Fully homomorphic encryption provides a new idea on the solution of many problems

such as the whole life cycle of data privacy protection on cloud computing.Currently

the existing fully homomorphic encryption schemes share a common flaw of large size public keys.We construct an identity-based fully homomorphic encryption which compromises the merits of both kinds of encryption from ring learning with errors to work in arbitrary cyclotomic rings.To make user's identity as the unique public key

our scheme has advantage in computational efficiency and key management.The security of our scheme strictly reduces to hardness of decision ring learning with problem solving in the random oracle model.