Unbounded Predicate Inner Product Functional Encryption from Pairings

Abstract 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 $${\textbf {x}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>x</mml:mi> </mml:math> is encrypted under an attribute $${\textbf {w}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>w</mml:mi> </mml:math> and a secret key is generated for a pair $$({\textbf {y}}, {\textbf {v}})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>y</mml:mi> <mml:mo>,</mml:mo> <mml:mi>v</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> such that recovery of $$\langle {{\textbf {x}}}, {{\textbf {y}}}\rangle $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>⟨</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo>⟩</mml:mo> </mml:mrow> </mml:math> requires the vectors $${\textbf {w}}, {\textbf {v}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>w</mml:mi> <mml:mo>,</mml:mo> <mml:mi>v</mml:mi> </mml:mrow> </mml:math> to satisfy a linear relation. We call a P-IPFE unbounded if it can encrypt unbounded length attributes and message vectors. $$\bullet $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>∙</mml:mo> </mml:math> zero predicate IPFE . We construct the first unbounded zero predicate IPFE (UZP-IPFE) which recovers $$\langle {{\textbf {x}}}, {{\textbf {y}}}\rangle $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>⟨</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo>⟩</mml:mo> </mml:mrow> </mml:math> if $$\langle {{\textbf {w}}}, {{\textbf {v}}}\rangle =0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>⟨</mml:mo> <mml:mi>w</mml:mi> <mml:mo>,</mml:mo> <mml:mi>v</mml:mi> <mml:mo>⟩</mml:mo> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> . 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. $$\bullet $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>∙</mml:mo> </mml:math> non-zero predicate IPFE . We present the first unbounded non-zero predicate IPFE (UNP-IPFE) that successfully recovers $$\langle {{\textbf {x}}}, {{\textbf {y}}}\rangle $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>⟨</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo>⟩</mml:mo> </mml:mrow> </mml:math> if $$\langle {{\textbf {w}}}, {{\textbf {v}}}\rangle \ne 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>⟨</mml:mo> <mml:mi>w</mml:mi> <mml:mo>,</mml:mo> <mml:mi>v</mml:mi> <mml:mo>⟩</mml:mo> <mml:mo>≠</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> . 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.