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

# EIP-4844 -- Polynomial Commitments

## Table of contents

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- [Introduction](#introduction)
- [Custom types](#custom-types)
- [Constants](#constants)
- [Preset](#preset)
  - [Blob](#blob)
  - [Crypto](#crypto)
  - [Trusted setup](#trusted-setup)
- [Helper functions](#helper-functions)
  - [Bit-reversal permutation](#bit-reversal-permutation)
    - [`is_power_of_two`](#is_power_of_two)
    - [`reverse_bits`](#reverse_bits)
    - [`bit_reversal_permutation`](#bit_reversal_permutation)
  - [BLS12-381 helpers](#bls12-381-helpers)
    - [`bytes_to_bls_field`](#bytes_to_bls_field)
    - [`blob_to_polynomial`](#blob_to_polynomial)
    - [`compute_challenges`](#compute_challenges)
    - [`bls_modular_inverse`](#bls_modular_inverse)
    - [`div`](#div)
    - [`g1_lincomb`](#g1_lincomb)
    - [`poly_lincomb`](#poly_lincomb)
    - [`compute_powers`](#compute_powers)
  - [Polynomials](#polynomials)
    - [`evaluate_polynomial_in_evaluation_form`](#evaluate_polynomial_in_evaluation_form)
  - [KZG](#kzg)
    - [`blob_to_kzg_commitment`](#blob_to_kzg_commitment)
    - [`verify_kzg_proof`](#verify_kzg_proof)
    - [`verify_kzg_proof_impl`](#verify_kzg_proof_impl)
    - [`compute_kzg_proof`](#compute_kzg_proof)
    - [`compute_aggregated_poly_and_commitment`](#compute_aggregated_poly_and_commitment)
    - [`compute_aggregate_kzg_proof`](#compute_aggregate_kzg_proof)
    - [`verify_aggregate_kzg_proof`](#verify_aggregate_kzg_proof)

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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.

Functions flagged as "Public method" MUST be provided by the underlying KZG library as public functions. All other functions are private functions used internally by the KZG library.

## Custom types

| Name | SSZ equivalent | Description |
| - | - | - |
| `G1Point` | `Bytes48` | |
| `G2Point` | `Bytes96` | |
| `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` |
| `Polynomial` | `Vector[BLSFieldElement, FIELD_ELEMENTS_PER_BLOB]` | a polynomial in evaluation form |
| `Blob` | `ByteVector[BYTES_PER_FIELD_ELEMENT * FIELD_ELEMENTS_PER_BLOB]` | a basic blob data |

## Constants

| Name | Value | Notes |
| - | - | - |
| `BLS_MODULUS` | `52435875175126190479447740508185965837690552500527637822603658699938581184513` | Scalar field modulus of BLS12-381 |
| `BYTES_PER_FIELD_ELEMENT` | `uint64(32)` | Bytes used to encode a BLS scalar field element |

## Preset

### Blob

| Name | Value |
| - | - |
| `FIELD_ELEMENTS_PER_BLOB` | `uint64(4096)` |
| `FIAT_SHAMIR_PROTOCOL_DOMAIN` | `b'FSBLOBVERIFY_V1_'` |

### Crypto

| Name | Value | Notes |
| - | - | - |
| `ROOTS_OF_UNITY` | `Vector[BLSFieldElement, FIELD_ELEMENTS_PER_BLOB]` | Roots of unity of order FIELD_ELEMENTS_PER_BLOB over the BLS12-381 field |

### 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_G1` | `Vector[G1Point, FIELD_ELEMENTS_PER_BLOB]`, contents TBD |
| `KZG_SETUP_G2` | `Vector[G2Point, FIELD_ELEMENTS_PER_BLOB]`, contents TBD |
| `KZG_SETUP_LAGRANGE` | `Vector[KZGCommitment, FIELD_ELEMENTS_PER_BLOB]`, contents TBD |

## Helper functions

### Bit-reversal permutation

All polynomials (which are always given in Lagrange form) should be interpreted as being in
bit-reversal permutation. In practice, clients can implement this by storing the lists
`KZG_SETUP_LAGRANGE` and `ROOTS_OF_UNITY` in bit-reversal permutation, so these functions only
have to be called once at startup.

