ABE ⇒ Fuzzy IBE

Fully-black-box reduction · standard model · GPSW06 · security loss: none (the reduction preserves the advantage)

Statement

If a KP-ABE scheme whose policy class contains the threshold gates is KP-IND-CPA-secure, the following Fuzzy IBE scheme with threshold $t$ is IND-FIBE-CPA-secure, with the same advantage: the key for attribute set $\omega$ is the ABE key for the policy $[|x\cap \omega |\ge t]$, a single $t$-of-$|\omega |$ threshold gate over $\omega$, and encryption under ${\omega }^{\prime }$ is ABE encryption under attribute set ${\omega }^{\prime }$, so decryption succeeds iff $|\omega \cap {\omega }^{\prime }|\ge t$. Fuzzy IBE is KP-ABE restricted to single-threshold-gate policies — GPSW06.

Sketch

A Fuzzy IBE adversary is verbatim a KP-ABE adversary: $|\omega \cap {\omega }^{\prime }|\ge t$ is $f(x)=1$ for $f$ the threshold gate over $\omega$ and $x={\omega }^{\prime }$, so Fuzzy IBE admissibility ($|\omega \cap {\omega }^{\ast }|<t$ for every queried $\omega$) is KP-ABE admissibility (no queried policy satisfied by the challenge attribute set).