Federated learning (FL) is a collaborative learning paradigm for decentralized private data from mobile terminals (MTs). However, it suffers from issues in terms of communication, resource of MTs, and privacy. Existing privacy-preserving FL methods usually adopt the instance-level differential privacy (DP), which provides a rigorous privacy guarantee but with several bottlenecks: severe performance degradation, transmission overhead, and resource constraints of edge devices such as MTs. To overcome these drawbacks, we propose Fed-LTP, an efficient and privacy-enhanced FL framework with \underline{\textbf{L}}ottery \underline{\textbf{T}}icket \underline{\textbf{H}}ypothesis (LTH) and zero-concentrated D\underline{\textbf{P}} (zCDP). It generates a pruned global model on the server side and conducts sparse-to-sparse training from scratch with zCDP on the client side. On the server side, two pruning schemes are proposed: (i) the weight-based pruning (LTH) determines the pruned global model structure; (ii) the iterative pruning further shrinks the size of the pruned model's parameters. Meanwhile, the performance of Fed-LTP is also boosted via model validation based on the Laplace mechanism. On the client side, we use sparse-to-sparse training to solve the resource-constraints issue and provide tighter privacy analysis to reduce the privacy budget. We evaluate the effectiveness of Fed-LTP on several real-world datasets in both independent and identically distributed (IID) and non-IID settings. The results clearly confirm the superiority of Fed-LTP over state-of-the-art (SOTA) methods in communication, computation, and memory efficiencies while realizing a better utility-privacy trade-off.