Skyline queries is an advanced data mining algorithm suitable for multi-criteria decision-making scenarios (i.e., medical pre-diagnosis). Privacy-preserving skyline queries schemes are usually constructed by certain methods of cryptography such as additive homomorphic cryptosystem, secret sharing technology, etc. Interestingly, these secure skyline queries schemes require that skyline computations do not reveal any message details, including encrypted inter-tuple domination relations, among which privacy schemes based on homomorphic cryptosystems are the most popular due to their strong security. However, existing secure skyline queries schemes not only suffer from low computational efficiency, but also do not have sufficient security for privacy-key management in the system. To address the above issues, this paper designs an efficient and privacy-preserving skyline queries over encrypted data under a blockchain-based audit architecture. Firstly, we propose a blockchain-based audit architecture that not only provides error auditing functionality but also makes our scheme suitable for (distributed) multi-user scenarios while providing secure key management in the system. Secondly, we implement a series of secure sub-protocols using the CRT-Based Paillier encryption algorithm and construct a privacy sparse matrix elimination protocol to reduce the size of the dataset, leading to a significant reduction in computational cost without compromising privacy. Finally, we put forward our secure skyline queries protocol and prove its security. The performance evaluation shows that our proposed method our proposed method is significantly more efficient (at least 7.4 times faster) compared to current methods.

Fully Homomorphic Encryption (FHE) provides a good solution to directly operate on the ciphertext, and the decryption result is equivalent to the corresponding operation on the plaintext. As a technique suitable for distributed environments, multi-key Fully Homomorphic Encryption (MKFHE) scheme is the most common variant of the FHE scheme since it allows encrypted data to be computed under different keys. Unfortunately, the existing dynamic MKFHE schemes based on learning with errors (LWE) still suffer from the inefficiency of long public keys, which typically grow cube in size along the lattice dimension. Moreover, there current constructions fail to provide reliable and fast algorithms to simultaneously expand ciphertexts with multiple additional keys. In order to solve the above problems, a new faster dynamic MKFHE scheme with shorter public key in asymmetric key setting from LWE is proposed in this paper, in which the size of the public key is further reduced from $\tilde O\left( {{n^3}{{(K + L)}^2}} \right)$ to $\tilde O\left( {{n^2}{{(K + L)}^2}} \right)$. In addition, our scheme cleverly adopts the dual-user cooperation method in distributed system to realize the ciphertext expansion locally, thereby reducing the computing overhead of the cloud server. More interestingly, we design a flexible parallel ciphertext expansion algorithm for the first time based on the basic algorithm. This algorithm realizes the ciphertext expansion when multiple keys are added at the same time, thus significantly improving the computational efficiency of ciphertext expansion in the dynamic MKFHE scheme. Finally, the CPA-secure of our scheme based on standard LWE assumptions is proven.