#### `is_power_of_two`

```python
def is_power_of_two(value: int) -> bool:
    """
    Check if ``value`` is a power of two integer.
    """
    return (value > 0) and (value & (value - 1) == 0)
```

#### `reverse_bits`

```python
def reverse_bits(n: int, order: int) -> int:
    """
    Reverse the bit order of an integer ``n``.
    """
    assert is_power_of_two(order)
    # Convert n to binary with the same number of bits as "order" - 1, then reverse its bit order
    return int(('{:0' + str(order.bit_length() - 1) + 'b}').format(n)[::-1], 2)
```

#### `bit_reversal_permutation`

```python
def bit_reversal_permutation(sequence: Sequence[T]) -> Sequence[T]:
    """
    Return a copy with bit-reversed permutation. The permutation is an involution (inverts itself).

    The input and output are a sequence of generic type ``T`` objects.
    """
    return [sequence[reverse_bits(i, len(sequence))] for i in range(len(sequence))]
```

### BLS12-381 helpers

#### `bytes_to_bls_field`

```python
def bytes_to_bls_field(b: Bytes32) -> BLSFieldElement:
    """
    Convert 32-byte value to a BLS field scalar. The output is not uniform over the BLS field.
    """
    return int.from_bytes(b, ENDIANNESS) % BLS_MODULUS
```

#### `blob_to_polynomial`

```python
def blob_to_polynomial(blob: Blob) -> Polynomial:
    """
    Convert a blob to list of BLS field scalars.
    """
    polynomial = Polynomial()
    for i in range(FIELD_ELEMENTS_PER_BLOB):
        value = int.from_bytes(blob[i * BYTES_PER_FIELD_ELEMENT: (i + 1) * BYTES_PER_FIELD_ELEMENT], ENDIANNESS)
        assert value < BLS_MODULUS
        polynomial[i] = value
    return polynomial
```

#### `compute_challenges`

```python
def compute_challenges(polynomials: Sequence[Polynomial],
                       commitments: Sequence[KZGCommitment]) -> BLSFieldElement:
    """
    Return the Fiat-Shamir challenges required by the rest of the protocol.
    The Fiat-Shamir logic works as per the following pseudocode:

       hashed_data = hash(DOMAIN_SEPARATOR, polynomials, commitments)
       r = hash(hashed_data, 0)
       r_powers = [1, r, r**2, r**3, ...]
       eval_challenge = hash(hashed_data, 1)

    Then return `r_powers` and `eval_challenge` after converting them to BLS field elements.
    The resulting field elements are not uniform over the BLS field.
    """
    # Append the number of polynomials and the degree of each polynomial as a domain separator
    num_polynomials = int.to_bytes(len(polynomials), 8, ENDIANNESS)
    degree_poly = int.to_bytes(FIELD_ELEMENTS_PER_BLOB, 8, ENDIANNESS)
    data = FIAT_SHAMIR_PROTOCOL_DOMAIN + degree_poly + num_polynomials

    # Append each polynomial which is composed by field elements
    for poly in polynomials:
        for field_element in poly:
            data += int.to_bytes(field_element, BYTES_PER_FIELD_ELEMENT, ENDIANNESS)

    # Append serialized G1 points
    for commitment in commitments:
        data += commitment

    # Transcript has been prepared: time to create the challenges
    hashed_data = hash(data)
    r = hash(hashed_data + b'\x00')
    r_powers = compute_powers(bytes_to_bls_field(r), len(commitments))
    eval_challenge = hash(hashed_data + b'\x01')

    return r_powers, bytes_to_bls_field(eval_challenge)
```

#### `bls_modular_inverse`

```python
def bls_modular_inverse(x: BLSFieldElement) -> BLSFieldElement:
    """
    Compute the modular inverse of x
    i.e. return y such that x * y % BLS_MODULUS == 1 and return 0 for x == 0
    """
    return pow(x, -1, BLS_MODULUS) if x != 0 else 0
```

#### `div`

```python
def div(x: BLSFieldElement, y: BLSFieldElement) -> BLSFieldElement:
    """
    Divide two field elements: ``x`` by `y``.
    """
    return (int(x) * int(bls_modular_inverse(y))) % BLS_MODULUS
```

