257 lines
8.6 KiB
Markdown
257 lines
8.6 KiB
Markdown
# EIP-4844 -- Polynomial Commitments
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## Table of contents
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<!-- TOC -->
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<!-- START doctoc generated TOC please keep comment here to allow auto update -->
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<!-- DON'T EDIT THIS SECTION, INSTEAD RE-RUN doctoc TO UPDATE -->
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- [Introduction](#introduction)
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- [Custom types](#custom-types)
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- [Constants](#constants)
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- [Preset](#preset)
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- [Trusted setup](#trusted-setup)
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- [Helper functions](#helper-functions)
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- [Bit-reversal permutation](#bit-reversal-permutation)
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- [`is_power_of_two`](#is_power_of_two)
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- [`reverse_bits`](#reverse_bits)
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- [`bit_reversal_permutation`](#bit_reversal_permutation)
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- [BLS12-381 helpers](#bls12-381-helpers)
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- [`bytes_to_bls_field`](#bytes_to_bls_field)
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- [`bls_modular_inverse`](#bls_modular_inverse)
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- [`div`](#div)
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- [`g1_lincomb`](#g1_lincomb)
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- [`vector_lincomb`](#vector_lincomb)
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- [KZG](#kzg)
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- [`blob_to_kzg_commitment`](#blob_to_kzg_commitment)
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- [`verify_kzg_proof`](#verify_kzg_proof)
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- [`compute_kzg_proof`](#compute_kzg_proof)
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- [Polynomials](#polynomials)
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- [`evaluate_polynomial_in_evaluation_form`](#evaluate_polynomial_in_evaluation_form)
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<!-- END doctoc generated TOC please keep comment here to allow auto update -->
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<!-- /TOC -->
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## Introduction
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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.
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## Custom types
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| Name | SSZ equivalent | Description |
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| - | - | - |
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| `G1Point` | `Bytes48` | |
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| `G2Point` | `Bytes96` | |
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| `BLSFieldElement` | `uint256` | `x < BLS_MODULUS` |
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| `KZGCommitment` | `Bytes48` | Same as BLS standard "is valid pubkey" check but also allows `0x00..00` for point-at-infinity |
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| `KZGProof` | `Bytes48` | Same as for `KZGCommitment` |
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## Constants
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| Name | Value | Notes |
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| - | - | - |
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| `BLS_MODULUS` | `52435875175126190479447740508185965837690552500527637822603658699938581184513` | Scalar field modulus of BLS12-381 |
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| `ROOTS_OF_UNITY` | `Vector[BLSFieldElement, FIELD_ELEMENTS_PER_BLOB]` | Roots of unity of order FIELD_ELEMENTS_PER_BLOB over the BLS12-381 field |
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## Preset
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### Trusted setup
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The trusted setup is part of the preset: during testing a `minimal` insecure variant may be used,
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but reusing the `mainnet` settings in public networks is a critical security requirement.
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| Name | Value |
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| - | - |
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| `KZG_SETUP_G1` | `Vector[G1Point, FIELD_ELEMENTS_PER_BLOB]`, contents TBD |
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| `KZG_SETUP_G2` | `Vector[G2Point, FIELD_ELEMENTS_PER_BLOB]`, contents TBD |
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| `KZG_SETUP_LAGRANGE` | `Vector[KZGCommitment, FIELD_ELEMENTS_PER_BLOB]`, contents TBD |
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## Helper functions
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### Bit-reversal permutation
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All polynomials (which are always given in Lagrange form) should be interpreted as being in
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bit-reversal permutation. In practice, clients can implement this by storing the lists
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`KZG_SETUP_LAGRANGE` and `ROOTS_OF_UNITY` in bit-reversal permutation, so these functions only
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have to be called once at startup.
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#### `is_power_of_two`
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```python
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def is_power_of_two(value: int) -> bool:
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"""
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Check if ``value`` is a power of two integer.
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"""
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return (value > 0) and (value & (value - 1) == 0)
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```
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#### `reverse_bits`
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```python
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def reverse_bits(n: int, order: int) -> int:
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"""
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Reverse the bit order of an integer n
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"""
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assert is_power_of_two(order)
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# Convert n to binary with the same number of bits as "order" - 1, then reverse its bit order
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return int(('{:0' + str(order.bit_length() - 1) + 'b}').format(n)[::-1], 2)
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```
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#### `bit_reversal_permutation`
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```python
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def bit_reversal_permutation(l: Sequence[T]) -> Sequence[T]:
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"""
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Return a copy with bit-reversed permutation. The permutation is an involution (inverts itself).
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The input and output are a sequence of generic type ``T`` objects.
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"""
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return [l[reverse_bits(i, len(l))] for i in range(len(l))]
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```
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### BLS12-381 helpers
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#### `bytes_to_bls_field`
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```python
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def bytes_to_bls_field(b: Bytes32) -> BLSFieldElement:
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"""
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Convert bytes to a BLS field scalar. The output is not uniform over the BLS field.
