Unbounded Predicate Inner Product Functional Encryption from Pairings

Predicate inner product functional encryption (P-IPFE) is essentially attribute-based IPFE (AB-IPFE) which additionally hides attributes associated to ciphertexts. In a P-IPFE, a message $\mathbf{x}$ is encrypted under an attribute $\mathbf{w}$ and a secret key is generated for a pair $(\mathbf{y},\mathbf{v})$ such that recovery of $⟨\mathbf{x},\mathbf{y}⟩$ requires the vectors $\mathbf{w},\mathbf{v}$ to satisfy a linear relation. We call a P-IPFE unbounded if it can encrypt unbounded length attributes and message vectors. $\bullet$zero predicate IPFE. We construct the first unbounded zero predicate IPFE (UZP-IPFE) which recovers $⟨\mathbf{x},\mathbf{y}⟩$ if $⟨\mathbf{w},\mathbf{v}⟩=0$. This construction is inspired by the unbounded IPFE of Tomida and Takashima (ASIACRYPT 2018) and the unbounded zero inner product encryption of Okamoto and Takashima (ASIACRYPT 2012). The UZP-IPFE stands secure against general attackers capable of decrypting the challenge ciphertext. Concretely, it provides full attribute-hiding security in the indistinguishability-based semi-adaptive model under the standard symmetric external Diffie–Hellman assumption. $\bullet$non-zero predicate IPFE. We present the first unbounded non-zero predicate IPFE (UNP-IPFE) that successfully recovers $⟨\mathbf{x},\mathbf{y}⟩$ if $⟨\mathbf{w},\mathbf{v}⟩\ne 0$. We generically transform an unbounded quadratic FE (UQFE) scheme to weak attribute-hiding UNP-IPFE in both public and secret key setting. Interestingly, our secret key simulation secure UNP-IPFE has succinct secret keys and is constructed from a novel succinct UQFE that we build in the random oracle model. We leave the problem of constructing a succinct public key UNP-IPFE or UQFE in the standard model as an important open problem.