# Privacy-Preserving Data Aggregation with Dynamic Billing in Fog-Based Smart Grid
## 1 Introduction
### Related Works  
Homomorphic encryption techniques are widely employed in privacy-preserving data aggregation schemes due to their homomorphic properties. In this article, we briefly review some data aggregation articles based on homomorphic encryption.  
In traditional homomorphic encryption-based data aggregation schemes, SMs send reports to a gateway node, then the gateway node aggregates all reports and transmits the results to the SP, thus allowing the SP to obtain total electricity consumption data [6]. [PERSON] et al. [4] proposed a multi-dimensional data aggregation scheme in which they used a super-increasing sequence to construct multi-dimensional data and encrypted the structured data using the Paillier cryptosystem. Some researchers constructed multi-subset data aggregation schemes [26; 27] using super-increasing sequences and homomorphic encryption techniques. In multi-subset data aggregation schemes, the SP is able to calculate the total electricity consumption of all users and the number of users in different electricity consumption ranges. However, in [27], the computation overhead at the SM side is high, which is impractical for SMs which have limited computational power. [PERSON] et al. [28] utilized the Paillier cryptosystem and secret sharing techniques to achieve privacy protection and flexible management of users. However, their scheme has high computation overhead since the keys need to be updated at each time slot. [PERSON] et al. [29] proposed a new type of attack, namely, the malicious data mining attack. Considering this attack, they proposed a privacy-preserving data aggregation scheme based on the Paillier cryptosystem.  
Compared with traditional data aggregation architectures, FC-based aggregation architectures can reduce latency and improve bandwidth utilization in time-sensitive applications [30]. So far, researchers have proposed a number of data aggregation schemes based on FC. [PERSON] et al. [6] came up with a multi-dimensional data aggregation scheme. They utilized binary encoding functions to construct multi-dimensional data into one-dimensional data and encrypt the constructed data using the Paillier cryptosystem. Compared with the super-increasing sequence mechanism, their scheme avoids the use of large coefficients and significantly reduces the ciphertext space. [PERSON] et al. [10] suggested a scheme that focuses on both communication aggregation and function query. They utilized the double trapdoor cryptosystem to encrypt data and outsourced the ciphertext to the cloud, so users and the SP can initiate function queries on data in the cloud while protecting privacy. In [11], [PERSON] et al. proposed a data aggregation scheme supporting multi-party computation in which the SP and users can make functional computation queries on the data. [PERSON] et al. [12] achieved privacy protection of data by using the Boneh-Goh-Nissam cryptosystem and authenticated the source of data via the Elliptic Curve Digital Signature Algorithm. [PERSON] et al. [31] submitted a multi-dimensional data aggregation scheme based on fog-cloud architecture, named PP-MDA. In PP-MDA, they employed two outsourced clouds to reduce the computation and storage overhead. [PERSON] et al. [32] added blinding factors to the encrypted data, so in their scheme, the adversary cannot recover individual data even if the symmetric key is compromised.