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"""
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return int.from_bytes(b, "little") % BLS_MODULUS
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```
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#### `bls_modular_inverse`
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```python
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def bls_modular_inverse(x: BLSFieldElement) -> BLSFieldElement:
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"""
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Compute the modular inverse of x
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i.e. return y such that x * y % BLS_MODULUS == 1 and return 0 for x == 0
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"""
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return pow(x, -1, BLS_MODULUS) if x != 0 else 0
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```
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#### `div`
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```python
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def div(x: BLSFieldElement, y: BLSFieldElement) -> BLSFieldElement:
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"""Divide two field elements: `x` by `y`"""
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return (int(x) * int(bls_modular_inverse(y))) % BLS_MODULUS
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```
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#### `g1_lincomb`
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```python
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def g1_lincomb(points: Sequence[KZGCommitment], scalars: Sequence[BLSFieldElement]) -> KZGCommitment:
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"""
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BLS multiscalar multiplication. This function can be optimized using Pippenger's algorithm and variants.
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"""
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assert len(points) == len(scalars)
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result = bls.Z1
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for x, a in zip(points, scalars):
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result = bls.add(result, bls.multiply(bls.bytes48_to_G1(x), a))
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return KZGCommitment(bls.G1_to_bytes48(result))
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```
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#### `vector_lincomb`
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```python
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def vector_lincomb(vectors: Sequence[Sequence[BLSFieldElement]],
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scalars: Sequence[BLSFieldElement]) -> Sequence[BLSFieldElement]:
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"""
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Given a list of ``vectors``, interpret it as a 2D matrix and compute the linear combination
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of each column with `scalars`: return the resulting vector.
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"""
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result = [0] * len(vectors[0])
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for v, s in zip(vectors, scalars):
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for i, x in enumerate(v):
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result[i] = (result[i] + int(s) * int(x)) % BLS_MODULUS
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return [BLSFieldElement(x) for x in result]
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```
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### KZG
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KZG core functions. These are also defined in EIP-4844 execution specs.
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#### `blob_to_kzg_commitment`
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```python
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def blob_to_kzg_commitment(blob: Blob) -> KZGCommitment:
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return g1_lincomb(bit_reversal_permutation(KZG_SETUP_LAGRANGE), blob)
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```
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#### `verify_kzg_proof`
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```python
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def verify_kzg_proof(polynomial_kzg: KZGCommitment,
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z: BLSFieldElement,
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y: BLSFieldElement,
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kzg_proof: KZGProof) -> bool:
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"""
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Verify KZG proof that ``p(z) == y`` where ``p(z)`` is the polynomial represented by ``polynomial_kzg``.
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"""
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# Verify: P - y = Q * (X - z)
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X_minus_z = bls.add(bls.bytes96_to_G2(KZG_SETUP_G2[1]), bls.multiply(bls.G2, BLS_MODULUS - z))
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P_minus_y = bls.add(bls.bytes48_to_G1(polynomial_kzg), bls.multiply(bls.G1, BLS_MODULUS - y))
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return bls.pairing_check([
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[P_minus_y, bls.neg(bls.G2)],
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[bls.bytes48_to_G1(kzg_proof), X_minus_z]
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])
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```
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#### `compute_kzg_proof`
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```python
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def compute_kzg_proof(polynomial: Sequence[BLSFieldElement], z: BLSFieldElement) -> KZGProof:
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"""
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Compute KZG proof at point `z` with `polynomial` being in evaluation form
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"""
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# To avoid SSZ overflow/underflow, convert element into int
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polynomial = [int(i) for i in polynomial]
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z = int(z)
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# Shift our polynomial first (in evaluation form we can't handle the division remainder)
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y = evaluate_polynomial_in_evaluation_form(polynomial, z)
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polynomial_shifted = [(p - int(y)) % BLS_MODULUS for p in polynomial]
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# Make sure we won't divide by zero during division
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assert z not in ROOTS_OF_UNITY
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denominator_poly = [(x - z) % BLS_MODULUS for x in bit_reversal_permutation(ROOTS_OF_UNITY)]
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# Calculate quotient polynomial by doing point-by-point division
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quotient_polynomial = [div(a, b) for a, b in zip(polynomial_shifted, denominator_poly)]
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return KZGProof(g1_lincomb(bit_reversal_permutation(KZG_SETUP_LAGRANGE), quotient_polynomial))
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```
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### Polynomials
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#### `evaluate_polynomial_in_evaluation_form`
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```python
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def evaluate_polynomial_in_evaluation_form(polynomial: Sequence[BLSFieldElement],
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z: BLSFieldElement) -> BLSFieldElement:
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"""
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Evaluate a polynomial (in evaluation form) at an arbitrary point `z`
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Uses the barycentric formula:
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f(z) = (1 - z**WIDTH) / WIDTH * sum_(i=0)^WIDTH (f(DOMAIN[i]) * DOMAIN[i]) / (z - DOMAIN[i])
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"""
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width = len(polynomial)
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assert width == FIELD_ELEMENTS_PER_BLOB
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inverse_width = bls_modular_inverse(width)
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# Make sure we won't divide by zero during division
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assert z not in ROOTS_OF_UNITY
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roots_of_unity_brp = bit_reversal_permutation(ROOTS_OF_UNITY)
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result = 0
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for i in range(width):
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result += div(int(polynomial[i]) * int(roots_of_unity_brp[i]), (z - roots_of_unity_brp[i]))
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result = result * (pow(z, width, BLS_MODULUS) - 1) * inverse_width % BLS_MODULUS
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return result
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```
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