ABE ⇒ IBE

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

Statement

Take a KP- or CP-ABE scheme over attribute universe $[\ell ]\times \{0,1\}$ whose policy class contains conjunctions. Encode identity $id\in \{0,1{\}}^{\ell }$ as the attribute set $x_{id}=\{(i,{id}_i){\}}_{i\le \ell }$ and the policy $f_{id}={\bigwedge }_{i\le \ell }(i,{id}_i)$, so $f_{id}(x_{{id}^{\prime }})=1$ iff $id={id}^{\prime }$; KP-ABE carries $f_{id}$ in the key and $x_{id}$ in the ciphertext, CP-ABE the reverse. If the scheme is KP-IND-CPA-secure (resp. CP-IND-CPA-secure), the resulting IBE for identities in $\{0,1{\}}^{\ell }$ is IND-ID-CPA-secure: every IBE adversary is verbatim an ABE adversary with the same advantage — folklore.

Sketch

Distinct identities differ in some position, so a key for $id\ne {id}^{\ast }$ is a policy–attribute pair unsatisfied by the challenge encoding; IBE admissibility (no key for ${id}^{\ast }$) is ABE admissibility.