A polynomial commitment is a fundamental cryptographic primitive that allows someone to commit to a polynomial without revealing its actual form. It has three key properties: binding, which means you cannot change the polynomial after making the commitment; hiding, which ensures the commitment reveals no information about the polynomial itself; and provable, which allows you to prove specific evaluations without exposing the entire polynomial.
The polynomial commitment process works in three main steps. First, in the commitment phase, the committer takes their secret polynomial P of x and computes a short commitment value C. This commitment is binding, meaning they cannot later claim a different polynomial produced the same commitment. Second, when someone wants to verify that the polynomial evaluates to a specific value y at point z, the committer generates an evaluation proof without revealing the polynomial itself. Finally, the verifier can check this proof using only the commitment C, the point z, the claimed value y, and the proof, without ever seeing the original polynomial.
The KZG polynomial commitment scheme is one of the most popular implementations. It works using elliptic curve cryptography. In the setup phase, a trusted setup generates parameters including a secret value tau and an elliptic curve generator g. The commitment is computed as g raised to the power of P evaluated at the secret tau. For evaluation proofs, we compute a quotient polynomial Q where Q of x equals P of x minus y, all divided by x minus z. The proof is g raised to Q of tau. Verification uses bilinear pairings to check the relationship between the commitment, proof, and claimed evaluation without revealing the polynomial or the secret tau.
Polynomial commitments have numerous important applications in modern cryptography. They are fundamental building blocks in zero-knowledge proof systems like ZK-SNARKs and ZK-STARKs, enabling privacy-preserving computations. In blockchain scaling, they power rollups and Layer 2 solutions by providing efficient data availability proofs. For verifiable computation, they allow outsourcing of computations to untrusted parties while maintaining verifiability, which is crucial for cloud computing scenarios. They also enable advanced cryptographic primitives like vector commitments and accumulators for membership proofs.
To summarize what we have learned about polynomial commitments: They are cryptographic primitives that allow binding and hiding commitments to polynomials. The KZG scheme provides an efficient implementation using elliptic curves and trusted setup. They enable proving polynomial evaluations without revealing the polynomial itself. These commitments are foundational for zero-knowledge proofs and blockchain scaling solutions, making them a critical primitive in modern cryptographic protocols and privacy-preserving systems.