## 1 Introduction
### Related Work  
The damage that smart meters can cause to user privacy has previously been studied from both a technical and a legal perspective [2; 12]. These studies propose the enforcement of privacy based on organizational means, codes of conduct, and regulations, subject to current legislation. Of the technical solutions that have been offered, several recent schemes have suggested protecting user privacy by aggregating individual metering data at a range of levels [3; 5; 7; 8] or by anonymizing high-frequency metering data through the use of pseudonymous identities [4]. However, the importance of data integrity has not drawn a great deal of attention in previous smart grid security literature, despite it being one of the most important security requirements (along with availability and confidentiality) in smart grid cyber security [13]. [PERSON] _et al._[8] introduced a smart metering privacy model to measure the degree of privacy that a smart metering application can provide, and proposed a privacy-preserving aggregation scheme. The basic principle of the scheme is that each electricity meter takes its current reading and adds a random value drawn from a known distribution with a known variance \(\sigma^{2}\) and expectation \(\mu\). The sent value is then distributed around the sum of the actual measurement and the expectation of the distribution. However, since the measurements are randomized, there is a trade-off between the precision of individual and aggregated readings. With small aggregation groups, it is not easy to achieve the desired level of privacy, which is a major drawback with regard to the practical deployment of this scheme. [PERSON] _et al._[3] suggested a multiparty computation protocol that allows multiple smart meters in a neighborhood to compute a partial aggregation of their data without disclosing their individual measurements by taking advantage of [PERSON]'s additive homomorphic encryption [9]. The partially aggregated data of every meter is then sent to the local substation, which is the collection unit in the neighborhood, and fully aggregated. This scheme requires that each meter compute \(O(N)\) encryptions, and that additional \(O(N^{2})\) communications between each meter and the local substation are made before sending the fully aggregated result to the electricity utility, where \(N\) is the number of smart meters in the neighborhood. Thus, it lacks efficiency in terms of computation and communication overheads. It is also vulnerable to data forgery at the local substation due to the malleability of homomorphic encryption. [PERSON] _et al._[5; 6] proposed a distributed in-network aggregation scheme for smart grid networks. Data aggregation is performed at all smart meters involved in routing the data from the source meter to the collection unit using [PERSON]'s homomorphic encryption [9]. When \(\{m\}_{K}\) represents the encryption of measurement \(m\) under the collection unit's public key \(K\), each intermediate smart meter \(s_{j}\) on the routing path performs \(\{m_{i}\}_{K}\cdot\{m_{j}\}_{K}=\{m_{i}+m_{j}\}_{K}\) and forwards it to the next smart meter,where \(m_{j}\) is the measurement to be sent and \(\{m_{i}\}_{K}\) is the received ciphertext from the previous smart meter \(s_{i}\). The collection unit can then decrypt the final ciphertext with its private key, and obtain the total sum of the measurements from all of the smart meters on the routing path without knowing individual plain measurements. This scheme is efficient in terms of computation and communication cost because the in-network aggregation is integrated into the data routing process. However, this scheme requires additional cryptographic mechanisms to allow the collection unit to verify the integrity of the aggregated data and authenticate the participating meters because intermediate malicious meters or outside adversaries can easily forge intermediate aggregation data en route. [PERSON] _et al._[14] proposed a metering data aggregation scheme using Paillier's homomorphic encryption [9].