Tightly Secure Inner Product Functional Encryption: Multi-input and Function-Hiding Constructions

Abstract

Tightly secure cryptographic schemes have been extensively studied in the fields of chosen-ciphertext secure public-key encryption, identity-based encryption, signatures and more. We extend tightly secure cryptography to inner product functional encryption (IPFE) and present the first tightly secure schemes related to IPFE. We first construct a new IPFE scheme that is tightly secure in the multi-user and multi-challenge setting. In other words, the security of our scheme does not degrade even if an adversary obtains many ciphertexts generated by many users. Our scheme is constructible on a pairing-free group and secure under the matrix decisional Diffie-Hellman (MDDH) assumption, which is the generalization of the decisional Diffie-Hellman (DDH) assumption. Applying the known conversions by Lin (CRYPTO 2017) and Abdalla et al. (CRYPTO 2018) to our scheme, we can obtain the first tightly secure function-hiding IPFE scheme and multi-input IPFE (MIPFE) scheme respectively. Our second main contribution is the proposal of a new generic conversion from function-hiding IPFE to function-hiding MIPFE, which was left as an open problem by Abdalla et al. (CRYPTO 2018). We obtain the first tightly secure function-hiding MIPFE scheme by applying our conversion to the tightly secure function-hiding IPFE scheme described above. Finally, the security reductions of all our schemes are fully tight, which means that the security of our schemes is reduced to the MDDH assumption with a constant security loss.