Fully homomorphic encryption allows valid operation on encrypted data without decrypting, providing a new solution to data confidentiality and privacy protection. However, current fully homomorphic encryption schemes are faced with challenges like large size of public key or low efficiency in calculation. To achieve an efficient fully homomorphic encryption scheme, this work provides an identity-based fully homomorphic encryption scheme employing the idea of eigenvector and arbitrary cyclotomic rings. Compared with existing scheme, this identity-based fully homomorphic encryption with eigenvector is able to successfully avoid the evaluation key, resulting a true identity-based scheme. Compared with special cyclotomic rings whose degree is power of 2, utilizing arbitrary cyclotomic rings may double the efficiency of encryption schemes and further improve the efficiency of calculation and memory using SIMD technique.