HIBE ⇒ IBE

Fully-black-box reduction · standard model · folklore · security loss: tight (advantage-preserving)

Statement

A HIBE scheme over alphabet $\Sigma \supseteq \{0,1{\}}^{\lambda }$, set up with depth $d=1$ and with $\mathsf{Delegate}$ discarded, is an IBE scheme with identity space $\Sigma$, which contains $\{0,1{\}}^{\lambda }$ as the IBE syntax requires. If the HIBE is IND-HIBE-CPA-secure, the IBE is IND-ID-CPA-secure with the same advantage — folklore.

Sketch

The only prefix of a depth-1 identity vector is itself, so at depth 1 the IND-HIBE-CPA game is the IND-ID-CPA game and every IBE adversary is, unchanged, an admissible HIBE adversary.