# Privacy-Preserving Federated Learning Using Homomorphic Encryption
## 1 Introduction  
Artificial intelligence (AI) is a technology that enables machines to realize human learning and reasoning abilities. This technology has been rapidly advancing and playing a significant role in our daily lives. In AI technology, data acquisition is crucial because AI technologies require model training using a certain amount of data for reliable AI-based services, and the performance of AI-based services is considerably affected by the training data quality. However, there are difficulties in data collection because the data may contain sensitive private information. In order to overcome these difficulties, federated learning (FL), in which training is performed without sharing sensitive local data, has been proposed in [1]. In FL, a centralized server sends a global model for AI learning to many distributed devices, which return local model parameters to the centralized server after training the model with local data. The centralized server updates the global model parameters using the locally trained model parameters from the distributed devices and sends the updated global model parameters to the distributed devices. This procedure is repeated until convergence is achieved. FL has the advantage of preventing the leakage of sensitive private information because it does not require local data sharing. However, recent research has shown that the local data of distributed devices can be leaked through the trained local model parameters, and attackers can exploit this loophole to infer sensitive information on the FL participant in [2, 3].  
Homomorphic encryption (HE) is a technology that enables arithmetic operations on ciphertexts without decryption. [PERSON] et al. utilized an HE scheme to protect local gradients trained with local data in [2]. Using the HE scheme, the centralized servercan update the global model parameters with the encrypted local gradients based on the homomorphic operation. Therefore, the distributed devices participating in FL, which we refer to as clients, do not have to concern about data leakage through local gradients because they deliver encrypted local gradients to the server. However, the clients must share the same private key in the FL-based system because homomorphic operations can only be performed between values encrypted with the same public key. In FL-based systems where many distributed devices, such as smartphones and Internet of Things (IoT) devices participate, the same private key for decryption can be distributed to many clients. Suppose the same private key is shared with many clients. Then, the probability of the private key being leaked or a malicious participant accessing other participants' data increases, which can weaken privacy protections in FL-based systems. As the result, stealing one client's private key can nullify the data privacy protection of all clients participating in FL systems. To overcome this vulnerability, this paper proposes a privacy-preserving federated learning (PPFL) algorithm that allows a cloud server to update global model parameters by aggregating local parameters encrypted by different HE keys in the same FL-based system using homomorphic operations based on a distributed cryptosystem.  
This paper is organized as follows. In Section 2, we present related works on FL and the privacy issues of FL. Section 3 describes the preliminaries for understanding the FL algorithm and the cryptosystem for homomorphic operations. In Section 4, we describe the system and attack models for the proposed PPFL algorithm. Next, Section 5 explains the proposed PPFL algorithm using the distributed cryptosystem based on an additive homomorphic encryption (AHE) scheme. Afterwards, Section 6 presents a theoretical analysis of the proposed PPFL algorithm, and Section 7 presents experimental results to verify the performance of the proposed PPFL scheme. Finally, Section 8 concludes the paper.