BDH ⇒ IBE

BDH implies IBE.

Statement

Migrated verbatim from bilinear-map-assumptions § Bilinear map assumptions:

Bilinear map (pairing) assumptions concern the computational hardness of certain problems in groups ${\mathbb{G}}_1,{\mathbb{G}}_2,{\mathbb{G}}_T$ equipped with a bilinear pairing $e:{\mathbb{G}}_1\times {\mathbb{G}}_2\to {\mathbb{G}}_T$, where $e(g_1^a,g_2^b)=e(g_1,g_2{)}^{ab}$ for generators $g_i$. Pairings enable cryptographic primitives not known to be constructible from DDH alone, including identity-based encryption and short signatures.

Migrated verbatim from attribute-based-encryption § Other results:

• The dual system encryption technique of Wat09 gives adaptively secure IBE, HIBE, and ABE under simple pairing-based assumptions — Wat09

Migrated verbatim from bilinear-map-assumptions § Known Results:

• BDH → IBE from the Weil pairing (Boneh-Franklin scheme) — standard

Notes

class: unstated: no citing page says which notion of reduction is meant. Recording a class the wiki does not state would add a claim.

This relation is stated on 3 pages; the statements above are all of them.

Citations disagree across pages: [object Object]

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

• Uncited, and the parent is uncited too (BF01 has no reference page linked here).
• The hypothesis is ‘pairings exist’ rather than a named hardness assumption; bilinear-map-assumptions is a page bundling BDH/BDDH/DLIN, so the node is coarser than the actual hypothesis.
• Model recorded as unstated: the standard Boneh-Franklin IBE is proved in the random-oracle model, which the page never says.
• Two conclusions (IBE and short signatures) in one sentence.
• No citation (BF01 and BLS01 missing).
• Hypothesis is pairings generically rather than a named assumption.
• The hypothesis bilinear-map-assumptions is a real page but an umbrella of several assumptions; Wat09 uses decisional bilinear Diffie-Hellman and decisional linear specifically, so the edge cannot name the assumption it depends on.
• IBE is not wikilinked on this bullet although content/Primitives/identity-based-encryption.md exists.
• ‘adaptively secure’ is a security level the conclusion node cannot express.
• Multi-conclusion bullet (IBE, HIBE, ABE) rather than a chain — must still be split into three reductions, one per conclusion. ‘isComposite’ is used here in that sense.
• ‘simple pairing-based assumptions’ is vague and unlinked (Wat09 uses decisional bilinear Diffie-Hellman and decisional linear); the hypothesis cannot be pinned to a specific assumption node.
• IBE and HIBE targets are not wikilinked on this bullet even though both pages exist.
• Marked standard, but Boneh-Franklin is an attributable result (BF01) — the folklore exception is misapplied.
• Boneh-Franklin IBE is proved secure in the random oracle model; the model is unstated on the bullet.
• BDH has no page of its own; the slug bilinear-map-assumptions bundles BDH, DBDH, DLIN, SXDH and k-Lin.