#### `g1_lincomb`

```python
def g1_lincomb(points: Sequence[KZGCommitment], scalars: Sequence[BLSFieldElement]) -> KZGCommitment:
    """
    BLS multiscalar multiplication. This function can be optimized using Pippenger's algorithm and variants.
    """
    assert len(points) == len(scalars)
    result = bls.Z1
    for x, a in zip(points, scalars):
        result = bls.add(result, bls.multiply(bls.bytes48_to_G1(x), a))
    return KZGCommitment(bls.G1_to_bytes48(result))
```

#### `poly_lincomb`

```python
def poly_lincomb(polys: Sequence[Polynomial],
                 scalars: Sequence[BLSFieldElement]) -> Polynomial:
    """
    Given a list of ``polynomials``, interpret it as a 2D matrix and compute the linear combination
    of each column with `scalars`: return the resulting polynomials.
    """
    assert len(polys) == len(scalars)
    result = [0] * FIELD_ELEMENTS_PER_BLOB
    for v, s in zip(polys, scalars):
        for i, x in enumerate(v):
            result[i] = (result[i] + int(s) * int(x)) % BLS_MODULUS
    return [BLSFieldElement(x) for x in result]
```

#### `compute_powers`

```python
def compute_powers(x: BLSFieldElement, n: uint64) -> Sequence[BLSFieldElement]:
    """
    Return ``x`` to power of [0, n-1], if n > 0. When n==0, an empty array is returned.
    """
    current_power = 1
    powers = []
    for _ in range(n):
        powers.append(BLSFieldElement(current_power))
        current_power = current_power * int(x) % BLS_MODULUS
    return powers
```


### Polynomials

#### `evaluate_polynomial_in_evaluation_form`

```python
def evaluate_polynomial_in_evaluation_form(polynomial: Polynomial,
                                           z: BLSFieldElement) -> BLSFieldElement:
    """
    Evaluate a polynomial (in evaluation form) at an arbitrary point ``z``.
    Uses the barycentric formula:
       f(z) = (z**WIDTH - 1) / WIDTH  *  sum_(i=0)^WIDTH  (f(DOMAIN[i]) * DOMAIN[i]) / (z - DOMAIN[i])
    """
    width = len(polynomial)
    assert width == FIELD_ELEMENTS_PER_BLOB
    inverse_width = bls_modular_inverse(width)

    # Make sure we won't divide by zero during division
    assert z not in ROOTS_OF_UNITY

    roots_of_unity_brp = bit_reversal_permutation(ROOTS_OF_UNITY)

    result = 0
    for i in range(width):
        result += div(int(polynomial[i]) * int(roots_of_unity_brp[i]), (int(z) - int(roots_of_unity_brp[i])))
    result = result * (pow(z, width, BLS_MODULUS) - 1) * inverse_width % BLS_MODULUS
    return result
```

### KZG

KZG core functions. These are also defined in EIP-4844 execution specs.

#### `blob_to_kzg_commitment`

```python
def blob_to_kzg_commitment(blob: Blob) -> KZGCommitment:
    """
    Public method.
    """
    return g1_lincomb(bit_reversal_permutation(KZG_SETUP_LAGRANGE), blob_to_polynomial(blob))
```

#### `verify_kzg_proof`

```python
def verify_kzg_proof(polynomial_kzg: KZGCommitment,
                     z: Bytes32,
                     y: Bytes32,
                     kzg_proof: KZGProof) -> bool:
    """
    Verify KZG proof that ``p(z) == y`` where ``p(z)`` is the polynomial represented by ``polynomial_kzg``.
    Receives inputs as bytes.
    Public method.
    """
    return verify_kzg_proof_impl(polynomial_kzg, bytes_to_bls_field(z), bytes_to_bls_field(y), kzg_proof)
```


#### `verify_kzg_proof_impl`

```python
def verify_kzg_proof_impl(polynomial_kzg: KZGCommitment,
                          z: BLSFieldElement,
                          y: BLSFieldElement,
                          kzg_proof: KZGProof) -> bool:
    """
    Verify KZG proof that ``p(z) == y`` where ``p(z)`` is the polynomial represented by ``polynomial_kzg``.
    """
    # Verify: P - y = Q * (X - z)
    X_minus_z = bls.add(bls.bytes96_to_G2(KZG_SETUP_G2[1]), bls.multiply(bls.G2, BLS_MODULUS - z))
    P_minus_y = bls.add(bls.bytes48_to_G1(polynomial_kzg), bls.multiply(bls.G1, BLS_MODULUS - y))
    return bls.pairing_check([
        [P_minus_y, bls.neg(bls.G2)],
        [bls.bytes48_to_G1(kzg_proof), X_minus_z]
    ])
```

