Functional Encryption & Inner-Product Predicates Studio

Architect fine-grained, non-all-or-nothing cryptographic access control. Simulate DDH-based Inner-Product Functional Encryption (IPFE), derive function-specific decryption keys sky, and evaluate dot products ⟨x, y⟩ with mathematical zero leakage.

The Decisional Diffie-Hellman (DDH) Construction

The Abdalla-Bourse-De Caro-Pointcheval (ABDP15) scheme provides efficient IPFE under standard DDH assumptions:

• Public Generators: Two generators $g, h in mathbb{G}$ with unknown relative discrete logarithm.
• Master Secret: Vector $s = (s_1, ldots, s_d) in mathbb{Z}_p^d$. Public key $h_i = g^{s_i}$.
• Encryption: For randomness $r in mathbb{Z}_p$, the ciphertext consists of $C_0 = g^r$ and $C_i = h_i^r cdot h^{x_i} = g^{r cdot s_i} cdot h^{x_i}$.
• Homomorphic Cancellation: Exponentiating each $C_i$ by $y_i$ and dividing by $C_0^{sk_y}$ exactly cancels $g^{r sum s_i y_i}$, leaving $h^{langle x, y angle}$.

Collusion Resistance & Privacy Guarantees

Functional Encryption enforces strict mathematical boundaries against adversarial query exploitation:

• Linear Algebraic Isolation: Possessing $sk_{y_1}$ and $sk_{y_2}$ reveals only $langle x, y_1 angle$ and $langle x, y_2 angle$. The adversary cannot recover $x$ unless they obtain $d$ linearly independent functional keys.
• Discrete Log Bound: Because the output appears in the exponent $h^{langle x, y angle}$, decryption requires solving a small discrete logarithm via Pollard's rho or baby-step giant-step (feasible when $langle x, y angle < 2^{32}$).
• Zero Untrusted Infrastructure Risk: Cloud inference engines can hold $sk_y$ and evaluate classification models on encrypted user data without ever possessing permission to decrypt raw inputs.