Fully Secure Functional Encryption for Inner Products, from Standard Assumptions

Functional encryption is a modern public-key paradigm where a master secret key can be used to derive sub-keys $SK_F$ associated with certain functions $F$ in such a way that the decryption operation reveals $F(M)$, if $M$ is the encrypted message, and nothing else. Recently, Abdalla et al. gave simple and efficient realizations of the primitive for the computation of linear functions on encrypted data: given an encryption of a vector $\vec{y}$ over some specified base ring, a secret key $SK_{\vec{x}}$ for the vector $\vec{x}$ allows computing $\langle \vec{x} ,\vec{y} \rangle$. Their technique surprisingly allows for instantiations under standard assumptions, like the hardness of the Decision Diffie-Hellman (DDH) and Learning-with-Errors (LWE) problems. Their constructions, however, are only proved secure against selective adversaries, which have to declare the challenge messages $M_0$ and $M_1$ at the outset of the game. In this paper, we provide constructions that provably achieve security against more realistic adaptive attacks (where the messages $M_0$ and $M_1$ may be chosen in the challenge phase, based on the previously collected information) for the same inner product functionality. Our constructions are obtained from hash proof systems endowed with homomorphic properties over the key space. They are (almost) as efficient as those of Abdalla et al. and rely on the same hardness assumptions. In addition, we obtain a solution based on Paillier\u27s composite residuosity assumption, which was an open problem even in the case of selective adversaries. We also propose $\LWE$-based schemes that allow evaluation of inner products modulo a prime $p$, as opposed to the schemes of Abdalla et al. that are restricted to evaluations of integer inner products of short integer vectors. We finally propose a solution based on Paillier\u27s composite residuosityassumption that enables evaluation of inner products modulo an RSA integer $N = pq$. We demonstrate that the functionality of inner products over a prime field is powerful and can be used to construct bounded collusion FE for all circuits