## 1 Introduction  
Fully homomorphic encryption is an ideal privacy protection scheme for the cloud environment. It can compute encrypted data without a secret key, and the resulting ciphertext can be decrypted correctly. This allows the cloud server to complete the processing of the data without disclosing sensitive information, while protecting the users' data and privacy security and making full use of the cloud server's computing power. At present, fully homomorphic encryption technology has been widely used in deep learning [1, 2], secure multiparty computation [3, 4], and other fields. The first fully homomorphic encryption scheme was proposed by [PERSON] [5], which was constructed based on the bounded distance decoding problem (BDDP) and sparse subset sum problem (SSSP) on the ideal lattice. Firstly, an encryption scheme that can perform finite degree homomorphic evaluation is constructed. Then, the noise growth is controlled using \"bootstrapping\" technology to achieve an arbitrary degree of homomorphic evaluation. [PERSON] and [PERSON] [6] proposed a variant of [PERSON]'s scheme [5] SV10 with a relatively small key and ciphertext size. Then, [PERSON] and [PERSON] [7] optimized the key generation process for SV10, but the limited condition of the finite field made the key generation algorithm very complex. The first-generation FHE (fully homomorphic encryption) scheme based on the ideal lattice has many disadvantages: (1) the complex construction; (2) the low efficiency of the encryption and decryption processes; (3) the security proof of the SSSP assumption is not sufficient. The construction of the second-generation FHE scheme is based on the LWE(learning with errors) or RLWE (ring LWE) assumption [8; 9], whereby the security can be reduced to standard problems in the lattice without additional assumptions. In 2011, [PERSON] and [PERSON] [10] first used the LWE problem to construct the BV11 scheme, which uses a re-linearization method to achieve fully homomorphic encryption, breaking the construction framework used by [PERSON]. In 2012, [PERSON] et al. [11] constructed the BGV12 scheme. They sorted out the re-linearization technology in the BV11 scheme and proposed the key switching technology and module switching technology schemes. They constructed a leveled FHE scheme with these technologies. Then, [PERSON] [12] constructed a scale-invariant FHE scheme using tensor product technology, which no longer requires the module switching technology to compress the scheme noise. Later, [PERSON] et al. [13] proposed the FV12 scheme to transplant the LWE-based Bra12 scheme to the RLWE problem. Two optimization methods were proposed to solve the problem of the low efficiency of the RLWE-based key switching technology. However, they rely on the circular security assumption without changing the key after re-linearization, which risks leaking the key. The second-generation FHE scheme is simple and efficient. However, bootstrapping technology is needed to gain an unbound number of computations for the ciphertext. The improvement of the bootstrapping process is one of the current research directions [14; 15] because of its low efficiency. The leveled fully homomorphic encryption scheme can evaluate arbitrary polynomials homomorphically, which can meet the requirements of most applications. The research on the leveled FHE scheme mainly focuses on how to reduce the ciphertext size effectively [16], control the noise growth after the homomorphic evaluation [17], and explore new methods to construct the FHE scheme [18]. The key switching technology is an important technology scheme used to control the ciphertext dimensions, and its optimization is also an important direction to improve the leveled fully homomorphic encryption scheme. Unlike the optimization of the above fully homomorphic encryption self-construction technology, researchers have tried to make breakthroughs in parallel computing. Under the construction of the first-generation fully homomorphic scheme, [PERSON] et al. [19] improved the SV10 scheme.