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The first identity-based fully homomorphic encryption (IBFHE) scheme was constructed from identity-based encryption (IBE) and lattice-based cryptography by Gentry, Sahai, and Waters in CRYPTO 2013. Their IBFHE scheme is improved in this paper, exploiting Alperin-Sheriff and Peikert's tight and simple noise analysis method when evaluating homomorphically and Micciancio and Peikert's powerful and novel trapdoor. Furthermore, using the masking scheme(Mukherjee and Wichs in EUROCRYPT 2016), we construct an efficient multi-identity fully homomorphic encryption (MIFHE) scheme by expanding a ''fresh'' ciphertext under a single identity key to an ''expanded'' one under a combined key that enables ciphertexts under different identities to be homomorphically evaluated.INDEX TERMS Fully homomorphic encryption, identity-based encryption, identity-based fully homomorphic encryption, learning with errors, multi-identity fully homomorphic encryption.