## 1 Introduction
[PERSON] et al. [6] proposed a privacy-preserving, multi-dimensional metering aggregation scheme in a neighborhood-wide grid with piallier encryption, bilinear pairing and computational Diffie-Hellman (DH) methods. For resisting against internal attackers possessing private keys, [PERSON] [8] introduced a spatial and temporal aggregation and authentication scheme by randomizing Paillier encryption with Lagrange interpolation. Their protocol requires \(O(n^{2})\) bytes of inter-action between the individual meters as well as relatively expensive cryptography on the meters (public key encryption). [PERSON] [9] also improved Paillier encryption and proposed a privacy-preserving aggregation scheme resisting at most \(t\) compromised servers in a control center with threshold protocol. [PERSON] et al. [20] provided a verifiable publicly aggregation scheme against dishonest users that attempt to provide fraudulent data. Any user node in the community can prove its computation accuracy by zero-knowledge proof that the two encrypted message with different public keys corresponding to the same plaintext message. While we can prove our scheme costs lower overhead to resist fraudulent report from internal nodes. [PERSON] et al. [23] adopted a stream cipher (e.g. RC4) to generate pseudo-random keys as masking keys between nodes to prevent internal nodes from possessing private keys. During the aggregation within a neighborhood, all masking random keys cancelled out and the aggregation value is revealed without compromising individual privacy based on the security properties of the Paillier encryption and stream cipher. We follow its Pseudo-Random Function and combine it with modular addition. The main difference from ours is they impose the random keys from PRF on the plaintext before encrypting it with Paillier cryptography, and send the encrypted message to all nodes. We set a security parameter \(k\) to represent the number of communicate nodes in a neighborhood and improve the encryption method by replacing the costly Paillier encryption with the simple and lightweight combination. More significantly, we supplement a publicly verifiable property to detect the fraudulent profile from malicious aggregators or dishonest user nodes. [PERSON] et al. [19] protected individual data by imposing masking keys from RC4 on the plaintext data under the multi-level wireless sensors network model, However, the protection protocol cannot resist malicious aggregators, as the session keys are generated by the sink as the aggregator. We extend its PRF method into the peer-to-peer system model and propose a privacy-preserving scheme against maliciously internal attack. In addition, traditional modular addition was adopted in [7; 24] by partitioning individual plaintext data into \(n\) shares and exchanging them between nodes (\(n\) is the number size of users in a neighborhood). [PERSON] et al. [7] adopted Paillier encryption and modular addition, in which every user node partitions its meter reading into \(n\) shares and transmits the encrypted shares with different public keys to the aggregator, which aggregates the data with the same public key before sending the aggregation to the users. Finally, the aggregator collects the plaintext sums to obtain the final aggregation. The method is privacy-preserving; however, during each spatial aggregation, three message exchanges are required between every user and the aggregator. Thus, the number of homomorphic encryption per user increases linearly with \(n\) increases, and the communication overhead is \(O(n^{2})\) messages [20]. [PERSON] et al. [24] also generated partitioned data with modular addition and imposed them on a high-order polynomial coefficient. The values of the polynomial at different points are transmitted to the aggregator which finds the coefficients of the polynomial with the private key and gains the aggregation, so the scheme is under the semi-trusted model and the aggregator is trustable. In addition, the computation overhead is relatively higher when \(k\) is increasing.