No reduction from IND-sHIBE-CPA Security (Selective) to IND-HIBE-CPA Security

A reduction of class unstated from IND-sHIBE-CPA Security (Selective) to IND-HIBE-CPA Security would imply a contradiction.

Statement

Migrated verbatim from hierarchical-identity-based-encryption:

In the selective variant, the adversary commits to the challenge identity ${\overrightarrow{id}}^{\ast }$ before $\mathsf{Setup}$ runs. The selective–adaptive separation is known to be strict for HIBE: there is no black-box complexity-leveraging argument that avoids an exponential loss in the depth $d$.

Notes

source: folklore: the claim carried no citation on the page it was migrated from, and none was invented.

Recorded during migration and not fixed — these are claims about the source text, not changes to it:

• No citation for a strong claim (‘the selective-adaptive separation is known to be strict for HIBE’).
• SUSPECT: conflates two different statements - (i) a strict separation between selective and adaptive HIBE security, and (ii) the loss of complexity leveraging. Complexity leveraging for HIBE loses a factor exponential in d*log|Sigma| (identity-vector length), not merely ‘exponential in the depth d’. Report only; do not fix.
• ‘hibe-selective-security’/‘hibe-adaptive-security’ are security notions, not wiki pages - no slug exists.