V God released the "polynomial commitment" diagram, what benefits will it bring to Ethereum?

Editor's Note: This article comes fromBitpushNews（ID：BitpushNews）, by Amy Liu

Editor's Note: This article comes from

, by Amy Liu

The polynomial commitment is related to Halo and future upgrades of Ethereum, which can reduce the witness data (witnesses) size of stateless Ethereum clients to close to zero and achieve better scalability.

Wall Street analyst Jacob said: "The problem Vitalik mentioned is to add a power of a complex number to the general-purpose elliptic curve code, so as to achieve more confidentiality, larger scale, etc. This idea is very good. Following this line of thought, you can Make a lot of new codes. For example, you can choose two coding systems X, Y to form a Cartesian product (x, y) and a relationship P between x, y to make a new coding system."

Vitalik calls it "magic mathematics," and polynomial commitments are being viewed as a way to verify the state of the network at a lower computational cost, which is a key goal for future networks.

It's similar to the polynomial we learned and loved in elementary school: a mathematical expression that has both variables and coefficients (i.e. Y=2X).

Buterin describes polynomial commitments as "some sort of 'hash' of polynomials P(x), with the property that arithmetic checks can be performed on the hash."

He has said that polynomial commitment cryptography is the key to stateless clients, but it is far from perfect. Likewise, work on many other advanced encryption techniques and improved virtual machines is still in its early stages.

In order to deal with the state explosion problem of Ethereum, Vitalik proposed to use a polynomial commitment scheme to replace the Merkle tree (Merkle tree). A major advantage of polynomial commitments is that it is much easier to generate complex proofs due to their mathematical structure.

We can replace current Merkle roots of block data (e.g. Ethereum 2.0 shard blocks) with polynomial commitments, and replace Merkle branches with open proofs.

This brings two big advantages. First, data availability checks are easy and fraud-free, since you can simply request opens in a random fashion. Non-interactive proofs of custody may also become easier.

This technique can provide some benefits for multiple access of block data. However, it is much more advantageous for a different use case: proving block transaction account witness data. On average, hundreds of accounts and storage keys are accessed per block, resulting in a witness data size of 0.5 MB for a potentially stateless client. Polynomial commitments can reduce the size of block witness data from tens of thousands of bytes to hundreds of bytes.