#### `compute_kzg_proof`

```python
def compute_kzg_proof(polynomial: Polynomial, z: BLSFieldElement) -> KZGProof:
    """
    Compute KZG proof at point `z` with `polynomial` being in evaluation form
    Do this by computing the quotient polynomial in evaluation form: q(x) = (p(x) - p(z)) / (x - z)
    """

    # To avoid SSZ overflow/underflow, convert element into int
    polynomial = [int(i) for i in polynomial]
    z = int(z)

    y = evaluate_polynomial_in_evaluation_form(polynomial, z)
    polynomial_shifted = [(p - int(y)) % BLS_MODULUS for p in polynomial]

    # Make sure we won't divide by zero during division
    assert z not in ROOTS_OF_UNITY
    denominator_poly = [(int(x) - z) % BLS_MODULUS for x in bit_reversal_permutation(ROOTS_OF_UNITY)]

    # Calculate quotient polynomial by doing point-by-point division
    quotient_polynomial = [div(a, b) for a, b in zip(polynomial_shifted, denominator_poly)]
    return KZGProof(g1_lincomb(bit_reversal_permutation(KZG_SETUP_LAGRANGE), quotient_polynomial))
```

#### `compute_aggregated_poly_and_commitment`

```python
def compute_aggregated_poly_and_commitment(
        blobs: Sequence[Blob],
        kzg_commitments: Sequence[KZGCommitment]) -> Tuple[Polynomial, KZGCommitment, BLSFieldElement]:
    """
    Return (1) the aggregated polynomial, (2) the aggregated KZG commitment,
    and (3) the polynomial evaluation random challenge.
    This function should also work with blobs == [] and kzg_commitments == []
    """
    assert len(blobs) == len(kzg_commitments)

    # Convert blobs to polynomials
    polynomials = [blob_to_polynomial(blob) for blob in blobs]

    # Generate random linear combination and evaluation challenges
    r_powers, evaluation_challenge = compute_challenges(polynomials, kzg_commitments)

    # Create aggregated polynomial in evaluation form
    aggregated_poly = Polynomial(poly_lincomb(polynomials, r_powers))

    # Compute commitment to aggregated polynomial
    aggregated_poly_commitment = KZGCommitment(g1_lincomb(kzg_commitments, r_powers))

    return aggregated_poly, aggregated_poly_commitment, evaluation_challenge
```

#### `compute_aggregate_kzg_proof`

```python
def compute_aggregate_kzg_proof(blobs: Sequence[Blob]) -> KZGProof:
    """
    Given a list of blobs, return the aggregated KZG proof that is used to verify them against their commitments.
    Public method.
    """
    commitments = [blob_to_kzg_commitment(blob) for blob in blobs]
    aggregated_poly, aggregated_poly_commitment, evaluation_challenge = compute_aggregated_poly_and_commitment(
        blobs,
        commitments
    )
    return compute_kzg_proof(aggregated_poly, evaluation_challenge)
```

#### `verify_aggregate_kzg_proof`

```python
def verify_aggregate_kzg_proof(blobs: Sequence[Blob],
                               expected_kzg_commitments: Sequence[KZGCommitment],
                               kzg_aggregated_proof: KZGProof) -> bool:
    """
    Given a list of blobs and an aggregated KZG proof, verify that they correspond to the provided commitments.
    
    Public method.
    """
    aggregated_poly, aggregated_poly_commitment, evaluation_challenge = compute_aggregated_poly_and_commitment(
        blobs,
        expected_kzg_commitments,
    )

    # Evaluate aggregated polynomial at `evaluation_challenge` (evaluation function checks for div-by-zero)
    y = evaluate_polynomial_in_evaluation_form(aggregated_poly, evaluation_challenge)

    # Verify aggregated proof
    return verify_kzg_proof_impl(aggregated_poly_commitment, evaluation_challenge, y, kzg_aggregated_proof)
```