods for scenarios involving bitwise operations, such as tree-based vertical federated learning (VFL), are still underexplored. Traditional mechanisms, including Shamir's secret sharing and multi-party computation (MPC), are not optimized for bitwise operations over binary data, particularly in settings where each participant holds a different part of the binary vector. This paper addresses the limitations of existing methods by proposing a novel binary multi-party computation (BiMPC) framework. The BiMPC mechanism facilitates privacy-preserving bitwise operations, with a particular focus on dot product computations of binary vectors, ensuring the privacy of each individual bit. The core of BiMPC is a novel approach called Dot Product via Modular Addition (DoMA), which uses regular and modular additions for efficient binary dot product calculation. To ensure privacy, BiMPC uses random masking in a higher field for linear computations and a three-party oblivious transfer (triot) protocol for non-linear binary operations. The privacy guarantees of the BiMPC framework are rigorously analyzed, demonstrating its efficiency and scalability in distributed settings.