Multi-key Fully Homomorphic Encryption (MKFHE) enables computation between ciphertexts encrypted with different keys. Multi-key bootstrapping is the key techniques to construct TFHE-like MKFHE. In this paper, we proposed two efficient multi-key bootstrapping algorithms under NTRU and (R)LWE assumptions. They are based on a new multi-key blind rotation framework, which uses NTRU-based GSW-like encryption to perform single-key blind rotation and extends it to multi-key blind rotation by re-encryption and key-switching. Compared to the state of the art, our methods achieve significant gains in both runtime and key size efficiency. Concretely, for the number of keys \(k = 16\) at the same security level, our method is 2.5-9.0 times faster than existing methods and reduces key size by a factor of up to 17.