15 KiB
EIP-4844 -- Polynomial Commitments
Table of contents
- Introduction
- Custom types
- Constants
- Preset
- Helper functions
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
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
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
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
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
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
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 = [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
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
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
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
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
def compute_powers(x: BLSFieldElement, n: uint64) -> Sequence[BLSFieldElement]:
"""
Return ``x`` to power of [0, n-1].
"""
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
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
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
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
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
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
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
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
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)