IND-ID-CPA Security ⇒ IND-sID-CPA Security (Selective)

Fully-black-box reduction · standard model · folklore · security loss: tight: the two advantages are equal

Statement

Every IND-ID-CPA-secure IBE scheme $\mathsf{IBE}$ is IND-sID-CPA-secure: for every efficient admissible selective adversary $\mathcal{A}$ there is an efficient admissible adaptive adversary $\mathcal{B}$ with ${\mathbf{Adv}}_{\mathsf{IBE},\mathcal{B}}^{\mathrm{id}\text{-}\mathrm{cpa}}(\lambda )={\mathbf{Adv}}_{\mathsf{IBE},\mathcal{A}}^{\mathrm{sid}\text{-}\mathrm{cpa}}(\lambda )$ — folklore.

Sketch

$\mathcal{B}$ runs $\mathcal{A}(1^{\lambda })$ to obtain ${id}^{\ast }$ before passing on $\mathsf{pp}$, forwards $\mathcal{A}$‘s extraction queries, and submits ${id}^{\ast }$ with $\mathcal{A}$‘s messages as its challenge; this simulates the selective game perfectly, and $\mathcal{B}$ queries ${id}^{\ast }$ only if $\mathcal{A}$ does.