## 1 Introduction  
Wireless sensor networks (WSNs), which can have hundreds or thousands of sensor nodes deployed over a monitored region, are being increasingly used in several applications such as military reconnaissance, target tracking, environmental monitoring, and medical monitoring [1; 2; 3; 4]. In these large-scale wireless sensor networks (LWSNs), the amount of data processed is very large and the data redundancy is very high, but the sensing devices (nodes) in LWSNs have limited computing power and communication resources, leading to network performance degradation or collapse once the nodes' limited power is depleted [5]. As a result, energy saving technologies must be considered. In-network data aggregation can reduce communication overhead and hence reduce the energy consumption, especially in LWSNs. Since many sensor nodes are deployed in hostile environments, data aggregation faces many security issues such as data confidentiality and integrity [6], and various attacks like selective forwarding attack, replay attack, false data injection _etc._[7; 8; 9]. When sensor nodes are compromised, it is easy for the adversary to change the aggregation result and inject false data into the LWSNs. Hence, data aggregation protocols in LWSNs should be designed with security in mind while achieving energy efficiency, even in the presence of malicious nodes in networks. The standard method to preserve confidentiality is to encrypt the data. Secure data aggregation protocols can be categorized as hop-by-hop encryption and end-to-end encryption [10; 11]. The hop-by-hop secure data aggregation protocols cannot provide data confidentiality at aggregators because the aggregators are required to share keys with their neighboring nodes [8; 9]. In end-to-end secure data aggregation protocols, intermediate nodes aggregate data directly without decrypting the received data. When they are captured, an adversary cannot get the original information. The most common method, named privacy homomorphic cryptography, has been studied for data aggregation in WSNs to achieve end-to-end confidentiality [12; 13; 14; 15]. However, this method only deals with limited arithmetic operations on ciphertexts because the noise contained in ciphertexts would become larger with successive homomorphic multiplications and make the decryption fail. An optimized method which can evaluate any function on encrypted data is fully homomorphic encryption (FHE) [16; 17; 18]. To the best of our knowledge, this is the first paper that combines the concept of FHE with secure data aggregation instead of using privacy homomorphism in WSNs. Confidentiality itself is not enough since an adversary is still able to add some fragments or falsify content to change the data although it knows nothing about the data. For this reason, Message Authentication Code (MAC) protocols are often used to detect false data and protect data integrity [14; 15]. However, these protocols only detect the false data in BS, which might lead to large communication overhead. To solve the problems mentioned above, this paper presents a secure data aggregation with fully homomorphic encryption for LWSNs. Our scheme can provide false data detection and secure data aggregation against up to \(T\) compromised sensor nodes. In particular, the main contributions of this paper may be summarized as follows:  
* To address the drawbacks of privacy homomorphic cryptography, we focus on the investigation of achievable FHE for end-to-end data confidentiality in LWSNs. The designed FESA can be implemented in sensor nodes, by which aggregators can do unlimited arithmetic aggregation functions on ciphertexts. * In order to detect false data during both data forwarding and aggregation processes as early as possible, we propose MFN-group network structure which consists of monitoring node, forwarding node and neighboring node of aggregator.