Fully Homomorphic Encryption ($\mathsf {FHE}$) allows computing over encrypted data without decrypting the corresponding ciphertexts, and it constitutes a promising cryptographic primitive to preserve data privacy in the big data computing environments. In general, $\mathsf {FHE}$ schemes can be constructed by using the standard Learning with Errors ($\mathsf {LWE}$) assumption, and the current crux lies in how to achieve efficient multi-bit $\mathsf {FHE}$ encryption while being leakage-resistent against attackers who may capture the information of cryptographic secret keys via side channel attacks. Based on Berkoff-Liu’s work at TCC’14, we aim to address this issue by giving a new structure of public key matrix with any number of $\mathsf {LWE}$ instances, thereby avoiding the use of a straightforward composition to achieve multi-bit $\mathsf {FHE}$ encryption under standard $\mathsf {LWE}$. Particularly, our scheme attains provable security.