# Applied Crypto #2: Polynomial Commitments

## Descriptionβ

Yi Sun talks about polynomial commitments.

π For polynomial $P$ and pair $(x, y)$, a

polynomial commitmentallows the prover to show that $P(x) = y$, such that the verifier can verify this evaluation without knowing what $P$ actually is.A polynomial commitment scheme starts with a trusted setup to generate public parameters used throughout the protocol. Then, the prover commits to a polynomial $P$ and is verified to be a valid commitment to $P$. To create a polynomial opening, the prover shows that $P(x) = y$ such that this evaluation can be checked by the verifier. To construct a polynomial $P$ with a series of desired points, we use

Lagrange interpolationβ which basically means that you draw a line through all of the points you want in the polynomial.There are different ways to implement polynomial commitment schemes. In this lecture, Yi discusses the

Kate polynomial commitment scheme(which uses elliptic curve pairings) and the Fast Reed-Solomon IOP (FRI)-based commitment scheme (which uses merkle trees). While the Kate commitment schemes has a trusted setup (and FRI-based doesnβt), the proof size is smaller and independent of the size of the polynomial, thus making verification time more efficient.

## Slides Link(s)β

https://drive.google.com/file/d/1vd5sgMUgZh3t3nP28zjfEE2h0oWOeCmw/view

## Additional resourcesβ

More on Polynomial commitment schemes.