Abstract. Gentry, Sahai and Waters recently presented the first (leveled) identity-based fully homomorphic (IBFHE) encryption scheme (CRYPTO 2013). Their scheme however only works in the single-identity setting; that is, homomorphic evaluation can only be performed on ciphertexts created with the same identity. In this work, we extend their results to the multi-identity setting and obtain a multi-identity IBFHE scheme that is selectively secure in the random oracle model under the hardness of Learning with Errors (LWE). We also obtain a multi-key fullyhomomorphic encryption (FHE) scheme that is secure under LWE in the standard model. This is the first multi-key FHE based on a wellestablished assumption such as standard LWE. The multi-key FHE of López-Alt, Tromer and Vaikuntanathan (STOC 2012) relied on a nonstandard assumption, referred to as the Decisional Small Polynomial Ratio assumption.

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