Multi-input Quadratic Functional Encryption from Pairings

We construct the first multi-input functional encryption (MIFE) scheme for quadratic functions from pairings. Our construction supports polynomial number of users, where user $i$, for $i \in [n]$, encrypts input $\bfx_i \in \mbZ^m$ to obtain ciphertext $\ct_i$, the key generator provides a key $\sk_\bfc$ for vector $\bfc \in \mbZ^{({mn})^2}$ and decryption, given $\ct_1,\ldots,\ct_n$ and $\sk_\bfc$, recovers $\ip{\bfc}{\bfx \otimes \bfx}$ and nothing else. We achieve indistinguishability-based (selective) security against unbounded collusions under the standard bilateral matrix Diffie-Hellman assumption. All previous MIFE schemes either support only inner products (linear functions) or rely on strong cryptographic assumptions such as indistinguishability obfuscation or multi-linear maps.