Two-input functional encryption for inner products from Bilinear maps

Functional encryption is a new paradigm of public-key encryption that allows a user to compute f (x) on encrypted data CT (x) with a private key SK_f to finely control the revealed information. Multi-input functional encryption is an important extension of (single-input) functional encryption that allows the computation f (x1, . . ., x_n) on multiple ciphertexts CT (x₁), . . ., CT (x_n) with a private key SK_f . Although multi-input functional encryption has many interesting applications like running SQL queries on encrypted database and computation on encrypted stream, current candidates are not yet practical since many of them are built on indistinguishability obfuscation. To solve this unsatisfactory situation, we show that practical two-input functional encryption schemes for inner products can be built based on bilinear maps. In this paper, we first propose a two-input functional encryption scheme for inner products in composite-order bilinear groups and prove its selective IND-security under simple assumptions. Next, we propose a two-client functional encryption scheme for inner products where each ciphertext can be associated with a time period and prove its selective IND-security. Furthermore, we show that our two-input functional encryption schemes in composite-order bilinear groups can be converted into schemes in prime-order asymmetric bilinear groups by using the asymmetric property of asymmetric bilinear groups.