## 1 Introduction
They used the polynomial CRT to construct the 'plaintext slot' and showed how to select parameters to achieve SIMD operations while maintaining the practicability of the key generation technology [7]. [PERSON], [PERSON], and [PERSON] [20] implemented the homomorphic evaluation of a multi-logarithm overhead in the RLWE-based FHE scheme using the packing technology from SV11. In 2013, [PERSON] et al. [21] proposed a packing method based on the LWE assumption, which can simplify the homomorphic scheme of the data movement on the plaintext slot. However, the asymptotic efficiency is not as good as the packing method based on the RLWE with the polynomial CRT. [PERSON] [22] proposed a multi-bit FHE scheme with a shorter public key size based on the improvement of [21]. Since then, researchers have proposed many methods to improve the parallel efficiency of the fully homomorphic encryption process [23; 24]. Regarding fully homomorphic encryption, many related studies have been focused on RLWE, relying on ring polynomials. Compared to the LWE problem, RLWE has many advantages: (1) the RLWE scheme has a great algebraic structure and simple lattice cipher characteristics; (2) the polynomial product in RLWE can be realized via fast Fourier transformation [25], and the multiplication evaluation is faster than that in LWE; (3) the batch technology based on RLWE is more efficient than the multi-bit technology based on LWE. Therefore, this paper studies how to improve the efficiency and security of fully homomorphic encryption schemes based on RLWE. The key switching technology is proposed to solve the problem of dimension expansion of the ciphertext after homomorphic evaluation, especially after homomorphic multiplication. Its essence is to encrypt the old key with a new key as the evaluation key, which is used as auxiliary information to convert the long ciphertext under the old key into the short ciphertext under the new key. The key replacement is performed to avoid the circular security assumption. The circular security assumption is that it is safe to encrypt the leveled FHE secret key under its public key. Some problems arise when encryption schemes rely on the circular security assumption. On the one hand, to design a scheme that satisfies the circular security, it is necessary to follow a specific structure that will make the scheme complex. On the other hand, the existing schemes [26, 27] that can satisfy the circular security assumption have problems with noncompact ciphertexts and low encryption and decryption efficiency rates. The key switching technology based on LWE compresses the noise introduced from the binary expansion, which is more complex for polynomials. The scheme based on RLWE achieves the compression of the ciphertext dimension using re-linearization technology. The technology also requires the evaluation key to help implement ciphertext updates, but the difference is that the key is not changed after re-linearization. The evaluation key is neither an RLWE instance nor a real encryption of the key, but rather a masking version with noise added. The scheme relies on the weak circular security assumption; that is, the scheme is still secure when the adversary obtains the evaluation key, although the security requires more proof. **Our Contribution. ** We first propose that the secret key \(s\) can be restricted to sampling from the error distribution on the ring via a transformation. This error distribution can be replaced by a smaller distribution with the variable parameter to make \(s\) smaller. Since the essence of the evaluation key is the encrypted secret key, selecting the key from the new distribution reduces the noise of the scheme and the size of the evaluation key. Second, we propose a new key switching technology to use the method of \"modulus scaling\" to reduce the introduced noise, which is simple and efficient. Our key switching technology is as efficient in the RLWE-based scheme as in the LWE-based scheme, without depending on the circular security assumption.