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.

A characterization of predicate encryption (PE) with support for homomorphic operations is presented and we describe the homomorphic properties of some existing PE constructions. Even for the special case of IBE, there are few known group-homomorphic cryptosystems. Our main construction is an XOR-homomorphic IBE scheme based on the quadratic residuosity problem (variant of the Cocks' scheme), which we show to be strongly homomorphic. We were unable to construct an anonymous variant that preserves this homomorphic property, but we achieved anonymity for a weaker notion of homomorphic encryption, which we call non-universal. A related security notion for this weaker primitive is formalized. Finally, some potential applications and open problems are considered.

Abstract. It has been an open problem for a number of years to construct an identity-based fully homomorphic encryption (IBFHE) scheme (first mentioned by Naccache at CHES/CRYPTO 2010). At CRYPTO 2013, Gentry, Sahai and Waters largely settled the problem by presenting leveled IBFHE constructions based on the Learning With Errors problem. However their constructions are not bootstrappable, and as a result, are not "pure" IBFHE schemes. The major challenge with bootstrapping in the identity-based setting is that it must be possible to non-interactively derive from the public parameters an "encryption" of the secret key for an arbitrary identity. All presently-known leveled IBFHE schemes only allow bootstrapping if such an "encryption" of the secret key is supplied out-of-band. In this work, we present a "pure" IBFHE scheme from indistinguishability obfuscation, and extend the result to the attribute-based setting. Our attribute-based scheme is the first to support homomorphic evaluation on ciphertexts with different attributes. Finally, we characterize presently-known leveled IBFHE schemes with a view to developing a "compiler" from a leveled IBFHE scheme to a bootstrappable IBFHE scheme, and sufficient conditions are identified.

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