KZG Polynomial Commitment Scheme
Last updated
Last updated
Let’s dive into the KZG polynomial commitment scheme — a standout in the world of cryptography for its efficiency and elegance. Named after its creators (Kate, Zaverucha, and Goldberg), KZG is a polynomial commitment scheme that’s become a go-to in areas like zero-knowledge proofs and blockchain tech. I’ll break it down in a way that’s clear and engaging, without getting lost in the weeds.
The KZG scheme lets someone commit to a polynomial — say, something like P(x) = 2x³ + x + 4 — in a way that’s compact and verifiable. The magic happens with a single, small commitment (think of it as a cryptographic fingerprint) that proves you’ve locked in your polynomial without showing it off entirely. Later, you can open up specific evaluations — like proving P(2) = 20 — and anyone can check it fast, without needing to see the whole polynomial. This mix of hiding and revealing is what makes KZG so powerful.
How does it work? It leans on something called bilinear pairings — a fancy math tool that lets you multiply points on elliptic curves in a way that’s hard to reverse. The setup starts with a trusted ceremony (more on that in a sec) that generates a structured reference string: a bunch of precomputed points like [G, αG, α²G, ..., αᵈG], where G is a base point on the curve, α is a secret number (called the trapdoor), and d is the max degree of polynomials you’ll commit to. This string is public, but α stays hidden — if someone learns it, the scheme collapses, so that initial setup has to be secure.
To commit to your polynomial P(x), you evaluate it at this secret α (symbolically, not literally since no one knows α) by computing C = P(α)G. This single point C is your commitment — tiny, constant-sized, no matter how big your polynomial is. Want to prove P(a) = b for some input a? You create a quotient polynomial Q(x) = (P(x) - b)/(x - a), which works because P(a) = b means x - a divides P(x) - b cleanly. Then you commit to Q(x) as W = Q(α)G, and the verifier uses the pairing magic to check e(C - bG, G) = e(W, (αG - aG)). If it holds, your claim’s legit — all in constant time, which is a big deal compared to older schemes that grew slower with polynomial size.
Why’s KZG cool? It’s super efficient: commitments are just one point, and verification is lightning-fast. It’s widely used in systems like zk-SNARKs and Ethereum’s scaling solutions because of this. The catch? That trusted setup — if the secret α leaks or the ceremony’s compromised, the whole thing unravels. Newer variants try to mitigate this with multi-party setups or updatable strings, but it’s still a hot topic.
In short, KZG is a sleek, pairing-based way to commit to polynomials, balancing compactness with provability. It’s a workhorse in modern crypto, and this chapter could dig deeper into its math, applications, or even how it stacks up against alternatives like FRI.
We would be looking into 3 flavors of the polynomial commitment scheme;