QR ⇒ HE

QR implies HE.

Statement

Migrated verbatim from quadratic-residuosity § Known Results:

• Goldwasser-Micali is multiplicatively homomorphic: $\mathsf{Enc}(b_1)\cdot \mathsf{Enc}(b_2)=\mathsf{Enc}(b_1\oplus b_2\mathrm{mod}2)$ — GM84

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.

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

• A property of the specific GM scheme rather than a reduction from the assumption; the target object (XOR-homomorphic bit encryption) is not the wiki homomorphic-encryption page and is not wikilinked at all.
• MINOR: “Enc(b1) . Enc(b2) = Enc(b1 XOR b2 mod 2)” is redundant — XOR is already mod 2 — and the operation is called “multiplicatively homomorphic” while the plaintext operation is additive mod 2.