## 8 Conclusions  
In this paper, we propose an efficient, verifiable, and privacy-preserving data aggregation scheme for federated learning. The scheme features lower computational and communication costs, making it highly resistant to inference attacks and suitable for scenarios involving IoT devices with relatively weak computational and communication capabilities. To effectively achieve privacy protection and ensure the verifiability of aggregated results, we devised a lightweight encryption protocol and leveraged fog nodes to mitigate the computational and communication costs for users. In addition, we designed a user management mechanism to ensure that the solution proposed in this paper is fault-tolerant. Even if some users drop out due to various issues, the solution proposed in this paper can still be successfully aggregated. Also, users can dynamically join the federated learning process. In the event of worst-case conditions, the incremental computational cost incurred by users remains capped at less than 1 millisecond, and the additional communication cost is maintained at under 0.05 MB. The experiments are conducted on three commonly used datasets in academia and compared with other existing schemes. The experimental results show that the federated learning scheme has high accuracy while protecting the privacy of users' data. In the future, we will consider how to support malicious user detection and prevention.  
Conceptualization, R.S. and L.W.; methodology, R.S. and L.W.; software, R.S.; validation, L.W. and L.Z.; formal analysis, R.S.; writing--original draft preparation, R.S.; writing--review and editing, L.W. and L.Z.; supervision, L.W.; funding acquisition, L.W. All authors have read and agreed to the published version of the manuscript.  
This work is supported in part by National Natural Science Foundation of China under Grant 61972241, Natural Science Foundation of Shanghai under Grant 22 ZR1427100 and Soft Science Project of Shanghai under Grant 23692106700.  
Not applicable.  
All datasets used in this experiment are publicly available datasets at the following URLs: MNIST. [[http://yann.lecun.com/exdb/mnist/](http://yann.lecun.com/exdb/mnist/)]([http://yann.lecun.com/exdb/mnist/](http://yann.lecun.com/exdb/mnist/)) (accessed on 29 April 2024); Fashion-MNIST. [[https://github.com/zalandoresearch/fashion-mnist](https://github.com/zalandoresearch/fashion-mnist)]([https://github.com/zalandoresearch/fashion-mnist](https://github.com/zalandoresearch/fashion-mnist)) (accessed on 29 April 2024); Cifar-10. [[https://www.cs.toronto.edu/~kriz/cifar.html](https://www.cs.toronto.edu/~kriz/cifar.html)]([https://www.cs.toronto.edu/~kriz/cifar.html](https://www.cs.toronto.edu/~kriz/cifar.html)) (accessed on 29 April 2024).  
The authors would like to express their gratitude for the support of Fishery Engineering and Equipment Innovation Team of Shanghai High-level Local University.  
The authors declare that this study was conducted without any commercial or financial relationships that could be considered potential conflicts of interest.  
Figure 9: Communication cost comparison of each user. (a) \(|\mathcal{U}|=100\), communication cost comparison with different numbers of gradients. (b) \(|\mathcal{G}|=1\times 10^{4}\), communication cost comparison with different numbers of users.