IPPE ⇒ HVE

IPPE implies HVE.

Statement

Migrated verbatim from hidden-vector-encryption § Other results:

• HVE is generalized by IPPE: a conjunctive pattern predicate over ${\Sigma }^n$ can be encoded as an inner product over ${\mathbb{Z}}_p$ with an appropriate alphabet embedding

Migrated verbatim from inner-product-predicate-encryption § Inner-product predicate encryption:

Inner-product predicate encryption (IPPE) is a form of predicate encryption in which keys and ciphertexts are each associated with vectors over ${\mathbb{Z}}_p$. A key for vector $v\in {\mathbb{Z}}_p^n$ can decrypt a ciphertext for vector $x\in {\mathbb{Z}}_p^n$ if and only if $⟨v,x⟩=0(\mathrm{mod}p)$. IPPE achieves full attribute-hiding: a ciphertext hides both the encrypted payload and the attribute vector $x$. It subsumes HVE and, via inner-product encodings, supports disjunctions, CNF/DNF formulas, and polynomial evaluation—making it the practical ceiling before full predicate encryption.

Migrated verbatim from inner-product-predicate-encryption § Other results:

• IPPE generalizes HVE: a conjunctive pattern predicate over ${\Sigma }^n$ can be encoded as an inner-product predicate by choosing an alphabet embedding into ${\mathbb{Z}}_p$

Notes

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

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.

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

• No citation (the encoding is from BW07/KSW08).
• ‘is generalized by’ reverses the arrow relative to word order (IPPE ⇒ HVE).
• The encoding sketch is incomplete: encoding a conjunction as a single inner product requires randomized coefficients to avoid accidental zero inner products (false accepts); as written the sketch suggests a deterministic embedding suffices.
• Intro claim (‘It subsumes HVE’); no citation here (KSW08 appears only at line 81).
• Same paragraph also claims support for disjunctions/CNF/DNF/polynomial evaluation - those are expressiveness claims with no object identifiers.
• No citation, though this encoding is due to KSW08 (cited two bullets later).
• Duplicate of the intro claim at line 13.