Selectively secure HIBE ⇒ Selectively secure IBE

Fully-black-box reduction · standard model · folklore · security loss: none

Statement

If $\mathsf{HIBE}=(\mathsf{Setup},\mathsf{Extract},\mathsf{Delegate},\mathsf{Enc},\mathsf{Dec})$ over alphabet $\Sigma \supseteq \{0,1{\}}^{\lambda }$ is IND-sHIBE-CPA-secure, then the IBE scheme with identity space $\Sigma$ that runs $\mathsf{Setup}(1^{\lambda },1)$ and uses $\mathsf{Extract}$, $\mathsf{Enc}$ and $\mathsf{Dec}$ on identity vectors of length $1$ is IND-sID-CPA-secure, with the same advantage: at maximum depth $1$ an identity vector is a single identity, an identity is a prefix of the challenge identity only if it equals it, and the two selective games coincide — folklore.