Kate-Zaverucha-Goldberg (KZG) Commitment

A Kate-Zaverucha-Goldberg (KZG) commitment is a cryptographic polynomial commitment scheme that creates a short, constant-sized proof for the evaluation of a polynomial at a specific point. It is a foundational component in modern zero-knowledge proofs and scalability solutions like Ethereum's Proto-Danksharding (EIP-4844). The scheme is based on pairing-based cryptography and requires a trusted setup to generate a Structured Reference String (SRS), which is a one-time public parameter used to create and verify commitments and proofs.

The core mechanism involves a prover committing to a polynomial f(x) by computing a commitment C = f(τ) * G, where τ is a secret value from the trusted setup and G is a generator of an elliptic curve group. To later prove that f(z) = y for a point z, the prover constructs a quotient polynomial q(x) = (f(x) - y) / (x - z) and provides a proof π = q(τ) * G. A verifier can then use a bilinear pairing to check the proof against the commitment C without knowing τ or f(x). This enables efficient verification of complex statements.

KZG commitments are prized for their constant proof size and constant verification time, regardless of the polynomial's degree. This makes them ideal for blockchain scaling, where they are used to commit to large batches of data (like blob data in danksharding) and allow any node to verify the availability and correctness of a single piece of that data with minimal computational overhead. Their properties directly enable technologies like data availability sampling (DAS).

The primary trade-off is the requirement for a trusted setup ceremony. If the secret τ is compromised, an attacker could create fraudulent commitments. However, this risk is mitigated through ceremonies with many participants (like the Perpetual Powers of Tau), which ensure security as long as at least one participant was honest and destroyed their secret. Compared to other commitment schemes like Merkle trees, KZG offers more efficient proofs for polynomial evaluations but introduces this setup complexity.

In practice, KZG is a critical enabler for layer-2 rollups and modular blockchain architectures. It allows validiums and volitions to prove data availability off-chain, and it is the basis for verkle trees, a proposed upgrade to Ethereum's state tree. Its efficiency in creating cryptographic accumulators for vector commitments makes it a versatile tool for proving statements about large datasets succinctly.

A Kate-Zaverucha-Goldberg (KZG) commitment is a cryptographic scheme, formally a polynomial commitment scheme, that allows a prover to succinctly commit to a polynomial and later reveal evaluations of that polynomial with a proof that can be efficiently verified. The scheme is named for its three authors—Aniket Kate, Gregory Zaverucha, and Ian Goldberg—who first described it in their 2010 paper, 'Constant-Size Commitments to Polynomials and Their Applications'. This foundational work provided a method to commit to a polynomial f(x) of degree d using a single group element, a significant efficiency breakthrough over prior schemes.

The cryptographic core of KZG relies on pairing-based cryptography and requires a trusted setup to generate a Structured Reference String (SRS), often called a Common Reference String or powers-of-tau. This setup produces public parameters that include encrypted powers of a secret value τ (tau), which is then discarded. The security of the scheme depends on the trusted setup ceremony ensuring this τ remains unknown and is not reused. The ability to create a constant-sized proof for any polynomial evaluation made KZG particularly attractive for verifiable computing and, later, for blockchain scalability.

KZG's journey from academic paper to blockchain staple was catalyzed by its application in data availability sampling (DAS) and proto-danksharding (EIP-4844) on Ethereum. Its properties—constant-sized proofs and efficient verification—are ideal for proving that all data for a block is available without downloading it entirely. While the need for a trusted setup was initially a point of contention, large-scale, participatory ceremonies (like the Ethereum KZG Ceremony) have established sufficient decentralized trust for production use, cementing KZG's role in the next generation of scalable blockchain architectures.

A KZG polynomial commitment is a cryptographic scheme where a prover commits to a polynomial f(x) by computing a single elliptic curve point, C = [f(τ)]G. Here, τ is a secret value (the toxic waste) from a trusted setup ceremony, and G is a generator point. This commitment C is a compact, fixed-size representation of the entire polynomial. The core property is that the prover can later generate a short proof, known as an opening proof or witness, for the claim that f(u) = v for any point u, without revealing the polynomial itself.

The verification of a KZG opening is remarkably efficient. Given the commitment C, a claimed evaluation point u and value v, and the accompanying proof π, a verifier checks a single pairing equation: e(π, [τ - u]G) == e(C - [v]G, G). This check confirms the polynomial's consistency. KZG commitments are binding (the prover cannot change the committed polynomial) and hiding (the commitment reveals nothing about the polynomial). Their power lies in enabling proofs about polynomial evaluations without transmitting the full polynomial.

KZG commitments are a cornerstone of modern scalable blockchain architectures. They are the cryptographic engine behind Ethereum's Proto-Danksharding (EIP-4844), where they commit to large data blobs, allowing for efficient data availability sampling. In zero-knowledge rollups like zkSync and Polygon zkEVM, KZG is used within Polynomial IOPs and Plonkish proof systems to commit to execution traces and constraint polynomials, enabling the generation of succinct validity proofs. Their ability to support batch openings and proof aggregation is critical for these high-throughput applications.

A Kate-Zaverucha-Goldberg (KZG) commitment is a cryptographic scheme that allows a prover to commit to a polynomial and later generate a short proof that the polynomial evaluates to a specific value at a given point. This proof can be verified by anyone holding the commitment, without needing the original polynomial data. The scheme is based on pairing-based cryptography and provides constant-sized proofs and commitments, a property that makes it exceptionally efficient for systems like Ethereum's data availability sampling (DAS) in danksharding.

The core mechanism involves a trusted setup ceremony that generates a structured reference string (SRS), consisting of powers of a secret value hidden within elliptic curve groups. The prover uses this SRS to create a commitment, which is a single elliptic curve point representing the entire polynomial. To prove an evaluation, the prover provides a witness—another elliptic curve point—that convinces the verifier the evaluation is correct. The security relies on the discrete logarithm assumption within bilinear groups.

In blockchain contexts, KZG commitments are pivotal for data availability proofs and verifiable secret sharing. For example, they allow a block producer to commit to all data in a block with a single KZG commitment. Light clients can then request random chunks of the data and use the commitment to verify the chunks are correct and part of the whole, without downloading the entire block. This is the cornerstone of scaling solutions like proto-danksharding (EIP-4844) on Ethereum, which uses KZG to commit to large data blobs.

Compared to other commitment schemes like Merkle trees, KZG offers superior efficiency for proving properties of polynomial-structured data. While a Merkle proof size grows logarithmically with the data size, a KZG proof is constant. However, KZG's requirement for a one-time trusted setup is a notable disadvantage, as compromise of the secret 'toxic waste' could allow the creation of fake proofs. Despite this, its properties make it the preferred tool for modern cryptographic protocols requiring succinct verification.