Bilinear pairing + q-SDH ⇒ KZG (Kate-Zaverucha-Goldberg)

Bilinear pairing together with q-SDH implies KZG (Kate-Zaverucha-Goldberg).

Statement

Migrated verbatim from KZG10 - Constant-size commitments to polynomials and their applications:

Introduced the KZG polynomial commitment scheme, which allows a prover to commit to a polynomial $f\in {\mathbb{F}}_p[X]$ with a single group element and later prove evaluations $f(z)=y$ with a single group element proof (O(1) size). Security relies on the $q$-Strong Diffie-Hellman assumption in a bilinear group and requires a structured reference string (trusted setup) of the form $(g,g^{\tau },g^{{\tau }^2},\dots ,g^{{\tau }^d})$ for secret $\tau$. KZG is the polynomial commitment underlying most practical SNARKs, including Plonk and Marlin, and is used in Ethereum’s KZG ceremony (EIP-4844).

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:

• STRUCTURAL: no Abstract heading; unlabelled editorial paragraph.
• The trusted-setup requirement (an SRS of powers of a secret tau) is a THIRD hypothesis in substance but is phrased as a caveat, so the conjunction {q-SDH, bilinear group, SRS} is under-recorded as a 2-node conjunction plus model:crs.
• The second sub-edge (KZG underlies Plonk and Marlin) has NO citation and neither Plonk nor Marlin has a reference page in content/References/ — sources[] left empty rather than fabricated.
• ‘most practical SNARKs’ is an unquantified superlative; the EIP-4844 / Ethereum ceremony remark is a deployment claim, not a relation, and is uncited.
• DUPLICATION: the inventory already has Primitives/polynomial-commitment.md:45 [q-strong-diffie-hellman+pairings ⇒ polynomial-commitment] and :76 [kzg-polynomial-commitment ⇒ succinct-argument], both sourced to KZG10 — this page restates both.