We propose a symmetric key homomorphic encryption scheme based on the evaluation of multivariate polynomials over a finite field. The proposed scheme is somewhat homomorphic with respect to addition and multiplication. Further, we define a generalization of the Learning with Errors problem called the Hidden Subspace Membership problem and show that the semantic security of the proposed scheme can be reduced to the hardness of this problem.

The security of most early fully homomorphic encryption schemes was based on the hardness of the Learning with Errors (LWE) problem. These schemes were inefficient in terms of per gate computations and public‐key size. More efficient schemes were later developed based on the hardness of the Ring‐LWE (RLWE) problem. While the hardness of the LWE problem is based on the hardness of the approximate shortest vector problem (GapSVPγ) over regular lattices, the hardness of the RLWE problem is based on the hardness of the approximate shortest vector problem over ideal lattices. As of now, it has not been proved that the (GapSVPγ) problem over ideal lattices is as difficult as the corresponding problem over regular lattices. In this work, the authors propose a multi‐bit levelled fully homomorphic encryption scheme using multivariate polynomial evaluations whose security depends on the hardness of the LWE problem. In terms of per gate computation cost, this scheme is more efficient than existing LWE‐based schemes. Further, for an appropriate choice of parameters, the per computation cost for homomorphic multiplication can be made asymptotically comparable to RLWE‐based schemes in a parallel computing environment. For homomorphic multiplication, the scheme uses a polynomial‐based technique that does not require relinearization (and key switching).

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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 $${\textbf {x}}$$ x is encrypted under an attribute $${\textbf {w}}$$ w and a secret key is generated for a pair $$({\textbf {y}}, {\textbf {v}})$$ ( y , v ) such that recovery of $$\langle {{\textbf {x}}}, {{\textbf {y}}}\rangle $$ ⟨ x , y ⟩ requires the vectors $${\textbf {w}}, {\textbf {v}}$$ w , 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 $$\langle {{\textbf {x}}}, {{\textbf {y}}}\rangle $$ ⟨ x , y ⟩ if $$\langle {{\textbf {w}}}, {{\textbf {v}}}\rangle =0$$ ⟨ w , 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 $$\langle {{\textbf {x}}}, {{\textbf {y}}}\rangle $$ ⟨ x , y ⟩ if $$\langle {{\textbf {w}}}, {{\textbf {v}}}\rangle \ne 0$$ ⟨ w , v ⟩ ≠ 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.

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