KZG Commitments

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Pronunciation

[kay-zee-gee kuh‑mit-muhnts]

Analogy

Like sealing a detailed spreadsheet in a tamper‑evident envelope, with a tiny stub that lets anyone verify any single line without opening the envelope.

Definition

Kate‑Zaverucha‑Goldberg polynomial commitments: a cryptographic scheme that allows one to commit to a polynomial and later open it at any point with a short proof, used for data availability and vector commitments in .

Key Points Intro

KZG commitments provide succinct, constant‑size proofs of data held in large sets or polynomials.

Key Points

Single commitment: Commits to entire polynomial or data vector.

Constant‑size proof: Opening proof is O(1) regardless of data size.

Homomorphic: Commitments can be combined for aggregated proofs.

Trusted setup: Requires a one‑time ceremony to generate evaluation keys.

Example

A zk‑rollup uses KZG commitments to commit to its entire root polynomial and provides proofs that individual entries are included.

Technical Deep Dive

Given polynomial \(f(x)\), commitment \(C = g^{f(\tau)}\) in pairing group. To open at point \(z\), prover computes witness polynomial \(h(x) = (f(x)-f(z))/(x-z)\) and proof \(\pi = g^{h(\tau)}\). Verifier checks pairing equation \(e(C/g^{f(z)}, g) = e(\pi, g^{\tau-z})\).

Security Warning

Toxic waste from trusted setup must be destroyed; compromise breaks binding.

Caveat

Trusted setup ceremonies are complex; newer transparent schemes may avoid it.

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