This paper investigates the problem of minimizing total energy consumption for secure federated learning (FL) in wireless edge networks, a key paradigm for decentralized big data analytics. To tackle the high computational cost and privacy challenges of processing large-scale distributed data with conventional neural networks, we propose an FL with hyperdimensional computing and differential privacy (FL-HDC-DP) framework. Each edge device employs hyperdimensional computing (HDC) for lightweight local training and applies differential privacy (DP) noise to protect transmitted model updates. The total energy consumption is minimized through a joint optimization of the HDC dimension, transmit power, and CPU frequency. An efficient hybrid algorithm is developed, combining an outer enumeration search for HDC dimensions with an inner one-dimensional search for resource allocation. Simulation results show that the proposed framework achieves up to 83.3% energy reduction compared with baseline schemes, while maintaining high accuracy and faster convergence.