## 1 Introduction
To address the aforementioned issues, the methods commonly used to enhance privacy preservation in federated learning are primarily divided into homomorphic encryption [9; 10; 11], differential privacy [12; 13], and secure multi-party computation [14; 15; 16]. In addition, it is also important to ensure that the server aggregation results are correct. A malicious federated learning server may return incorrect aggregation results. For example, a malicious federated learning server may intentionally return the aggregation results of the previous round to the user during a certain round of federated learning for the purpose of saving its own computational cost. Therefore, it is of interest to design a federated learning protocol that can both protect user privacy and verify the correctness of the server's aggregation results. Recently, some researchers have proposed the verifiable privacy-preserving aggregation schemes [17; 18; 19; 20; 21]. These schemes either require additional trusted hardware support or have high computational power requirements on the user side, rendering them inappropriate for IoT devices characterized by limited processing capability. The properties that the related work satisfies are shown in Table 1.  
In this paper, we propose a privacy-preserving federated learning data aggregation scheme that is both efficient and verifiable, specifically designed for IoT devices with lower computational and communication costs to accommodate their weaker performance capabilities. The specific contributions of this paper are as follows. * We propose a novel, efficiency-focused federated learning scheme that facilitates privacy-preserving data aggregation, while notably reducing computational and communication costs. One of the distinguishing features of our solution is that it allows users to perform low-cost validation of the aggregation results provided by the server to ensure their integrity and reliability. This neither compromises computational efficiency nor strict privacy measures, making it a secure and effective data aggregation solution. * We also devised a dynamic user management mechanism. This mechanism is constructed to enable the seamless onboarding and departure of participants within the federated learning process. Through its implementation, it ensures that the federated learning process remains robust against user attrition. When some users dropout, the remaining active participants experience minimal disruption in terms of both computational load and communication bandwidth. Despite these departures, the integrity of the aggregated encrypted gradients is preserved, maintaining the overall efficacy of the federated learning process. * We conduct comprehensive tests on numerous real-world datasets and thoroughly analyze accuracy and performance. The outcomes of these tests have been strikingly positive. Our scheme demonstrates a significant reduction in computational and communication costs, achieving at least 85% and 47%, respectively, when compared to current schemes in existence. Furthermore, our scheme has proven its mettle in terms of accuracy as well. It has managed to attain a level of precision that is highly comparable to those achieved using plaintext baselines, thereby affirming its reliability and robustness. These findings not only highlight the superiority of our scheme but also underscore its potential for practical application in real-world scenarios, particularly where computational resources are limited.