eth2.0-specs/specs/eip4844/polynomial-commitments.md

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# EIP-4844 -- Polynomial Commitments
## Table of contents
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- [Introduction](#introduction)
- [Custom types](#custom-types)
- [Constants](#constants)
- [Preset](#preset)
- [Trusted setup](#trusted-setup)
- [Helper functions](#helper-functions)
- [BLS12-381 helpers](#bls12-381-helpers)
- [`bls_modular_inverse`](#bls_modular_inverse)
- [`div`](#div)
- [`lincomb`](#lincomb)
- [KZG](#kzg)
- [`blob_to_kzg`](#blob_to_kzg)
- [`verify_kzg_proof`](#verify_kzg_proof)
- [Polynomials](#polynomials)
- [`evaluate_polynomial_in_evaluation_form`](#evaluate_polynomial_in_evaluation_form)
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## Introduction
This document specifies basic polynomial operations and KZG polynomial commitment operations as they are needed for the EIP-4844 specification. The implementations are not optimized for performance, but readability. All practical implementations should optimize the polynomial operations.
## Custom types
| Name | SSZ equivalent | Description |
| - | - | - |
| `BLSFieldElement` | `uint256` | `x < BLS_MODULUS` |
| `KZGCommitment` | `Bytes48` | Same as BLS standard "is valid pubkey" check but also allows `0x00..00` for point-at-infinity |
| `KZGProof` | `Bytes48` | Same as for `KZGCommitment` |
## Constants
| Name | Value | Notes |
| - | - | - |
| `BLS_MODULUS` | `52435875175126190479447740508185965837690552500527637822603658699938581184513` | Scalar field modulus of BLS12-381 |
| `ROOTS_OF_UNITY` | `Vector[BLSFieldElement, FIELD_ELEMENTS_PER_BLOB]` | Roots of unity of order FIELD_ELEMENTS_PER_BLOB over the BLS12-381 field |
## Preset
### Trusted setup
The trusted setup is part of the preset: during testing a `minimal` insecure variant may be used,
but reusing the `mainnet` settings in public networks is a critical security requirement.
| Name | Value |
| - | - |
| `KZG_SETUP_G2` | `Vector[G2Point, FIELD_ELEMENTS_PER_BLOB]`, contents TBD |
| `KZG_SETUP_LAGRANGE` | `Vector[KZGCommitment, FIELD_ELEMENTS_PER_BLOB]`, contents TBD |
## Helper functions
### BLS12-381 helpers
#### `bls_modular_inverse`
```python
def bls_modular_inverse(x: BLSFieldElement) -> BLSFieldElement:
"""
Compute the modular inverse of x using the eGCD algorithm
i.e. return y such that x * y % BLS_MODULUS == 1 and return 0 for x == 0
"""
if x == 0:
return 0
lm, hm = 1, 0
low, high = x % BLS_MODULUS, BLS_MODULUS
while low > 1:
r = high // low
nm, new = hm - lm * r, high - low * r
lm, low, hm, high = nm, new, lm, low
return lm % BLS_MODULUS
```
#### `div`
```python
def div(x, y):
"""Divide two field elements: `x` by `y`"""
return x * inv(y) % BLS_MODULUS
```
#### `lincomb`
```python
def lincomb(points: List[KZGCommitment], scalars: List[BLSFieldElement]) -> KZGCommitment:
"""
BLS multiscalar multiplication. This function can be optimized using Pippenger's algorithm and variants.
"""
r = bls.Z1
for x, a in zip(points, scalars):
r = bls.add(r, bls.multiply(x, a))
return r
```
### KZG
KZG core functions. These are also defined in EIP-4844 execution specs.
#### `blob_to_kzg`
```python
def blob_to_kzg(blob: Blob) -> KZGCommitment:
return lincomb(KZG_SETUP_LAGRANGE, blob)
```
#### `verify_kzg_proof`
```python
def verify_kzg_proof(polynomial_kzg: KZGCommitment,
x: BLSFieldElement,
y: BLSFieldElement,
quotient_kzg: KZGProof) -> bool:
"""Verify KZG proof that `p(x) == y` where `p(x)` is the polynomial represented by `polynomial_kzg`"""
# Verify: P - y = Q * (X - x)
X_minus_x = bls.add(KZG_SETUP_G2[1], bls.multiply(bls.G2, BLS_MODULUS - x))
P_minus_y = bls.add(polynomial_kzg, bls.multiply(bls.G1, BLS_MODULUS - y))
return bls.pairing_check([
[P_minus_y, bls.neg(bls.G2)],
[quotient_kzg, X_minus_x]
])
```
### Polynomials
#### `evaluate_polynomial_in_evaluation_form`
```python
def evaluate_polynomial_in_evaluation_form(poly: List[BLSFieldElement], x: BLSFieldElement) -> BLSFieldElement:
"""
Evaluate a polynomial (in evaluation form) at an arbitrary point `x`
Uses the barycentric formula:
f(x) = (1 - x**WIDTH) / WIDTH * sum_(i=0)^WIDTH (f(DOMAIN[i]) * DOMAIN[i]) / (x - DOMAIN[i])
"""
width = len(poly)
assert width == FIELD_ELEMENTS_PER_BLOB
inverse_width = bls_modular_inverse(width)
for i in range(width):
r += div(poly[i] * ROOTS_OF_UNITY[i], (x - ROOTS_OF_UNITY[i]) )
r = r * (pow(x, width, BLS_MODULUS) - 1) * inverse_width % BLS_MODULUS
return r
```